lemmaLemma #1 claimClaim #1 theoremTheorem #1 corollaryCorollary #1
propositionProposition #1 conjectureConjecture #1 definitionDefinition #1
questionQuestion #1 problemProblem #1 remarkRemark #1 exampleExample #1
equation(#1) figureFigure #1 tableTable #1 chapterChapter #1 appendixAppendix
#1 sectionSection #1 subsectionSubsection #1

# Posets arising from decompositions of objects in a monoidal category

Kevin I. Piterman  Philipps-Universität Marburg
Fachbereich Mathematik und Informatik
35032 Marburg, Germany<EMAIL_ADDRESS>and  Volkmar Welker
Philipps-Universität Marburg
Fachbereich Mathematik und Informatik
35032 Marburg, Germany<EMAIL_ADDRESS>
###### Abstract.

Given a symmetric monoidal category ${\mathcal{C}}$ with product $\sqcup$,
where the neutral element for the product is an initial object, we consider
the poset of $\sqcup$-complemented subobjects of a given object $X$. When this
poset has finite height, we define decompositions and partial decompositions
of $X$ which are coherent with $\sqcup$, and order them by refinement. From
these posets, we define complexes of frames and partial bases, augmented
Bergman complexes and related ordered versions. We propose a unified approach
to the study of their combinatorics and homotopy type, establishing various
properties and relations between them. Via explicit homotopy formulas, we will
be able to transfer structural properties, such as Cohen-Macaulayness.

In well-studied scenarios, the poset of $\sqcup$-complemented subobjects
specializes to the poset of free factors of a free group, the subspace poset
of a vector space, the poset of non-degenerate subspaces of a vector space
with a non-degenerate form, and the lattice of flats of a matroid. The
decomposition and partial decomposition posets, the complex of frames and
partial bases together with the ordered versions, either coincide with well-
known structures, generalize them, or yield new interesting objects. In these
particular cases, we provide new results along with open questions and
conjectures.

###### Contents

  1. 1 Introduction
  2. 2 Poset of decompositions into subobjects
    1. 2.1 Monoidal categories and restricted decompositions
  3. 3 Properties of decomposition posets
  4. 4 Simplicial complexes and posets derived from decompositions
    1. 4.1 Frames and partial bases
    2. 4.2 Ordered versions
    3. 4.3 The augmented Bergman complex
    4. 4.4 The Charney poset
  5. 5 Homotopy equivalences and isomorphisms
    1. 5.1 General homotopy formulas
    2. 5.2 Homotopy formulas and results for posets from 2 \- 4
    3. 5.3 Unique complementation
  6. 6 Application to classical examples
    1. 6.1 Finitely generated free groups
    2. 6.2 Vector spaces
    3. 6.3 Vector spaces with forms
    4. 6.4 Matroids
    5. 6.5 Modules over Dedekind domains

##  1\. Introduction

In diverse areas of mathematics, posets and simplicial complexes arising from
algebraic, combinatorial or geometric objects are widely studied. Usually, one
tries to answer questions or determine intrinsic properties of a mathematical
object by looking into the combinatorics and topology of associated
combinatorial structures. One of the most classical examples is the poset of
subspaces of a finite-dimensional vector space, which has been extensively
studied in connection with the combinatorics, representation theory, and
cohomology of linear groups. Sometimes one even endeavors to associate several
posets or simplicial complexes with a given object, establish relations among
them, and then achieve stronger results. In this direction, many works provide
resembling constructions and morphisms between the various combinatorial
structures, analyzed often via “Quillen’s fiber theorem” type tools. In some
cases, these yield wedge/join-decompositions, allowing us to translate or
interchange combinatorial and homotopy properties from one structure to
another.

This paper aims to present a general framework for which many of these
constructions are particular cases. We discuss fundamental combinatorial
properties which are intrinsic to our approach and show how they can be used
to establish comparison results between the different posets and simplicial
complexes arising from the same mathematical object. These results provide a
toolset for verifying homotopical properties such as high connectivity,
sphericity or even Cohen-Macaulayness, and also for computing Euler
characteristics (or, more generally, the module afforded in the homology).
These are goals prevalent in the literature.

The starting point of our constructions is the subobject poset of a given
object $X$ in some category ${\mathcal{C}}$. However, from a combinatorial
point of view, this poset is often not suitable since it may contain infinite
chains, as it is, for example, the case of the subgroup poset of a free group,
or even the subgroup poset of a free Abelian group. Infinite chains obstruct
the construction of “natural” posets of decompositions and frame complexes
associated with $X$. For that reason, we will keep only the subobjects which
are complemented with respect to a suitable product. Thus, we shift gears and
work instead with an initial category ${\mathcal{C}}$ equipped with a
symmetric monoidal product $\sqcup$, which will often be the coproduct of
${\mathcal{C}}$. With this additional structure at hand, we can extract more
“interesting” subobject posets and decompositions that encode different ways
of resembling the original object from sets or tuples of subobjects whose
$\sqcup$-product is isomorphic to $X$.

To be more precise, the constructions in this article are built up from a so-
called poset of complemented subobjects
$\operatorname{{\mathcal{S}}}(X,\sqcup)$ of a fixed object $X$ in our initial
category ${\mathcal{C}}$ with symmetric monoidal product $\sqcup$. A prominent
example to keep in mind is the above-mentioned poset of subspaces
${\mathcal{S}}(V)$ of a vector space $V$ of dimension $n$ over a field
${\mathbb{K}}$. In this case, we consider the category of vector spaces over
${\mathbb{K}}$ with $\sqcup$ being just the usual coproduct. The order complex
of ${\mathcal{S}}(V)$ is the Tits building of
$\operatorname{SL}_{n}({\mathbb{K}})$, and it has been largely studied in many
areas of mathematics. Then we define the posets of $\sqcup$-decompositions and
partial $\sqcup$-decompositions (see 2.0). In our example case, the former
corresponds to the poset ${\mathcal{D}}(V)$ of collections of non-zero
subspaces which are in direct sum and span the whole space $V$, and the
latter, denoted by $\operatorname{\mathcal{PD}}(V)$, consists of collections
of non-zero subspaces in direct sum, the span being just a subspace. Both
posets are ordered by refinement. Direct sum decompositions of vector spaces
over finite fields were shown to be homotopically Cohen-Macaulay in [We1].
Indeed, the same result for arbitrary fields is already implicit in [C], as we
explain later in the article (see 4.0 and 6.2). For partial direct sum
decompositions of vector spaces over finite fields, Cohen-Macaulayness was
established in the unpublished work [HHS]. For infinite fields, the proper
part of $\operatorname{\mathcal{PD}}(V)$ is spherical by results of [BPW,
MPW], as we explain in 6.2. However, Cohen-Macaulayness was not established
yet for the infinite field case. Then we define a simplicial complex of
frames. In the vector space case, a frame is a partial decomposition into
$1$-dimensional subspaces. A further construction is obtained as the inflation
of the frame complex. That is, we replace vertices of the frame complex with
sets (interpreted as sets of generators or bases). This yields the complex of
partial bases. From subobject posets and frame complexes, one can also build
the augmented Bergman complex – which in the matroid case recently gained a
lot of attention through [BHMPW].

For (partial) decompositions, frames and partial bases, there also exist
ordered versions. These are posets whose elements are all linear orders on
sets and simplices in the unordered poset or simplicial complex (see 4.0). Our
key result regarding the ordered versions is that they exhibit a nice wedge
decomposition in terms of their unordered counterparts. Therefore, properties
such as sphericity or Cohen-Macaulayness can be transferred back and forth. In
our vector space example, the ordered partial decompositions are linearly
ordered sets of subspaces that form direct sums, and ordered decompositions
are in close relationship to a complex studied in [C], which in our setting we
call the Charney complex. The ordered versions of the frame complex and the
partial basis complex essentially correspond to the associated complex of
injective words (see [JW]).

Our results about properties and connections among these different posets and
complexes allow us to specialize them to important cases. We summarize the
main examples to which we apply our theory in 1. In the row of vector spaces
with forms of this table, we take the category of vector spaces with suitable
forms, and morphisms are isometries. Even though this category misses most of
the coproducts, the natural candidate for it is the orthogonal sum, and that
is the monoidal product we consider. Also observe that in the matroid case,
there is no categorical approach developed so far that suits our purposes. The
problem relies on the fact that the main poset we want to study is the lattice
of flats, but the matroid obtained from the join of two flats depends
specifically on the matroid we are working with. Hence, in that case, we
regard the lattice of flats as a poset-category, and the monoidal product is
just the join.

Finally, we would like to emphasize that our results are based on
combinatorial and topological methods that apply mainly to finite-dimensional
complexes and posets.

Object | Category | Monoidal product | $\operatorname{{\mathcal{S}}}(X,\sqcup)$
---|---|---|---
Free group of finite rank | Groups | Free product | Poset of free factors
4cmFinite-dimensional ${\mathbb{K}}$-vector space |  ${\mathbb{K}}$-vector spaces | Direct sum | Poset of subspaces
4cmFinite-dimensional ${\mathbb{K}}$-vector space with a non-degenerate form |  3.5cm${\mathbb{K}}$-vector spaces with forms and isometries | Orthogonal sum |  3.5cmNon-degenerate subspace poset
Matroid |  3.5cmLattice of flats (of the matroid) | Join | Lattice of flats
4cmFree module of finite rank over a Dedekind domain ${\mathcal{O}}$ |  (Free) ${\mathcal{O}}$-modules | Direct sum | Poset of summands
Table 1. Main examples of the paper.

A related categorical approach is developed in [GKRW, RW, RWW], where certain
complexes are constructed from complemented subobjects. While our focus is the
analysis of the combinatorics and the homotopy types of our constructions,
these works address homological stability problems for quite general families
of groups by starting out with high connectivity assumptions. Thus, in a
certain sense, our work complements [GKRW, RW, RWW] by establishing a set of
posets and simplicial complexes together with properties and relations among
them that can be used to prove Cohen-Macaulayness or high connectivity. We
expect that our results yield new classes of complexes which then can be fed
into the sophisticated machinery devised in these works.

A different but related problem where our results may provide new input data,
is the search for dualizing modules (see e.g., [CP, CFP]). If $V$ is an
$n$-dimensional vector space over ${\mathbb{K}}$, the order complex of the
proper part of the poset of subspaces of $V$ is the Tits building of
$\operatorname{SL}_{n}({\mathbb{K}})$. The top-dimensional homology group of
this complex is the well-known Steinberg module of
$\operatorname{SL}_{n}({\mathbb{K}})$, and by Borel-Serre [BS73] it is a
rational dualizing module for $\operatorname{SL}_{n}({\mathcal{O}})$, when
${\mathbb{K}}$ is a number field and ${\mathcal{O}}$ is its ring of integers.
The explicit determination of a dualizing module may be a difficult and often
unsolved problem. For example, a question raised by Hatcher and Vogtmann
[HV98a] asked whether the top dimensional cohomology group of the free factor
complex $\operatorname{FC}(F_{n})$ of the free group $F_{n}$ of rank $n$
yields a dualizing module for $\operatorname{Aut}(F_{n})$. However, a negative
answer to this question was recently given in the case $n=5$ in [HMNP]. In
general, the homology (or even the “Lefschetz module”) of
$\operatorname{{\mathcal{S}}}(X,\sqcup)$ can be regarded as a Steinberg module
for a suitable group of automorphisms of this poset, prompting consideration
for a similar interpretation in other contexts.

From the combinatorial perspective, this paper explores a unified approach to
many well-studied poset constructions (see e.g., [Sta1, Sta2, Wa]). It
comprises important complexes and posets associated with matroids including
the lattice of flats, the independence complex and the recently studied
augmented Bergman complex (see e.g., [Bjo2, BHMPW]). In both cases, new
constructions and results are added. These yield results on homotopy types
about “natural” combinatorially defined posets and complexes. When looking at
the reduced Euler characteristic or equivalently the Möbius number, new
enumerative identities follow. Experimentally we observe that even more of
such identities should hold, and leave these and related challenging questions
and conjectures open.

The paper is organized as follows. In 2 we give precise definitions of the
framework we want to work with and provide definitions of complemented
elements and decompositions arising from subobjects in a certain monoidal
category. In 3 we establish basic properties of poset of decompositions and
partial decompositions, and in 4 we define other posets that can also be
constructed from decompositions. In particular, we include here the definition
of the frame complex, the partial basis complex, their ordered versions, the
augmented Bergman complex, and the Charney complex. In 5, we prove some
homotopy results that usually set homotopical relations among the different
posets. Finally, 6 explores examples where our framework fits into the
classical constructions appearing in the literature. Throughout that section,
we pose some questions and present open problems.

Acknowledgments. We are very grateful to Benjamin Brück for the many
discussions during his research visit to Marburg and for providing us with
several motivational examples and references.

The first author is supported by a Postdoctoral fellowship granted by the
Alexander von Humboldt Stiftung, and the FWO grant 12K1223N.

##  2\. Poset of decompositions into subobjects

We start our poset constructions with a purely combinatorial approach. We will
see that this approach, even though vital for the subsequent developments,
fails to cover important classes of posets which otherwise perfectly fit our
setting. For that reason, we will later in this section use category theory
language.

We recall first some basic notation and definitions. A poset ${\mathcal{S}}$
is a (not necessarily finite) set with a partial order $\leq$. We call
${\mathcal{S}}$ bounded if it has a unique minimal element $0_{{\mathcal{S}}}$
and a unique maximal element $1_{{\mathcal{S}}}$. When the reference poset
${\mathcal{S}}$ is clear from the context, we just write $0$ and $1$ for the
minimum and maximum of ${\mathcal{S}}$. For a bounded poset ${\mathcal{S}}$,
we denote by
${\mathcal{S}}^{*}={\mathcal{S}}\setminus\\{0_{{\mathcal{S}}},1_{{\mathcal{S}}}\\}$
its proper part. Also if ${\mathcal{S}}$ has just a unique maximum element
$1_{{\mathcal{S}}}$, we set
${\mathcal{S}}^{\circ}={\mathcal{S}}\setminus\\{1_{{\mathcal{S}}}\\}$. For
$x\in{\mathcal{S}}$ we write ${\mathcal{S}}_{\leq x}$ for the subposet
$\\{\,y\in{\mathcal{S}}~{}|~{}y\leq x\\}$. Analogously defined are
${\mathcal{S}}_{<x}$, ${\mathcal{S}}_{\geq x}$ and ${\mathcal{S}}_{>x}$.

The order complex of a poset ${\mathcal{S}}$ is the simplicial complex
$\Delta({\mathcal{S}})$ whose $i$-simplices are the finite chains
$x_{0}<\cdots<x_{i}$ of $i+1$ distinct elements from ${\mathcal{S}}$. When we
speak of topological properties of a poset we mean the corresponding property
of the geometric realization of its order complex. For an element $x$ of
${\mathcal{S}}$, we let $h(x)$ be the dimension of the order complex of
${\mathcal{S}}_{\leq x}$ and call $h(x)$ the height of $x$ in ${\mathcal{S}}$.
The height of ${\mathcal{S}}$ is the supremum of the heights of its elements
or, equivalently, the dimension of its order complex. All poset constructions
we want to unify in this paper yield posets of finite height. Nevertheless,
during the construction process, we will sometimes have to go through infinite
height posets.

In a poset ${\mathcal{S}}$ we say for a subset $\sigma$ that the join of
$\sigma$ exists in ${\mathcal{S}}$ if $\\{y\in{\mathcal{S}}~{}|~{}y\geq x$ for
all $x\in\sigma\\}$ has a unique minimal element. In this case, we write
$\bigvee_{x\in\sigma}x$ for the join of $\sigma$. Analogously, we say that the
meet of $\sigma$ exists in ${\mathcal{S}}$ if
$\\{y\in{\mathcal{S}}~{}|~{}y\leq x$ for all $x\in\sigma\\}$ has a unique
maximal element, and write $\bigwedge_{x\in\sigma}x$ for the meet of $\sigma$.
Here, we adopt the usual convention that in a bounded poset the join over the
empty set is $0_{{\mathcal{S}}}$ and the meet over the empty set if
$1_{{\mathcal{S}}}$.

We denote by $|\sigma|$ the size of a set $\sigma$.

######  Definition 2.0.

Let ${\mathcal{S}}$ be a bounded poset. We say that a non-empty subset
$\sigma\subseteq{\mathcal{S}}$ is a (full) decomposition of ${\mathcal{S}}$ if

  * (1)

for all $\tau\subseteq\sigma$ the join $\bigvee_{x\in\tau}x$ exists in
${\mathcal{S}}$ and

$h\big{(}\bigvee_{x\in\tau}x\big{)}=\sum_{x\in\tau}h(x),$

  * (2)

$\big{\\{}\,\bigvee_{x\in\tau}x\,\big{|}\,\tau\subseteq\sigma\,\big{\\}}\subseteq{\mathcal{S}}$
with suprema and infima taken in ${\mathcal{S}}$ is a Boolean lattice on
$|\sigma|$ elements with maximal element $1_{\mathcal{S}}$.

We say $\tau\subseteq{\mathcal{S}}$ is a partial decomposition if it is a
subset of a decomposition. We denote by
$\operatorname{\mathcal{PD}}({\mathcal{S}})$ the set of all partial
decompositions and by ${\mathcal{D}}({\mathcal{S}})$ the set of decompositions
of ${\mathcal{S}}$. We order $\operatorname{\mathcal{PD}}({\mathcal{S}})$ and
${\mathcal{D}}({\mathcal{S}})$ by refinement. That is, $\tau\leq\sigma$ if and
only if for all $x\in\tau$ there exists $y\in\sigma$ such that $x\leq y$.

Note that $0_{{\mathcal{S}}}$ cannot be part of a decomposition. It also
follows from the definition that $\operatorname{\mathcal{PD}}({\mathcal{S}})$
is a bounded poset where the empty set $\emptyset$ is the unique minimal
element, and $\\{1_{{\mathcal{S}}}\\}$ is the unique maximal element. The
subposet ${\mathcal{D}}({\mathcal{S}})$ is an upper-order ideal of
$\operatorname{\mathcal{PD}}({\mathcal{S}})$ which may not have a unique
minimal element. We study the case where ${\mathcal{D}}({\mathcal{S}})$ has a
unique minimal element in 5.0.

The following simple properties of full and partial decompositions are
immediate consequences of the definition.

######  Lemma 2.1.

Let ${\mathcal{S}}$ be a bounded poset and
$\sigma,\tau\in{\mathcal{D}}({\mathcal{S}})$.

  * •

if $\tau\subseteq\sigma$, then $\tau=\sigma$.

  * •

if $\tau\leq\sigma$ and $|\tau|=|\sigma|$ then $\tau=\sigma$.

######  Remark 2.0.

There are several natural orders on
$\operatorname{\mathcal{PD}}({\mathcal{S}})$:

  * •

the order $\leq$ we have defined, i.e., order by refinement;

  * •

the order $\leq^{\prime}$ where $\tau\leq^{\prime}\sigma$ if $\tau\leq\sigma$
and also there exist decompositions $\hat{\tau}\leq\hat{\sigma}$ with
$\tau\subseteq\hat{\tau}$ and $\sigma\subseteq\hat{\sigma}$;

  * •

the order $\subseteq$ induced by set-inclusion.

The ordering $\leq^{\prime}$ coincides with $\leq$ if
$\tau,\sigma\in{\mathcal{D}}({\mathcal{S}})$. Although in general
$\leq^{\prime}$ is more restrictive than $\leq$, they will coincide in many
important examples discussed later. See 6.

On the other hand, we will not address the poset structure obtained from the
ordering $\subseteq$. See 6.0 for an example comparing the inclusion and the
refinement orderings.

######  Example 2.0.

If ${\mathcal{S}}$ is the Boolean lattice of all subsets of a set $\tau$
ordered by refinement. Then ${\mathcal{D}}({\mathcal{S}})$ is the partition
lattice $\Pi(\tau)$. It is easily checked that for a decomposition
$\sigma\in{\mathcal{D}}({\mathcal{S}})$ the upper interval
${\mathcal{D}}({\mathcal{S}})_{\geq\sigma}$ is isomorphic to $\Pi(\sigma)$.
Recall that the Möbius number of $\Pi(\tau)$ is $(-1)^{|\tau|-1}(|\tau|-1)!$
if $|\tau|\geq 2$.

Indeed, one reason for including the height condition in the definition of a
decomposition is that we want to control upper intervals in
${\mathcal{D}}({\mathcal{S}})$ in the same way we do it in the partition
lattice.

######  Lemma 2.2.

For a bounded poset ${\mathcal{S}}$ and $\tau\in{\mathcal{D}}({\mathcal{S}})$,
we have ${\mathcal{D}}({\mathcal{S}})_{\geq\tau}\cong\Pi(\tau)$.

###### Proof.

By definition, it is clear that for every partition $\pi$ of $\tau$, the set
$\\{\bigvee_{y\in p}y\mathrel{{\>:\>}}p\in\pi\\}$ is a decomposition. Now
suppose that $\tau\leq\sigma$. For $x\in\sigma$, let
$x^{\prime}=\bigvee_{y\in\tau\mathrel{{\>:\>}}y\leq x}y$. Then
$\sigma^{\prime}=\\{x^{\prime}\mathrel{{\>:\>}}x\in\sigma\\}$ is a
decomposition since it is obtained from a suitable partition of $\tau$, with
$|\sigma^{\prime}|=|\sigma|$ and $\sigma^{\prime}\leq\sigma$. By 2.1,
$\sigma^{\prime}=\sigma$. This means that elements of $\sigma$ are obtained by
joining elements of $\tau$. Thus
${\mathcal{D}}({\mathcal{S}})_{\geq\tau}\cong\Pi(\tau)$. ∎

The lower intervals in ${\mathcal{D}}({\mathcal{S}})$ and
$\operatorname{\mathcal{PD}}({\mathcal{S}})$ are not necessarily direct
products of decomposition or partial decomposition posets if we do not require
additional “relative-extension” properties. We have the following
characterization.

######  Lemma 2.3.

Let ${\mathcal{S}}$ be a bounded poset such that for every
$\sigma\in\operatorname{\mathcal{PD}}({\mathcal{S}})$ it holds:

  * (E1)

$\operatorname{\mathcal{PD}}({\mathcal{S}}_{\leq
x})=\operatorname{\mathcal{PD}}({\mathcal{S}})_{\leq\\{x\\}}$ for all
$x\in\sigma$, and

  * (E2)

if $\tau_{x}\in\operatorname{\mathcal{PD}}({\mathcal{S}}_{\leq x})$ for
$x\in\sigma$, then
$\cup_{x\in\sigma}\tau_{x}\in\operatorname{\mathcal{PD}}({\mathcal{S}})_{\leq\sigma}$.

Then for every $\sigma\in\operatorname{\mathcal{PD}}({\mathcal{S}})$ we have
$\operatorname{\mathcal{PD}}({\mathcal{S}})_{\leq\sigma}=\prod_{x\in\sigma}\operatorname{\mathcal{PD}}({\mathcal{S}}_{\leq
x})$. If $\sigma\in{\mathcal{D}}({\mathcal{S}})$ then also
${\mathcal{D}}({\mathcal{S}})_{\leq\sigma}=\prod_{x\in\sigma}{\mathcal{D}}({\mathcal{S}}_{\leq
x})$.

Conversely, if we naturally have
$\operatorname{\mathcal{PD}}({\mathcal{S}})_{\leq\sigma}=\prod_{x\in\sigma}\operatorname{\mathcal{PD}}({\mathcal{S}}_{\leq
x})$ then (E1) and (E2) hold.

###### Proof.

Let $\tau\leq\sigma$ and define $\tau_{x}=\sigma\cap{\mathcal{S}}_{\leq x}$.
Then by (E1), $\tau_{x}\in\operatorname{\mathcal{PD}}({\mathcal{S}}_{\leq
x})$. Therefore we have a well-defined order-preserving map
$\operatorname{\mathcal{PD}}({\mathcal{S}})_{\leq\sigma}\to\prod_{x\in\sigma}\operatorname{\mathcal{PD}}({\mathcal{S}}_{\leq
x})$ given by $\tau\mapsto(\tau_{x})_{x\in\sigma}$. This map is clearly an
embedding by definition of the ordering. Property (E2) also implies that this
is surjective: indeed, if
$\tau_{x}\in\operatorname{\mathcal{PD}}({\mathcal{S}}_{\leq x})$, then
$\tau=\cup_{x\in\sigma}\tau_{x}\in\operatorname{\mathcal{PD}}({\mathcal{S}})_{\leq\sigma}$
by (E2), and $\tau\cap{\mathcal{S}}_{\leq x}=\tau_{x}$.

The structure of the lower interval in the decomposition poset follows
immediately from the result in the partial decompositions. We leave the proof
of the reverse implication for the reader. ∎

If we drop the requirement on the heights in 2.0, we may encounter situations
where we have two distinct comparable decompositions of the same size. In
particular, the conclusions of 2.2 may fail. This leads us to the definition
of weak decompositions.

######  Definition 2.0.

Let ${\mathcal{S}}$ be a bounded poset (not necessarily of finite height), and
let $\sigma\subseteq{\mathcal{S}}$. We say that $\sigma$ is a weak
decomposition if for all $\tau\subseteq\sigma$, $\bigvee_{x\in\tau}x$ exists
and the subposet
$\\{\bigvee_{x\in\tau}x\mathrel{{\>:\>}}\tau\subseteq\sigma\\}$, with suprema
and infima taken in ${\mathcal{S}}$, is a Boolean lattice on $|\sigma|$
elements with maximal element $1_{\mathcal{S}}$. We analogously define the
posets of weak decompositions ${\mathcal{D}}_{w}({\mathcal{S}})$ and weak
partial decompositions $\operatorname{\mathcal{PD}}_{w}({\mathcal{S}})$, both
ordered by refinement.

Clearly decompositions are weak decompositions, so
${\mathcal{D}}({\mathcal{S}})\subseteq{\mathcal{D}}_{w}({\mathcal{S}})$ and
$\operatorname{\mathcal{PD}}({\mathcal{S}})\subseteq\operatorname{\mathcal{PD}}_{w}({\mathcal{S}})$
are subposets.

The following example shows that indeed weak decompositions are not always
decompositions and that upper-intervals in ${\mathcal{D}}_{w}({\mathcal{S}})$
may not be partition lattices.

######  Example 2.0.

Let ${\mathcal{S}}$ be a bounded poset. Then
$d\in{\mathcal{D}}_{w}({\mathcal{S}})$ if and only if either $d=\\{x,y\\}$,
with $x$ and $y$ belonging to different connected components of
${\mathcal{S}}^{*}$, or else
$d\in{\mathcal{D}}_{w}(C\cup\\{0_{{\mathcal{S}}},1_{{\mathcal{S}}}\\})$ where
$C$ is a connected component of ${\mathcal{S}}^{*}$. Indeed, if
$d\in{\mathcal{D}}_{w}({\mathcal{S}})$ contains elements $x,y$ in different
connected components then $x\vee y=1$. Thus, for $z\in d$ we have
$z\wedge(x\vee y)=z$, which implies that $z=x$ or $y$. Note also that if
$C_{1},C_{2}$ are two different connected components of ${\mathcal{S}}^{*}$
and $x,y,z\in{\mathcal{S}}^{*}$ are elements such that $x\leq y\in C_{1}$ and
$z\in C_{2}$, then $\\{x,z\\}\leq\\{y,z\\}$ and both are weak decompositions
of ${\mathcal{S}}$. In particular,
${\mathcal{D}}_{w}({\mathcal{S}})_{\geq\\{x,z\\}}={\mathcal{S}}_{\geq
x}\times{\mathcal{S}}_{\geq z}$, so the upper interval is not necessarily a
partition lattice.

Indeed, suppose that ${\mathcal{S}}=\\{0,a,b,c,d,1\\}$ with minimum $0$,
maximum $1$, $a<c$ and $b<d$. We have
${\mathcal{D}}_{w}({\mathcal{S}})=\\{\\{a,b\\},\\{b,c\\},\\{a,d\\},\\{c,d\\},\\{1\\}\\}$,
and the upper-interval ${\mathcal{D}}_{w}({\mathcal{S}})_{\geq\\{a,b\\}}$ is
not a partition lattice. See 1 for the Hasse diagrams of ${\mathcal{S}}$,
$\operatorname{\mathcal{PD}}_{w}({\mathcal{S}})$ and
$\operatorname{\mathcal{PD}}(S)$, and the corresponding decomposition posets
which are depicted in blue. Note that ${\mathcal{S}}$ is a lattice but
$\operatorname{\mathcal{PD}}({\mathcal{S}})$ is not.

This example also shows that
$(\operatorname{\mathcal{PD}}({\mathcal{S}}),\leq)\neq(\operatorname{\mathcal{PD}}({\mathcal{S}}),\leq^{\prime})$,
where $\leq^{\prime}$ is the ordering defined in 2.0 The latter poset is
displayed in 2.

$0$$1$$a$$b$$c$$d$${\mathcal{S}}$$\operatorname{\mathcal{PD}}_{w}({\mathcal{S}})$$\operatorname{\mathcal{PD}}({\mathcal{S}})$$1$$1$$a,b$$a,d$$b,c$$c,d$$a,d$$b,c$$b$$d$$a$$c$$c$$a$$b$$d$$\emptyset$$\emptyset$
Figure 1. Hasse diagrams of posets ${\mathcal{S}}$,
$\operatorname{\mathcal{PD}}_{w}({\mathcal{S}})$ and
$\operatorname{\mathcal{PD}}({\mathcal{S}})$. The corresponding decomposition
posets are displayed in blue. $a$$d$$b$$c$$b,c$$1$$a,d$$\emptyset$ Figure 2.
Poset $(\operatorname{\mathcal{PD}}({\mathcal{S}}),\leq^{\prime})$.

Let us analyze now some examples in subgroup posets.

######  Example 2.0.

Let $G=S_{3}$ be the symmetric group on $3$ letters. Then the poset of
subgroups of $G$, denoted by ${\mathcal{S}}(G)$, consists of a unique minimal
element (the trivial subgroup, denoted by $1$ in this example), a unique
maximal element (the whole group $G$), a subgroup of order $3$ generated by a
$3$-cycle, and 3 subgroups of order $2$ (one for each transposition). Then
${\mathcal{S}}(G)^{*}$ is a discrete poset with $4$ points. Hence, a
decomposition of ${\mathcal{S}}(G)$ different from $\\{G\\}$ is a set of two
non-trivial proper subgroups of $G$.

######  Example 2.0.

Let $G$ be a cyclic group of order $m$. If $m=p^{n}$ is a prime power, then
${\mathcal{S}}(G)$, the lattice of subgroups of $G$, has height $n$, and
$\operatorname{\mathcal{PD}}({\mathcal{S}}(G))={\mathcal{D}}({\mathcal{S}}(G))=\\{\\{G\\}\\}$.

Suppose $G=C_{p_{1}^{n_{1}}}\times\ldots\times C_{p_{r}^{n_{r}}}$ for
different primes $p_{j}$. Let $G_{j}=C_{p_{j}^{n_{j}}}$. Then the height of a
subgroup of $G$ equals the number of primes (counted with multiplicity) of its
order. In particular, $h({\mathcal{S}}(G))=\sum_{j}n_{j}$. Therefore, the only
possible decompositions of $G$ are those obtained by coarsening the
decomposition $\\{G_{1},\ldots,G_{r}\\}$. Hence
${\mathcal{D}}({\mathcal{S}}(G))$ is the partition lattice on the set
$\\{G_{1},\ldots,G_{r}\\}$.

###  2.1. Monoidal categories and restricted decompositions

The constructions above will not cover all the structures we would like to
study. For example, consider the case of a non-trivial finitely generated free
Abelian group $A$. Its poset of subgroups is a lattice of infinite height.
Therefore, we cannot talk about decompositions given the finite-dimensional
requirement in 2.0. Nevertheless, it is implicit here that decompositions
should be sets of subgroups $A_{1},\ldots,A_{r}$ which span a Boolean lattice
inside the poset of subgroups with join $A_{1}\oplus\ldots\oplus
A_{r}=\left\langle A_{1},\ldots,A_{r}\right\rangle=A$. Thus, if we instead
consider the subposet of complemented subgroups of $A$ (or summands), this is
a bounded poset of finite height. However, there is still a problem in this
subposet: even if $\\{A_{1},A_{2}\\}$ is a (weak) decomposition in the poset
sense, it is not clear that $A_{1}\vee A_{2}$ coincides with the direct sum
$A_{1}\oplus A_{2}$. For example, if $A={\mathbb{Z}}^{2}$, then
$A_{1}=\left\langle(1,2)\right\rangle$ and
$A_{2}=\left\langle(1,0)\right\rangle$ are summands of $A$ such that
$A_{1}\wedge A_{2}=0$ and $A_{1}\vee A_{2}=A$ in the poset of summands of $A$,
but $A_{1}+A_{2}=A_{1}\oplus A_{2}<A$.

A similar situation arises if we consider instead a free group of rank $\geq
2$. However, by a decomposition here we not only want the Boolean lattice
condition, but also “free” spans. That is, if $F$ is a free group of rank
$\geq 2$ and $H,K$ are two subgroups, then $H,K$ is a weak decomposition if
$H\cap K=1$ and $H\vee K=\left\langle H,K\right\rangle=F$. But the span
$\left\langle H,K\right\rangle$ forgets the relation between the elements of
$H$ and $K$, and we may actually want $\left\langle H,K\right\rangle$ to be
equal to the free product of $H$ and $K$. An adaptation of the Abelian-case
example with ${\mathbb{Z}}^{2}$ shows that there are free factors $H,K$ of the
free group $F_{2}$ of rank $2$ such that $H\cap K=1$ but $H\vee K=\left\langle
H,K\right\rangle=F$ is not the free product of $H$ and $K$.

To cover these and other cases we extend our purely combinatorial setting by
using concepts from category theory. We start with some notation.

For a category ${\mathcal{C}}$ and an object $X\in{\mathcal{C}}$, write
$\operatorname{Hom}_{\mathcal{C}}(Y,X)$ for the homomorphisms from an object
$Y$ to $X$, and $\operatorname{Aut}_{\mathcal{C}}(X)$ for the isomorphisms of
$X$. Let ${\mathcal{C}}\downarrow X$ be the comma category with objects
$(Y,f)$, where $Y\in{\mathcal{C}}$ and
$f\in\operatorname{Hom}_{{\mathcal{C}}}(Y,X)$, and morphisms between $(Y,f)$
and $(Z,g)$ being $h\in\operatorname{Hom}_{{\mathcal{C}}}(Y,Z)$ with $f=gh$.
We consider the subcategory
${\mathcal{C}}\mathrel{\rotatebox[origin={c}]{-90.0}{$\hookrightarrow$}}X$
with objects $(Y,i)$ such that $i$ is a monomorphism, and morphisms inherited
from ${\mathcal{C}}\downarrow X$. Note that a morphism
$h:(Y,i)\rightarrow(Z,j)$ in
${\mathcal{C}}\mathrel{\rotatebox[origin={c}]{-90.0}{$\hookrightarrow$}}X$
comes indeed from a monomorphism $h:Y\to Z$.

A subobject of $X$ is an isomorphism class of objects in
${\mathcal{C}}\mathrel{\rotatebox[origin={c}]{-90.0}{$\hookrightarrow$}}X$,
where $(Y,i)$ and $(Z,j)$ are isomorphic if and only if there exists an
isomorphism $\phi:Y\to Z$ such that $i=j\phi$. We write $[(Y,i)]$ for the
subobject represented by $(Y,i)$. Let $\operatorname{{\mathcal{S}}}(X)$ denote
the collection of subobjects of $X$, which is a poset-category (that is, there
is at most one arrow between any two objects and this defines an anti-
symmetric relation). We gather some properties in the following lemma.

######  Lemma 2.4.

Let $X$ be an object of a category ${\mathcal{C}}$. The following hold:

  1. (i)

${\mathcal{C}}\mathrel{\rotatebox[origin={c}]{-90.0}{$\hookrightarrow$}}X$ is
a pre-order with $x\lesssim y$ if there is an arrow from $x$ to $y$. Namely,
there is at most one arrow between any two objects.

  2. (ii)

$\operatorname{{\mathcal{S}}}(X)$ is a skeleton of the category
${\mathcal{C}}\mathrel{\rotatebox[origin={c}]{-90.0}{$\hookrightarrow$}}X$.

  3. (iii)

If ${\mathcal{C}}$ has an initial object $0$, then
$0\in\operatorname{{\mathcal{S}}}(X)$ is the unique minimum of this poset.

  4. (iv)

The poset $\operatorname{{\mathcal{S}}}(X)$ has a unique maximum given by the
class of $(X,i)$, where $i$ is any isomorphism of $X$. We denote this class
just by $X$.

  5. (v)

If $z\in\operatorname{{\mathcal{S}}}(X)$ then there is a poset isomorphism
$\operatorname{{\mathcal{S}}}(X)_{\leq z}\cong\operatorname{{\mathcal{S}}}(Z)$
for any representative $(Z,i)\in z$.

###### Proof.

These are straightforward facts from the definitions. ∎

Note that (v) also implies that if $i,i^{\prime}:Z\to X$ are two monomorphisms
and $z:=[(Z,i)]$, $z^{\prime}:=[(Z,i^{\prime})]$ then
$\operatorname{{\mathcal{S}}}(X)_{\leq
z}\cong\operatorname{{\mathcal{S}}}(Z)\cong\operatorname{{\mathcal{S}}}(X)_{\leq
z^{\prime}}$. In particular, if $i:X\to X$ is a monomorphism, then
$\operatorname{{\mathcal{S}}}(X)_{\leq[(X,i)]}\cong\operatorname{{\mathcal{S}}}(X)$.

######  Warning.

$\operatorname{{\mathcal{S}}}(X)$ is always a bounded poset but, in general,
of infinite height.

We want to restrict the collection of decompositions to those that have a
particular shape. To specify the “shape” of the decompositions, we define
beforehand a “product of objects”, and then we select the decompositions whose
join equals the product. This naturally leads us to the definition of a
monoidal product in our category. However, we will require this product to
have some particular properties closer to a coproduct.

######  Definition 2.0.

An initial symmetric monoidal category, or ISM-category for short, is an
initial monoidal category $({\mathcal{C}},\sqcup)$, where $\sqcup$ is a
naturally associative and commutative product, and the initial object of the
category is also a neutral element for $\sqcup$.

The notation “$\sqcup$” suggests a similarity with the coproduct. Indeed,
sometimes $\sqcup$ will be the coproduct of the category. However, in many
cases, the category ${\mathcal{C}}$ may not have all coproducts, and so
$\sqcup$ will serve as a remedy.

######  Remark 2.0.

Note that, for any two objects $X$, $Y$, we have a canonical map
$\iota_{X}:X\to X\sqcup Y$ arising from

$X\cong X\sqcup 0\overset{\operatorname{id}_{X}\sqcup
i_{0}}{\longrightarrow}X\sqcup Y,$

where $i_{0}:0\to Y$.

To define decompositions in $\operatorname{{\mathcal{S}}}(X)$ we need finite
height, and $\operatorname{{\mathcal{S}}}(X)$ may have infinite dimension in
general (for example, when $X$ is a non-trivial free group in the category of
groups, see 2.0). For that reason, we will work with the subposet
$\operatorname{{\mathcal{S}}}(X,\sqcup)$ of $\sqcup$-complemented subobjects,
which we expect to have finite height and whose elements are eligible to be
part of decompositions. To that end, we introduce first the notion of
$\sqcup$-compatible joins of elements.

######  Definition 2.0.

Let $({\mathcal{C}},\sqcup)$ be an ISM-category, $X\in{\mathcal{C}}$ and
$\sigma=\\{x_{1},\ldots,x_{r}\\}\subseteq{\mathcal{T}}\subseteq\operatorname{{\mathcal{S}}}(X)$.
We say that $\sigma$ is $\sqcup$-compatible in ${\mathcal{T}}$ if there are
suitable representatives $x_{i}=[(X_{i},j_{i})]$ such that the following
conditions hold:

  1. (i)

For all $\emptyset\neq I\subseteq I^{\prime}\subseteq[r]$, we have natural
monomorphisms 111In the sense they are induced by the canonical maps $Y\to
Y\sqcup Z\cong Z\sqcup Y$.

$j_{I,I^{\prime}}:\bigsqcup_{i\in I}X_{i}\rightarrow\bigsqcup_{i\in
I^{\prime}}X_{i}\qquad\text{ and }\qquad j_{I}:\bigsqcup_{i\in
I}X_{i}\rightarrow X;$

  2. (ii)

for any $\emptyset\neq I\subseteq[r]$, there exists the join $\bigvee_{i\in
I}x_{i}$ in ${\mathcal{T}}$ and

$\Big{[}\,\big{(}\,\bigsqcup_{i\in
I}X_{i},j_{I}\,\big{)}\,\Big{]}=\bigvee_{i\in I}x_{i};$

  3. (iii)

the following diagram commutes for all $\emptyset\neq I\subseteq
I^{\prime}\subseteq[r]$ and $k\in I$:

$\textstyle{X_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{\\{k\\},I}}$$\scriptstyle{j_{k}}$$\scriptstyle{j_{\\{k\\},I^{\prime}}}$$\textstyle{\bigsqcup_{i\in
I}X_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{I,I^{\prime}}}$$\scriptstyle{j_{I}}$$\textstyle{\bigsqcup_{i\in
I^{\prime}}X_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{I^{\prime}}}$$\textstyle{X}$

We now define posets of subobjects compatible with the monoidal product
$\sqcup$:

######  Definition 2.0.

Let $({\mathcal{C}},\sqcup)$ be an ISM-category and let $X\in{\mathcal{C}}$ be
an object.

  1. (1)

An element $x\in\operatorname{{\mathcal{S}}}(X)$ is $\sqcup$-complemented if
there exists a poset-complement $y\in\operatorname{{\mathcal{S}}}(X)$ of $x$
(that is, $x\wedge y=0$ and $x\vee y=1$) such that the set
$\\{x,y\\}\subseteq\operatorname{{\mathcal{S}}}(X)$ is $\sqcup$-compatible in
$\operatorname{{\mathcal{S}}}(X)$.

  2. (2)

We define $\operatorname{{\mathcal{S}}}(X,\sqcup)$ to be the subposet of
$\sqcup$-complemented elements of $\operatorname{{\mathcal{S}}}(X)$.

  3. (3)

If $\operatorname{{\mathcal{S}}}(X,\sqcup)$ has finite height, a
$\sqcup$-decomposition of $X$ is a decomposition
$\sigma\in{\mathcal{D}}(\operatorname{{\mathcal{S}}}(X,\sqcup))$ which is
$\sqcup$-compatible in $\operatorname{{\mathcal{S}}}(X,\sqcup)$.

  4. (4)

Denote by ${\mathcal{D}}(X,\sqcup)$ the subposet of
${\mathcal{D}}(\operatorname{{\mathcal{S}}}(X,\sqcup))$ consisting of
$\sqcup$-decompositions, and by $\operatorname{\mathcal{PD}}(X,\sqcup)$ the
subposet of
$\operatorname{\mathcal{PD}}(\operatorname{{\mathcal{S}}}(X,\sqcup))$
consisting of the subsets of $\sqcup$-decompositions.

Similarly, we define $\sqcup$-weak (partial) decompositions but without the
need to require finite height for $\operatorname{{\mathcal{S}}}(X,\sqcup)$.

Note that by definition $\operatorname{{\mathcal{S}}}(X,\sqcup)$ is bounded
with maximum $X$ and minimum $0$ (the initial object), and all its elements
are $\sqcup$-complemented, and in particular complemented. Also
$\operatorname{\mathcal{PD}}(X,\sqcup)$ is bounded with maximum $\\{X\\}$ and
minimum $\emptyset$. Moreover, ${\mathcal{D}}(X,\sqcup)$ is a non-empty order
ideal of $\operatorname{\mathcal{PD}}(X,\sqcup)$ and, in particular, $\\{X\\}$
is the unique maximal element of ${\mathcal{D}}(X,\sqcup)$.

######  Example 2.0.

Let ${\mathcal{C}}$ be the category of groups, so
$\operatorname{{\mathcal{S}}}(X)$ is the subgroup lattice for a group $X$. We
choose “$\sqcup$” to be the direct product of groups. In order to get a
finite-dimensional poset $\operatorname{{\mathcal{S}}}(X,\sqcup)$, we work
with a finite group $X$. Hence $\operatorname{{\mathcal{S}}}(X,\sqcup)$ is the
subposet of subgroups of $X$ which are direct-product factors of $X$. That is,
$H\leq X$ is $\sqcup$-complemented if and only if there exists a subgroup
$K\leq X$ such that $H\cap K=1$, $[H,K]=1$ and $H\times K\cong HK=X$. Recall
that, by the Krull-Schmidt theorem for finite groups, there exists a unique
direct product decomposition of a finite group into indecomposable factors, up
to permutation and isomorphism of the factors. In particular, the minimal
elements of $\operatorname{{\mathcal{S}}}(X,\sqcup)$ are the indecomposable
factors of $X$ and the height of an element
$H\in\operatorname{{\mathcal{S}}}(X,\sqcup)$ is the number of (non-trivial)
indecomposable direct product factors. Therefore,
${\mathcal{D}}_{w}(X,\sqcup)$ is the poset of all (internal) direct-product
decompositions of $X$ ordered by refinement, and ${\mathcal{D}}(X,\sqcup)$
equals ${\mathcal{D}}_{w}(X,\sqcup)$.

In view of the previous example and 2.0, we propose the following question:

######  Question 2.0.

If $G$ is the direct product of groups $G_{1},\ldots,G_{r}$ of relative prime
orders, do we have
${\mathcal{D}}({\mathcal{S}}(G),\times)=\Pi(r)\times{\mathcal{D}}(G_{1},\times)\times\ldots\times{\mathcal{D}}(G_{r},\times)$?

######  Example 2.0.

Recall that the coproduct in the category of groups is the free product. So we
choose “$\sqcup$” to be the free product of groups and $X$ a free group of
finite rank. Then $\operatorname{{\mathcal{S}}}(X)$ is a lattice with infinite
height, and we cannot apply our definition of decompositions. However,
$\operatorname{{\mathcal{S}}}(X,\sqcup)$ does have finite height. Indeed, we
claim that $\operatorname{{\mathcal{S}}}(X,\sqcup)$ is the poset
$\operatorname{FC}(X)$ of free factors of $X$. To see this, let
$H\in\operatorname{{\mathcal{S}}}(X,\sqcup)$ and let
$K\in\operatorname{{\mathcal{S}}}(X)$ be a $\sqcup$-complement for $H$. This
means that $H\cap K=1$ and $H*K$ is naturally isomorphic to $\left\langle
H,K\right\rangle=X$. That is, $X=H*K$, so $H,K$ are free factors of $X$. Since
a free factor of $X$ belongs to $\operatorname{{\mathcal{S}}}(X,\sqcup)$, we
conclude that $\operatorname{{\mathcal{S}}}(X,\sqcup)=\operatorname{FC}(X)$.
In particular, $\operatorname{FC}(X)$ is a graded poset and if
$H\in\operatorname{FC}(X)$ then its height equals $\operatorname{rk}(H)$, the
rank of $H$ as a (free) group. Moreover, since
$\operatorname{rk}(H*K)=\operatorname{rk}(H)+\operatorname{rk}(K)$, we
conclude that $\sqcup$-weak decompositions are $\sqcup$-decompositions.
Therefore, ${\mathcal{D}}_{w}(X,\sqcup)={\mathcal{D}}(X,\sqcup)$ is the poset
of decompositions $\\{H_{1},\ldots,H_{k}\\}$ of $X$ into free factors
$X=H_{1}*\ldots*H_{k}$.

######  Example 2.0.

Let ${\mathcal{S}}$ be a lattice. Then we have the induced poset-category
${\mathcal{C}}$ whose objects are the elements of ${\mathcal{S}}$ with arrows
$x\to y$ if and only if $x\leq y$. Then
$\operatorname{{\mathcal{S}}}(1_{{\mathcal{S}}})={\mathcal{S}}$. Moreover,
${\mathcal{C}}$ is an ISM-category with $\sqcup$ given by the join of elements
in ${\mathcal{S}}$, i.e. $x\sqcup y=x\vee y$. It is straightforward to verify
then that $\operatorname{{\mathcal{S}}}(1_{{\mathcal{S}}},\sqcup)$ is the
subposet of complemented elements of ${\mathcal{S}}$. In particular, if every
element of ${\mathcal{S}}$ is complemented then
${\mathcal{S}}=\operatorname{{\mathcal{S}}}(1_{{\mathcal{S}}},\sqcup)$,
${\mathcal{D}}({\mathcal{S}})={\mathcal{D}}({\mathcal{S}},\sqcup)$ and
$\operatorname{\mathcal{PD}}({\mathcal{S}})=\operatorname{\mathcal{PD}}(1_{{\mathcal{S}}},\sqcup)$.

##  3\. Properties of decomposition posets

From now on, we assume that $X$ is an object of an ISM-category
$({\mathcal{C}},\sqcup)$ such that $\operatorname{{\mathcal{S}}}(X,\sqcup)$
has finite height. Recall by 2.0 that we can regard a complemented lattice
${\mathcal{S}}$ as some poset ${\mathcal{S}}(X,\sqcup)$ where also
${\mathcal{D}}({\mathcal{S}})={\mathcal{D}}(X,\sqcup)$ and
$\operatorname{\mathcal{PD}}({\mathcal{S}})=\operatorname{\mathcal{PD}}(X,\sqcup)$.
Therefore, the results of this section are also valid for complemented
lattices of finite height.

The following proposition recovers the structure of upper intervals of
decompositions and it is an immediate consequence of the definitions.

######  Proposition 3.0.

For $\sigma\in{\mathcal{D}}(X,\sqcup)$ we have
${\mathcal{D}}(X,\sqcup)_{\geq\sigma}\cong\Pi(\sigma)$.

###### Proof.

This follows from 2.2 and the fact that ${\mathcal{D}}(X,\sqcup)$ is an upper
order ideal in ${\mathcal{D}}(\operatorname{{\mathcal{S}}}(X,\sqcup))$. ∎

In some important cases, we also have control over the lower intervals and the
following property holds:

(LIP) |  13cmFor all $\sigma\in\operatorname{\mathcal{PD}}(X,\sqcup)$ we have $\operatorname{\mathcal{PD}}(X,\sqcup)_{\leq\sigma}=\prod_{y\in\sigma}\operatorname{\mathcal{PD}}(Y,\sqcup)$, for suitable representatives $(Y,i_{y})\in y$.
---|---

In particular, (LIP) implies that
${\mathcal{D}}(X,\sqcup)_{\leq\sigma}=\prod_{y\in\sigma}{\mathcal{D}}(Y,\sqcup)$.
Conditions analogous to those from 2.3 imply (LIP).

The following proposition provides some information on the heights of the
posets of decompositions and partial decompositions.

######  Proposition 3.0.

Let $\operatorname{{\mathcal{S}}}(X,\sqcup)$ be of height $n$. Then every
decomposition in ${\mathcal{D}}(X,\sqcup)$ has size at most $n$. In
particular, $h(\operatorname{\mathcal{PD}}(X,\sqcup))\leq 2n-1$ and
$h({\mathcal{D}}(X,\sqcup))\leq n-1$. Moreover, the following are equivalent

  * (i)

there exists a decomposition of size $n$.

  * (ii)

$h(\operatorname{\mathcal{PD}}(X,\sqcup))=2n-1$.

  * (iii)

$h({\mathcal{D}}(X,\sqcup))=n-1$.

If one of the preceding conditions holds then the following are equivalent:

  * (i)’

for every decomposition $\sigma$ there exists a decomposition $\tau$ of size
$n$ with $\tau\leq\sigma$.

  * (ii)’

${\mathcal{D}}(X,\sqcup)$ is graded.

###### Proof.

The bound on the size of elements of ${\mathcal{D}}(X,\sqcup)$ is an immediate
consequence of the definitions. The bound on the height of
${\mathcal{D}}(X,\sqcup)$ then follows from 3.0. For
$\operatorname{\mathcal{PD}}(X,\sqcup)$ we argue as follows. Suppose that
$\sigma\in\operatorname{\mathcal{PD}}(X,\sqcup)$, $\sigma\neq\emptyset$, and
consider a saturated chain $\gamma$ in $\operatorname{\mathcal{PD}}(X,\sqcup)$
with maximal element $\sigma$. In particular, $\gamma$ contains the empty
partial decomposition. Then, a covering relation $\tau\prec\tau^{\prime}$ in
$\gamma$ means that either we “splitted” some elements of $\tau^{\prime}$
(meaning that for some $x\in\tau^{\prime}$ there are $y,z\in\tau$ with $x\geq
y\vee z$ and $x\not\in\tau$) or we eliminated an “unsplittable” element (i.e.
there is $x\in\tau^{\prime}$ which cannot be splitted and
$\tau=\tau^{\prime}\setminus\\{x\\}$). Therefore, for $x\in\tau$ we can
perform at most $h(x)-1$ splittings and eliminate at most $h(x)$ elements.
Thus, the maximal chain $\gamma$ has size at most

$1+\sum_{x\in\sigma}\big{(}\,h(x)-1+h(x)\,\big{)}=1-|\sigma|+2\sum_{x\in\sigma}h(x)\leq
1-|\sigma|+2n\leq 2n.$

This implies that $h(\operatorname{\mathcal{PD}}(X,\sqcup))\leq 2n-1$.

The equivalence of (i) and (iii) immediately follows from 3.0. The equivalence
of (i) and (ii) is a consequence of the proof of the bound
$h(\operatorname{\mathcal{PD}}(X,\sqcup))\leq 2n-1$.

Given the equivalent conditions (i),(ii), (iii) again 3.0 implies the
equivalence of (i)’ and (iii)’. ∎

######  Proposition 3.0.

The poset ${\mathcal{D}}(X,\sqcup)$ is a lattice after adding an extra minimum
element.

###### Proof.

We show that two $\sqcup$-decompositions $\sigma_{1},\sigma_{2}$, lower
bounded by some element $\tau\in{\mathcal{D}}(X,\sqcup)$, have a meet in
${\mathcal{D}}(X,\sqcup)$. Indeed, since
$\sigma_{1},\sigma_{2}\in{\mathcal{D}}(X,\sqcup)_{\geq\tau}\cong\Pi(\tau)$ is
a partition lattice, they have a meet in this poset, which we denote by
$\sigma$. Let $e:2^{\tau}\to{\mathcal{S}}(X,\sqcup)$ be the lattice embedding
of the Boolean lattice on $\tau$. If $x\in\sigma_{1}$ and $y\in\sigma_{2}$,
then there exist unique sets $\tau_{x},\tau_{y}\subseteq\tau$ such that
$x=\bigvee_{a\in\tau_{x}}a$ and $y=\bigvee_{b\in\tau_{y}}b$, where the join is
taken on the poset $\operatorname{{\mathcal{S}}}(X,\sqcup)$. Hence
$e(\tau_{x})=x$, $e(\tau_{y})=y$ and $e(\tau_{x}\cap\tau_{y})=x\wedge y$. The
decomposition $\sigma$ can be described as follows:

$\sigma=\\{e(\tau_{x}\cap\tau_{y})\mathrel{{\>:\>}}x\in\sigma_{1},y\in\sigma_{2}\\}\setminus\\{0\\}.$

But then, by definition of $e$,

$\sigma=\\{x\wedge
y\mathrel{{\>:\>}}x\in\sigma_{1},y\in\sigma_{2}\\}\setminus\\{0\\},$

where the meet is taken in $\operatorname{{\mathcal{S}}}(X,\sqcup)$. This
description of $\sigma$ does not depend on $\tau$, and clearly
$\sigma\geq\tau$. Therefore, $\sigma_{1}\wedge\sigma_{2}$ exists in
${\mathcal{D}}(X,\sqcup)$ and it is equal to the element $\sigma$ described
above. ∎

The poset $\operatorname{\mathcal{PD}}(X,\sqcup)$ is not a lattice in general,
as it is seen in the poset of 1 in 2.0. The next example also shows that weak
decomposition and weak partial decomposition posets might not be lattices.
Here we invoke 2.0.

######  Example 3.0.

Let $G=S_{4}$ be the symmetric group on $4$ letters. Then for
${\mathcal{S}}(G)$, the lattice of subgroups of $G$, we have that
${\mathcal{D}}_{w}({\mathcal{S}}(G))$ is not a lattice after adding a minimal
element, and hence nor is $\operatorname{\mathcal{PD}}_{w}({\mathcal{S}}(G))$.
This was shown by using GAP. To be more concrete, we have weak decompositions

$\\{\left\langle(3\,4)\right\rangle,\left\langle(2\,3)\right\rangle,\left\langle(1\,2)\right\rangle\\},\quad\text{and}\quad\\{\left\langle(3\,4)\right\rangle,\left\langle(2\,3)\right\rangle,\left\langle(1\,4)(2\,3)\right\rangle\\},$

for which their join in ${\mathcal{D}}_{w}({\mathcal{S}}(G))$ does not exist
since the elements above both of them in ${\mathcal{D}}_{w}({\mathcal{S}}(G))$
are

$\\{G\\},\,\,\\{\left\langle(3\,4),(1\,2)\right\rangle,\left\langle(2\,3),(1\,4)\right\rangle\\},\,\,\\{\left\langle(1\,4)(2\,3),(1\,3)(2\,4),(3\,4)\right\rangle,\left\langle(2\,3)\right\rangle\\}.$

Moreover, $\operatorname{\mathcal{PD}}({\mathcal{S}}(G))$ is not a lattice
either since the join fails for the partial decompositions

$\\{\left\langle(3\,4)\right\rangle\\},\quad\text{and}\quad\\{\left\langle(2\,3)\right\rangle,\left\langle(1\,2)\right\rangle\\}.$

In view of our definition of decompositions in terms of heights, it will be
useful to work with complements of elements which also have the correct
height.

######  Definition 3.0.

Let ${\mathcal{S}}$ be a poset.

  * •

Let $y\in{\mathcal{S}}$ and suppose that ${\mathcal{S}}_{\leq y}$ is bounded
of finite height. We say that $x\in{\mathcal{S}}_{\leq y}$ is $h$-complemented
in $y$ if there exists $x^{\prime}\leq y$ such that $x^{\prime}\wedge x=0$ and
$x\vee x^{\prime}=y$ (meet and join taken in ${\mathcal{S}}$) and
$h(x)+h(x^{\prime})=h(y)$.

Suppose now that ${\mathcal{S}}$ is bounded of finite height.

  * •

An element $x\in{\mathcal{S}}$ is called $h$-complemented if $x$ is
$h$-complemented in $1_{{\mathcal{S}}}$. We denote by ${\mathcal{S}}_{h}$ the
subposet of $h$-complemented elements.

  * •

We say that ${\mathcal{S}}$ is $h$-complemented if
${\mathcal{S}}={\mathcal{S}}_{h}$.

  * •

If for every $x\leq y$ in ${\mathcal{S}}$ there exists a (unique)
$h$-complement of $x$ in $y$, then we say that ${\mathcal{S}}$ is (uniquely)
downward $h$-complemented.

Next, we define the analogous concepts in posets
$\operatorname{{\mathcal{S}}}(X,\sqcup)$ by taking into account the monoidal
product $\sqcup$.

######  Definition 3.0.

Suppose that $\operatorname{{\mathcal{S}}}(X,\sqcup)$ has finite height.

  * •

If $z\leq y\in\operatorname{{\mathcal{S}}}(X,\sqcup)$, a
$(\sqcup,h)$-complement of $z$ in $y$ is an $h$-complement $w$ of $z$ in $y$
such that $\\{z,w\\}$ is $\sqcup$-compatible in
$\operatorname{{\mathcal{S}}}(X,\sqcup)$. We denote by
$\operatorname{{\mathcal{S}}}_{h}(X,\sqcup)$ the subposet of
$\operatorname{{\mathcal{S}}}(X,\sqcup)$ whose elements have a
$(\sqcup,h)$-complement in $\operatorname{{\mathcal{S}}}(X,\sqcup)$.

  * •

We say that $\operatorname{{\mathcal{S}}}(X,\sqcup)$ is
$(\sqcup,h)$-complemented if
$\operatorname{{\mathcal{S}}}_{h}(X,\sqcup)=\operatorname{{\mathcal{S}}}(X,\sqcup)$.

  * •

We say that $\operatorname{{\mathcal{S}}}(X,\sqcup)$ is (uniquely) downward
$(\sqcup,h)$-complemented if for all $z\leq
y\in\operatorname{{\mathcal{S}}}(X,\sqcup)$ there exists a (unique)
$(\sqcup,h)$-complement of $z$ in $y$. In the case of uniqueness, we denote by
$z^{\perp_{y}}$ the unique $(\sqcup,h)$-complement of $z$ in $y$, and by
$z^{\perp}$ the unique $(\sqcup,h)$-complement of $z$ in
$\operatorname{{\mathcal{S}}}(X,\sqcup)$.

Note that ${\mathcal{S}}_{h}$ (resp.
$\operatorname{{\mathcal{S}}}_{h}(X,\sqcup)$) contains the unique minimum
element and those elements which belong to some
$\sigma\in{\mathcal{D}}({\mathcal{S}})$ (reps.
$\sigma\in{\mathcal{D}}(X,\sqcup)$).

In the case $\operatorname{{\mathcal{S}}}(X,\sqcup)$ is uniquely
$(\sqcup,h)$-complemented, the complements in the purely poset-theoretic sense
might not be unique. For example, for the poset of non-degenerate subspaces of
a finite-dimensional vector space equipped with a non-degenerate sesquilinear
form and $\sqcup$ the orthogonal sum, complements and even $h$-complements of
non-degenerate subspaces are not unique. However, the poset is uniquely
downward $(\sqcup,h)$-complemented, i.e. $(\sqcup,h)$-complements are unique.
See 6.3 for more details.

The following lemma is an immediate consequence of the definitions.

######  Lemma 3.1.

The map
$\Phi:\operatorname{\mathcal{PD}}(X,\sqcup)\to\operatorname{{\mathcal{S}}}(X,\sqcup)$
sending $\sigma$ to $\Phi(\sigma)=\bigvee_{x\in\sigma}x$ is an order-
preserving map with $\Phi^{-1}(0)=\\{\emptyset\\}$ and
$\Phi^{-1}(1)={\mathcal{D}}(X,\sqcup)$. Its image is the poset
$\operatorname{{\mathcal{S}}}_{h}(X,\sqcup)$. Thus $\Phi$ is surjective if and
only if $\operatorname{{\mathcal{S}}}(X,\sqcup)$ is $(\sqcup,h)$-complemented.

Similarly, for a bounded poset ${\mathcal{S}}$ of finite height and
$x\in{\mathcal{S}}$, we have that $\\{x\\}$ is a partial decomposition of
${\mathcal{S}}$ if and only if $x\neq 0$ and it is $h$-complemented in
${\mathcal{S}}$.

We have the following necessary condition for the lattice property on
$\operatorname{\mathcal{PD}}(X,\sqcup)$.

######  Lemma 3.2.

If $\operatorname{\mathcal{PD}}(X,\sqcup)$ is a lattice, then
$\operatorname{{\mathcal{S}}}_{h}(X,\sqcup)$ is also a lattice.

###### Proof.

Suppose that $x,y\in\operatorname{{\mathcal{S}}}_{h}(X,\sqcup)$ have a lower
bound $0\neq z\in\operatorname{{\mathcal{S}}}_{h}(X,\sqcup)$. Then
$\\{z\\}\leq\\{x\\},\\{y\\}$ in $\operatorname{\mathcal{PD}}(X,\sqcup)$. Since
$\operatorname{\mathcal{PD}}(X,\sqcup)$ is a lattice, there exists
$\sigma\in\operatorname{\mathcal{PD}}(X,\sqcup)$ such that
$\sigma=\\{x\\}\wedge\\{y\\}$, and therefore $\\{z\\}\leq\sigma$. But this
implies that $\Phi(\sigma)\leq x,y$, so we must have $\sigma=\\{w\\}$. Hence
$z\leq w\in\operatorname{{\mathcal{S}}}_{h}(X,\sqcup)$. ∎

The converse of this lemma does not hold, as the lattice of 3.0 shows.

We will work with the following properties. These are motivated by the
question of whether a decomposition of a subobject of $X$ can be extended to a
decomposition of $X$. Essentially, they will give us natural ways of
constructing new partial decompositions and will help us to establish stronger
relations among the different posets and complexes, in particular when
combined with downward $(\sqcup,h)$-complementation. In well-known scenarios
(cf. 6), these properties are usually easy to verify, or else they are
consequences of known theorems.

(EP) |  12cmIf $y\in\sigma\in\operatorname{\mathcal{PD}}(X,\sqcup)$ and $\tau\in{\mathcal{D}}(\operatorname{{\mathcal{S}}}(X,\sqcup)_{\leq y})$ is $\sqcup$-compatible in $\operatorname{{\mathcal{S}}}(X,\sqcup)$, then $\left(\sigma\setminus\\{y\\}\right)\cup\tau\in\operatorname{\mathcal{PD}}(X,\sqcup)$.
---|---

(CP) |  12cmIf $\\{x,y\\},\\{x^{\prime},y^{\prime}\\}\in{\mathcal{D}}(X,\sqcup)$ are such that $x\leq x^{\prime}$ and $y^{\prime}\leq y$, then $x^{\prime}\wedge y$ exists and $\\{x^{\prime}\wedge y,x,y^{\prime}\\}\setminus\\{0\\}\in{\mathcal{D}}(X,\sqcup)$.
---|---

######  Remark 3.0.

Note that in (EP), if $\sigma\in{\mathcal{D}}(X,\sqcup)$ is a decomposition,
then $\left(\sigma\setminus\\{y\\}\right)\cup\tau\in{\mathcal{D}}(X,\sqcup)$
by height.

Moreover, in (CP), $\\{x^{\prime}\wedge
y,y^{\prime}\\}\in{\mathcal{D}}(\operatorname{{\mathcal{S}}}(X,\sqcup)_{\leq
y})$ since $(x^{\prime}\wedge y)\vee y^{\prime}$ exists, it is below $y$, and
the height condition $h(x^{\prime}\wedge y)+h(y^{\prime})=h(1)-h(x)=h(y)$
guarantees that $(x^{\prime}\wedge y)\vee y^{\prime}=y$.

######  Corollary 3.0.

If $\operatorname{{\mathcal{S}}}(X,\sqcup)$ is uniquely downward
$(\sqcup,h)$-complemented and (EP) holds, then the following hold:

  1. (1)

the map $x\mapsto x^{\perp}$ is an anti-isomorphism of
$\operatorname{{\mathcal{S}}}(X,\sqcup)$;

  2. (2)

if $x\leq y\leq z$, then
$\\{x,x^{\perp_{y}},y^{\perp_{z}},z^{\perp}\\}\setminus\\{0\\}\in{\mathcal{D}}(X,\sqcup)$
and $x^{\perp_{y}}\vee y^{\perp_{z}}=x^{\perp_{z}}$;

  3. (3)

(CP) is satisfied.

###### Proof.

We prove (2). The assertion is trivial for $z=0$, so we assume that $z\neq 0$
and let $x\leq y\leq z$ and $\sigma=\\{z,z^{\perp}\\}\setminus\\{0\\}$.
Clearly $\sigma\in{\mathcal{D}}(X,\sqcup)$ and
$\\{y,y^{\perp_{z}}\\}\setminus\\{0\\}\in{\mathcal{D}}(\operatorname{{\mathcal{S}}}(X,\sqcup)_{\leq
z})$ is $\sqcup$-compatible in $\operatorname{{\mathcal{S}}}(X,\sqcup)$ by
definition. Hence, by (EP),
$\\{z^{\perp},y,y^{\perp_{z}}\\}\setminus\\{0\\}\in{\mathcal{D}}(X,\sqcup)$.
Similarly,
$\\{x,x^{\perp_{y}}\\}\setminus\\{0\\}\in{\mathcal{D}}(\operatorname{{\mathcal{S}}}(X,\sqcup)_{\leq
y})$ and it is $\sqcup$-compatible in
$\operatorname{{\mathcal{S}}}(X,\sqcup)$, so
$\\{z^{\perp},x,x^{\perp_{y}},y^{\perp_{z}}\\}\setminus\\{0\\}\in{\mathcal{D}}(X,\sqcup)$.
Then $z=x\vee x^{\perp_{y}}\vee y^{\perp_{z}}$, and by uniqueness we conclude
that $x^{\perp_{z}}=x^{\perp_{y}}\vee y^{\perp_{z}}$. This establishes (2).
Moreover, by putting $z=1$ the maximal element, we see that
$x^{\perp}=x^{\perp_{y}}\vee y^{\perp}$, i.e., $x^{\perp}\geq y^{\perp}$.
Thus, the map $x\mapsto x^{\perp}$ is an anti-isomorphism, which proves (1).

For (3), we can take decompositions $\\{x,x^{\perp}\\}$ and
$\\{x^{\prime},x^{\prime\perp}\\}$ where $x\leq x^{\prime}$ (by uniqueness and
(1)). We show that $x^{\prime}\wedge x^{\perp}=x^{\perp_{x^{\prime}}}$. Let
$z\leq x^{\perp},x^{\prime}$. By (1), $z^{\perp}\geq x,x^{\prime\perp}$. Since
$\\{x,x^{\perp_{x^{\prime}}},x^{\prime\perp}\\}\setminus\\{0\\}\in{\mathcal{D}}(X,\sqcup)$
by (2), $x\vee x^{\prime\perp}$ exists and so $z^{\perp}\geq x\vee
x^{\prime\perp}$. Taking complements again by (1) we see that $z\leq(x\vee
x^{\prime\perp})^{\perp}$, and the latter is equal to $x^{\perp_{x^{\prime}}}$
by uniqueness, that is $z\leq x^{\perp_{x^{\prime}}}$. Therefore
$x^{\perp}\wedge x^{\prime}$ exists and it is $x^{\perp_{x^{\prime}}}$. Thus
(CP) holds. ∎

##  4\. Simplicial complexes and posets derived from decompositions

In this section, we define additional posets and complexes related to
decompositions, most of them inspired by analogous constructions in specific
cases. We continue with our usual assumptions, so $X$ is an object of an ISM-
category ${\mathcal{C}}$ with product $\sqcup$ and such that
$\operatorname{{\mathcal{S}}}(X,\sqcup)$ has finite height. At the end of this
section, the reader can find a table with the names of the combinatorial
structures defined here, except for the ordered versions. See 2.

###  4.1. Frames and partial bases

######  Definition 4.0 (Frame).

A partial frame of $X$ is $\sqcup$-partial decomposition
$\sigma\in\operatorname{\mathcal{PD}}(X,\sqcup)$ where all elements of
$\sigma$ are atoms of $\operatorname{{\mathcal{S}}}(X,\sqcup)$. We write
$\operatorname{\mathcal{PF}}(X,\sqcup)$ for the subposet of
$\operatorname{\mathcal{PD}}(X,\sqcup)$ consisting of all partial frames of
$X$.

A partial frame $\sigma$ is called a full frame of $X$ if
$\sigma\in{\mathcal{D}}(X,\sqcup)$. We write ${\mathcal{F}}(X,\sqcup)$ for the
subposet of $\operatorname{\mathcal{PF}}(X,\sqcup)$ consisting of partial
frames contained in full frames.

The following property follows immediately from the definition.

######  Lemma 4.1.

The posets
${\mathcal{F}}(X,\sqcup)\subseteq\operatorname{\mathcal{PF}}(X,\sqcup)$ are
lower intervals of $\operatorname{\mathcal{PD}}(X,\sqcup)$. Moreover, the
refinement ordering $\leq$ in ${\mathcal{F}}(X,\sqcup)$ and
$\operatorname{\mathcal{PF}}(X,\sqcup)$ coincides with the inclusion ordering
$\subseteq$, and hence these posets are simplicial complexes.

Now we use frames to recover the idea of bases. We propose the following
definition for the partial basis complexes.

######  Definition 4.0 (Partial bases).

Let $\mathcal{A}$ be the set of vertices of
$\operatorname{\mathcal{PF}}(X,\sqcup)$ and $(P_{x})_{x\in\mathcal{A}}$ be a
collection of non-empty sets. We call
$B=\\{(s_{1},x_{1}),\ldots,(s_{k},x_{k})\\}$, $s_{i}\in P_{x_{i}}$, a partial
basis of $X$ if
$p(B):=\\{x_{1},\ldots,x_{k}\\}\in\operatorname{\mathcal{PF}}(X,\sqcup)$ is a
partial frame. We say that $B$ extends to a full basis if
$p(B)\in{\mathcal{F}}(X,\sqcup)$. Denote by $\mathcal{PB}(X,\sqcup,P)$ the
complex of partial bases with respect to the collection $P$, and by
$\mathcal{B}(X,\sqcup,P)$ the complex of partial bases that extend to a full
basis.

Intuitively, we think of $P_{x}$ as a set of suitable bases of the atom $x$.
For example, if $x$ represents a $1$-dimensional vector space in the subspace
poset, then $P_{x}$ can represent the set of non-zero vectors in $x$. In the
case $x$ represents a $1$-dimensional non-degenerate subspace in the non-
degenerate subspace poset of a unitary space, we can take $P_{x}$ to be the
set of normal vectors $v\in x$.

######  Lemma 4.2.

$\mathcal{PB}(X,\sqcup,P)$ is a simplicial complex with vertex set
$\bigcup_{x\in\mathcal{A}}P_{x}\times\\{a\\}$. Also $\mathcal{B}(X,\sqcup,P)$
is a simplicial complex with vertex set
$\bigcup_{x\in\mathcal{A}_{0}}P_{x}\times\\{a\\}$, where $\mathcal{A}_{0}$ is
the set of vertices of ${\mathcal{F}}(X,\sqcup)$.

If each $P_{x}$ is finite then $\mathcal{PB}(X,\sqcup,P)$ is an inflation of
$\operatorname{\mathcal{PF}}(X,\sqcup)$ in the sense of [BWW, Section 6]. We
will see that results analogous to those from [BWW] hold in the general
situation.

To simplify the notation, we will usually drop the $P$ from the notation and
leave the collection implicit. When the stated results strongly depend on the
collection $P$, we will use the notation again.

###  4.2. Ordered versions

We define the ordered versions of the posets attached to
$\operatorname{{\mathcal{S}}}(X,\sqcup)$.

######  Definition 4.0.

Let ${\mathcal{T}}$ be one of the symbols
${\mathcal{D}},\operatorname{\mathcal{PD}},{\mathcal{F}},\operatorname{\mathcal{PF}},\mathcal{B},\mathcal{PB}$.
The ordered poset $\mathcal{O}{\mathcal{T}}(X,\sqcup)$ consists of distinct
element tuples $(x_{1},\ldots,x_{r})$ such that
$\\{x_{1},\ldots,x_{r}\\}\in{\mathcal{T}}(X,\sqcup)$. The order in
$\mathcal{O}{\mathcal{T}}(X,\sqcup)$ is given by order-preserving refinement,
that is:

$(x_{1},\ldots,x_{r})\leq(y_{1},\ldots,y_{s})\quad\Leftrightarrow\quad\forall
1\leq i\leq j\leq r,\exists k\leq l\mathrel{{\>:\>}}x_{i}\leq y_{k}\text{ and
}x_{j}\leq y_{l}.$

We denote by
$F_{\mathcal{T}}:\mathcal{O}{\mathcal{T}}(X,\sqcup)\to{\mathcal{T}}(X,\sqcup)$
the forgetful poset map.

######  Remark 4.0.

For a simplicial complex $K$, let $\mathcal{O}K$ be the complex of injective
words on $K$. That is, $\mathcal{O}K$ is a set whose elements are tuples
$z=(v_{1},\ldots,v_{r})$ consisting of distinct vertices of $K$ and such that
$F_{K}(z)=\\{v_{1},\ldots,v_{r}\\}$ is a simplex of $K$. Similar to the
previous definition, $\mathcal{O}K$ is a poset ordered by order-preserving
inclusion, and we have a forgetful poset map $F_{K}:\mathcal{O}K\to K$.
However, $\mathcal{O}K$ is not a simplicial complex in general.

We see then that the poset $\mathcal{O}{\mathcal{T}}(X,\sqcup)$, for
${\mathcal{T}}\in\\{{\mathcal{F}},\operatorname{\mathcal{PF}},\mathcal{B},\mathcal{PB}\\}$,
is exactly the complex of injective words over ${\mathcal{T}}(X,\sqcup)$. As a
consequence, we will be able to invoke results from [JW].

Since we aim to relate later the homotopy type of
$\mathcal{O}{\mathcal{T}}(X,\sqcup)$ with that of ${\mathcal{T}}(X,\sqcup)$,
in the following lemmas we take a closer look at the fibers by the canonical
forgetful maps and the intervals in the ordered versions.

######  Lemma 4.3.

The following hold:

  1. (1)

$F_{\mathcal{D}}^{-1}({\mathcal{D}}(X,\sqcup)_{\geq\sigma})$ is the ordered
partition lattice on the set $\sigma$, which is the face poset of a
permutohedron of order $|\sigma|$. Thus
$F_{\mathcal{D}}^{-1}({\mathcal{D}}(X,\sqcup)^{\circ}_{\geq\sigma})$ is a
sphere of dimension $|\sigma|-2$ (and also Cohen-Macaulay).

  2. (2)

If
$\sigma\in\operatorname{\mathcal{PD}}(X,\sqcup)\setminus{\mathcal{D}}(X,\sqcup)$
then
$F_{\operatorname{\mathcal{PD}}}^{-1}(\operatorname{\mathcal{PD}}(X,\sqcup)^{*}_{\geq\sigma})$
is contractible.

  3. (3)

For a simplicial complex $K$ and $\sigma\in K$, $F_{K}^{-1}(K_{\leq\sigma})$
is the ordered version of the Boolean lattice, which is the poset of injective
words. In particular, $F_{K}^{-1}(K_{\leq\sigma})$ is shellable of dimension
$|\sigma|-1$ and the number of spheres is the number of derangements on
$|\sigma|$ points:

$\sum_{i=0}^{|\sigma|}(-1)^{i}\frac{|\sigma|!}{i!}.$

###### Proof.

(1) follows from Proposition 1.4 of [BS96], and (3) from [BW].

For (2), we construct a homotopy between the identity and a constant map.
Since $\sigma$ is a $\sqcup$-partial decomposition (not full), there exists
$x=\bigvee_{y\in\sigma}y$ and this is different from $1$. Let $\tau\in
F^{-1}_{\operatorname{\mathcal{PD}}}(\operatorname{\mathcal{PD}}(X,\sqcup)^{*}_{\geq\sigma})$,
and write $\tau=(z_{1},\ldots,z_{s})$. Define

$z_{i}^{\prime}=\bigvee_{y\in\sigma\mathrel{{\>:\>}}y\leq z_{i}}y.$

and let $r(\tau)$ be the tuple $(z_{1}^{\prime},\ldots,z_{s}^{\prime})$ after
removing the $0$ elements. We see that
$F_{\operatorname{\mathcal{PD}}}(r(\tau))$ is a $\sqcup$-partial decomposition
obtained by joining elements of $\sigma$, so
$F_{\operatorname{\mathcal{PD}}}(r(\tau))\geq\sigma$. Thus we have well-
defined order-preserving maps

$\tau\geq r(\tau)\leq(x).$

This implies that
$F^{-1}_{\operatorname{\mathcal{PD}}}(\operatorname{\mathcal{PD}}(X,\sqcup)^{*}_{\geq\sigma})$
is contractible. ∎

We describe now some intervals in the ordered versions. Here $|z|$ denotes the
size of an ordered tuple $z$, that is, the number of (distinct) elements of
the tuple.

######  Lemma 4.4.

The following hold:

  1. (1)

For $z\in\mathcal{O}{\mathcal{D}}(X,\sqcup)$,
$\mathcal{O}{\mathcal{D}}(X,\sqcup)_{\geq z}$ is the Boolean lattice on a set
of size $|z|-1$. In particular, if ${\mathcal{D}}(X,\sqcup)$ is finite then

$\tilde{\chi}\left(\mathcal{O}{\mathcal{D}}(X,\sqcup)^{\circ}\right)=\sum_{k\geq
1}(-1)^{k}\cdot
k!\cdot\\#\\{\sigma\in{\mathcal{D}}(X,\sqcup)\mathrel{{\>:\>}}|\sigma|=k\\}.$

  2. (2)

Suppose that (LIP) holds. Then for
$z\in\mathcal{O}\operatorname{\mathcal{PD}}(X,\sqcup)$ we have

$\mathcal{O}\operatorname{\mathcal{PD}}(X,\sqcup)_{\leq z}\cong\prod_{y\in
z}\mathcal{O}\operatorname{\mathcal{PD}}(Y,\sqcup),$

and if $z\in\mathcal{O}{\mathcal{D}}(X,\sqcup)$ then also

$\mathcal{O}{\mathcal{D}}(X,\sqcup)_{\leq z}\cong\prod_{y\in
z}\mathcal{O}{\mathcal{D}}(Y,\sqcup).$

Here $(Y,i_{y})\in y$ denote suitable representatives for $y\in z$.

  3. (3)

Let $K$ be a simplicial complex and $z\in\mathcal{O}K$. Then
$\mathcal{O}K_{>z}\cong\mathcal{O}\operatorname{Lk}_{K}(z)$ and
$\mathcal{O}K_{\leq z}$ is the Boolean lattice on $F_{K}(z)$.

###### Proof.

These are straightforward. ∎

Finally, we establish a connection between the ordered version of the poset of
decompositions and the subposet of $(\sqcup,h)$-complemented subobjects.

######  Definition 4.0.

Let
$G:\mathcal{O}\operatorname{\mathcal{PD}}(X,\sqcup)^{*}\to{(\Delta\operatorname{{\mathcal{S}}}_{h}(X,\sqcup)^{*})}^{\mathrm{op}}\cup\\{\emptyset\\}$
be the function

$G(x_{1},\ldots,x_{r})=\\{x_{1}<x_{1}\vee x_{2}<\ldots<x_{1}\vee\ldots\vee
x_{r-1}\\}.$

It is clear that if $\\{x_{1},\ldots,x_{r}\\}$ is a $\sqcup$-partial
decomposition, then any join of its elements is $(\sqcup,h)$-complemented, and
hence belongs to $\operatorname{{\mathcal{S}}}_{h}(X,\sqcup)$. Thus $G$ is a
well-defined function. However, $G$ is not order-preserving for partial
decompositions: for example, if $x\leq y$ and $\\{x,z\\},\\{y,z\\}$ are
$\sqcup$-partial decompositions, then $(x,z)\leq(y,z)$ but
$G(x,z)=\\{x\\}\not\subseteq\\{y\\}=G(y,z)$. Nevertheless, this is a poset map
when restricted to decompositions:

######  Lemma 4.5.

$G:\mathcal{O}{\mathcal{D}}(X,\sqcup)^{\circ}\to{\left(\Delta\big{(}\operatorname{{\mathcal{S}}}_{h}(X,\sqcup)^{*}\big{)}\right)}^{\mathrm{op}}$
is an order-preserving map.

###### Proof.

This follows since elements of $\mathcal{O}{\mathcal{D}}(X,\sqcup)_{>\tau}$
are obtained by joining adjacent blocks of an ordered decomposition $\tau$.
That is, if $\sigma>\tau$ and $\tau=(x_{1},\ldots,x_{r})$, we can write

$\sigma=\bigg{(}{\bigvee_{i=1}^{t_{1}}}x_{i},\bigvee_{i=t_{1}+1}^{t_{2}}x_{i},\ldots,\bigvee_{i=t_{s-1}+1}^{r}x_{i}\bigg{)}$

for suitable $1\leq t_{1}<t_{2}<\ldots<t_{s}=r$. Thus $G(\tau)$ contains
$G(\sigma)$. ∎

We will see later that this map turns out to be an isomorphism under unique
complementation and suitable extension properties. See 5.5 and 5.6.

###  4.3. The augmented Bergman complex

We introduce now a (simplified) version of the augmented Bergman complex in
our context. This complex was first defined for matroids and it gained
substantial attention due to the works [BHMPW] where its Stanley-Reisner ring
is a key object in the proofs of the Top-Heavy conjecture (raised by Dowling
and Wilson), and the non-negativity of the coefficients of Kazhdan-Lusztig
polynomials for all matroids. Its topology in the matroid case was carefully
analyzed in [BKRRSSSZ].

######  Definition 4.0 (Bergman).

Take disjoint copies of the vertices of ${\mathcal{F}}(X,\sqcup)$ and the
atoms of $\operatorname{{\mathcal{S}}}(X,\sqcup)$.

In $\Delta(X,\sqcup)$ we collect all sets $\sigma\cup\\{x_{1}\lneq
x_{2}\lneq\cdots\lneq x_{r}\\}$ where $\sigma\in{\mathcal{F}}(X,\sqcup)$ and
$\Phi(\sigma)\leq
x_{1},\ldots,x_{r}\in\operatorname{{\mathcal{S}}}(X,\sqcup)^{\circ}$. We call
$\Delta(X,\sqcup)$ the augmented Bergman complex of $X$.

It is a direct consequence of the definition that the augmented Bergman
complex $\Delta(X,\sqcup)$ is indeed a simplicial complex. Note that if
$\sigma\in{\mathcal{F}}(X,\sqcup)$ is a full frame, then
$\sigma\in\Delta(X,\sqcup)$ is a maximal simplex. Also note that
$0\in\operatorname{{\mathcal{S}}}(X,\sqcup)$ is a vertex of
$\Delta(X,\sqcup)$, and a simplex containing $0$ must be a flag of
$\operatorname{{\mathcal{S}}}(X,\sqcup)^{\circ}$.

###  4.4. The Charney poset

A different version of the ordered decompositions was introduced by R. Charney
in [C]. There she works with the poset of summands of a finitely generated
free module over a Dedekind domain, and defines a poset whose elements are
ordered direct sum decompositions of size $2$ with a zigzag ordering. Another
version of Charney’s poset appears in the context of split buildings for
spaces with a certain form, and we interpret decompositions of size $2$ there
as direct sum decomposition pairs $(V_{1},V_{2})$ such that $V_{1}$ and the
orthogonal complement of $V_{2}$ are totally isotropic subspaces (see [LR]).

We adapt the definition of this poset to our context.

######  Definition 4.0.

The Charney poset associated with $X$ is the poset $\mathcal{G}(X,\sqcup)$
whose underlying set consists of ordered decompositions of size $2$:

$\mathcal{G}(X,\sqcup)=\\{z\in\operatorname{\mathcal{OD}}(X,\sqcup)\mathrel{{\>:\>}}|z|=2\\},$

and the ordering is given by

$(x,y)\leq(x^{\prime},y^{\prime})\quad\text{ if }\quad x\leq x^{\prime}\text{
and }y\geq y^{\prime}.$

The following proposition relates the ordered decompositions with the Charney
complex.

######  Proposition 4.0.

Let $\beta$ be the function

$\beta:\left\\{\begin{array}[]{ccc}{\operatorname{\mathcal{OD}}(X,\sqcup)^{\circ}}^{\mathrm{op}}&\to&\Delta\mathcal{G}(X,\sqcup)\\\
\tau=(z_{0},z_{1},\ldots,z_{r})&\mapsto&\\{\big{(}\bigvee_{i=0}^{j}z_{i},\bigvee_{k=j+1}^{r}z_{k}\big{)}\mathrel{{\>:\>}}0\leq
j<r\\}\end{array}\right.$

Then $\beta$ is a rank-preserving poset embedding whose image is downward
closed.

Moreover, if (EP) and (CP) hold then $\beta$ is surjective and therefore an
isomorphism.

###### Proof.

It is clear that $\beta$ is well-defined, order-preserving and rank-
preserving, and also injective: if
$\beta(z_{0},\ldots,z_{r})=\\{(x_{0},y_{0})<\ldots<(x_{r-1},y_{r-1})\\}$ then

(1)
$z_{j}=(\bigvee_{i=0}^{j}z_{i}\big{)}\wedge\big{(}\bigvee_{k=j}^{r}z_{k}\big{)}=x_{j}\wedge
y_{j-1}.$

Note that this meet always exists by definition of a decomposition. By
convention, set $x_{-1}=0,y_{-1}=1$ and $x_{r}=1,y_{r}=0$.

We show now that $\beta$ is an embedding. Suppose that
$\beta(\tau)\supseteq\beta(\sigma)$ with

$\beta(\tau)=\beta(z_{0},\ldots,z_{r})=\\{(x_{0},y_{0})<\ldots<(x_{r-1},y_{r-1})\\},$
$\beta(\sigma)=\beta(w_{0},\ldots,w_{s})=\\{(u_{0},v_{0})<\ldots<(u_{s-1},v_{s-1})\\}.$

For $0\leq i\leq s-1$ we take the unique index $0\leq j_{i}\leq r-1$ such that
$(u_{i},v_{i})=(x_{j_{i}},y_{j_{i}})$. Since $(u_{-1},v_{-1})=(x_{-1},y_{-1})$
and $(u_{s},v_{s})=(x_{r},y_{r})$, we set $j_{-1}=-1$ and $j_{s}=r$. It is
clear that $(j_{i})_{i=-1}^{s}$ is a strictly monotone increasing sequence.
Then, for all $0\leq i\leq s$ we have

$w_{i}=u_{i}\wedge v_{i-1}=x_{j_{i}}\wedge
y_{j_{i-1}}=(\bigvee_{k=0}^{j_{i}}z_{k}\big{)}\wedge\big{(}\bigvee_{k=j_{i-1}+1}^{r}z_{k}\big{)}=\bigvee_{k=j_{i-1}+1}^{j_{i}}z_{k}.$

This implies $\tau\leq\sigma$, so $\beta$ is a poset embedding. It is also
clear that if $\omega\subseteq\beta(z)$ then $\omega\in\mathrm{Im}(\beta)$.

Finally, we show that $\beta$ is surjective under (EP) and (CP). We argue by
induction on the size $r$ of a chain $s\in\Delta\mathcal{G}(X,\sqcup)$ and in
view of 1, we show that if

$s=\\{(x_{0},y_{0})<\ldots<(x_{r},y_{r})\\}$

then $y_{i-1}\wedge x_{i}$ exists for all $1\leq i\leq r$ and, if we set
$y_{-1}=1$ and $x_{r+1}=1$, then $\\{x_{i}\wedge y_{i-1}\mathrel{{\>:\>}}0\leq
i\leq r+1\\}\in{\mathcal{D}}(X,\sqcup)$.

The case $r=0$ is trivial. Suppose then that $r>0$ and that
$s=\\{(x_{0},y_{0})<\ldots<(x_{r},y_{r})\\}$. By induction, we have a
$\sqcup$-decomposition

$\sigma=\\{x_{0},x_{0}\wedge y_{1},\ldots,x_{r-1}\wedge
y_{r-2},y_{r-1}\\}\in{\mathcal{D}}(X,\sqcup).$

By (CP) applied to $(x_{r-1},y_{r-1})<(x_{r},y_{r})$, we have that
$\tau:=\\{x_{r}\wedge y_{r-1},x_{r-1},y_{r}\\}\in{\mathcal{D}}(X,\sqcup)$.
Hence $\\{x_{r}\wedge
y_{r-1},y_{r}\\}\in{\mathcal{D}}(\operatorname{{\mathcal{S}}}(X,\sqcup)_{\leq
y_{r-1}})$ by 3.0 and it is $\sqcup$-compatible in
$\operatorname{{\mathcal{S}}}(X,\sqcup)$, so by (EP) we have

$\sigma^{\prime}:=\left(\sigma\setminus\\{y_{r-1}\\}\right)\cup\tau\in{\mathcal{D}}(X,\sqcup).$

But note that

$\sigma^{\prime}=\\{x_{0},x_{1}\wedge y_{0},\ldots,x_{r-1}\wedge
y_{r-2},x_{r}\wedge y_{r-1},y_{r}\\}=\\{x_{i}\wedge
y_{i-1}\mathrel{{\>:\>}}0\leq i\leq r+1\\}.$

This concludes the induction, and thus $\beta$ is surjective since

$\beta(x_{0},x_{1}\wedge y_{0},\ldots,x_{r}\wedge
y_{r-1},y_{r})=\\{(x_{0},y_{0})<\ldots<(x_{r},y_{r})\\}.$

∎

We summarize the non-ordered constructions in 2.

Structure | Name
---|---
$\operatorname{{\mathcal{S}}}(X)$ |  poset of subobjects
$\operatorname{{\mathcal{S}}}(X,\sqcup)$ |  poset of $\sqcup$-complemented subobjects
${\mathcal{D}}(X,\sqcup)$ |  poset of $\sqcup$-decompositions
$\operatorname{\mathcal{PD}}(X,\sqcup)$ |  poset of $\sqcup$-partial decomposition
$\operatorname{\mathcal{PF}}(X,\sqcup)$ |  complex of partial frames
${\mathcal{F}}(X,\sqcup)$ |  frame complex
$\mathcal{PB}(X,\sqcup,P)$ |  complex of partial bases
$\mathcal{B}(X,\sqcup,P)$ |  (pure) complex of partial bases
$\Delta(X,\sqcup)$ |  augmented Bergman complex
$\mathcal{G}(X,\sqcup)$ |  Charney poset
Table 2. Structures associated to an object $X$ of an ISM-category
$({\mathcal{C}},\sqcup)$ such that $\operatorname{{\mathcal{S}}}(X,\sqcup)$
has finite height.

######  Remark 4.0.

Most of the structures we defined for an object of an ISM-category, along with
their corresponding properties, can be translated to the context of posets
according to 2.0. In the case of lattices, they coincide with the definitions
in the categorical context as discussed in 2.0. However, we will not present
those analogous results here; the interested reader can verify the validity of
the results presented in this article within the context of posets.

##  5\. Homotopy equivalences and isomorphisms

###  5.1. General homotopy formulas

In this section, we first recall some known facts about homotopy types of
posets and then apply them to the posets we defined in the previous sections.
When writing down explicit formulas for the homotopy type we identify the
poset with its order complex. For example
${\mathcal{T}}\simeq{\mathcal{S}}\vee{\mathcal{S}}$ means that the geometric
realization of the order complex of ${\mathcal{T}}$ is homotopy equivalent to
a wedge sum of two copies of the geometric realization of the order complex of
${\mathcal{S}}$. If we need to emphasize the topological spaces behind our
results and constructions we write $|{\mathcal{S}}|$ for the geometric
realization of the order complex of a poset ${\mathcal{S}}$.

Assume that ${\mathcal{S}}$ is a poset of finite height $n$. We say that
${\mathcal{S}}$ is spherical if it is $(n-1)$-connected or, equivalently,
homotopy equivalent to a wedge of $n$-spheres. The poset ${\mathcal{S}}$ is
Cohen-Macaulay if ${\mathcal{S}}$ is spherical and for all
$x,y\in{\mathcal{S}}$, the intervals ${\mathcal{S}}_{>x}$,
${\mathcal{S}}_{<y}$ and ${\mathcal{S}}_{>x}\cap{\mathcal{S}}_{<y}$ (if $x<y$)
are spherical of height $n-h(x)-1$, $h(y)-1$ and $h(y)-h(x)-2$ respectively.
This is the classical notion of homotopically Cohen-Macaulayness of a poset or
simplicial complex. This implies Cohen-Macaulayness over any field, which is
defined by the corresponding high homological connectedness.

Recall that if $f,g:{\mathcal{T}}\to{\mathcal{S}}$ are order-preserving maps
such that $f(x)\leq g(x)$ for all $x\in{\mathcal{T}}$ then $f$ and $g$ are
homotopy equivalent. We will repeatedly use this fact throughout this paper to
obtain homotopy equivalences and establish the contractibility of some posets.

The first tool we will need is a consequence of Propositions 3.1, 3.5, and 3.7
from [WZZ]. Note that in [WZZ] Proposition 3.1 was formulated only for
diagrams over finite posets but the proof only relies on paracompactness of
the spaces involved (see e.g., [Se]). As a consequence, the results from [WZZ]
can be applied to finite height posets.

######  Proposition 5.0.

Let ${\mathcal{S}},{\mathcal{T}}$ be posets of finite height and
$f:{\mathcal{T}}\to{\mathcal{S}}$ a poset map such that for all
$x\in{\mathcal{S}}$, the inclusion
$f^{-1}\big{(}\,{\mathcal{S}}_{<x}\,\big{)}\hookrightarrow
f^{-1}\big{(}\,{\mathcal{S}}_{\leq x}\,\big{)}$ is homotopy equivalent to the
constant map with image $c_{x}\in f^{-1}({\mathcal{S}}_{\leq x})$. Then there
is a homotopy equivalence

${\mathcal{T}}\simeq{\mathcal{S}}\vee\bigvee_{x\in{\mathcal{S}}}f^{-1}\big{(}\,{\mathcal{S}}_{\leq
x}\,\big{)}*{\mathcal{S}}_{>x},$

where the wedge identifies $x\in{\mathcal{S}}$ with $c_{x}\in
f^{-1}({\mathcal{S}}_{\leq x})$.

The following well-known fact will be useful for verifying the requirements of
5.0.

######  Lemma 5.1.

Let ${\mathcal{Q}}\subseteq{\mathcal{S}}$ be posets such that ${\mathcal{S}}$
is $k$-connected and ${\mathcal{Q}}$ has dimension at most $k$. Then the
inclusion ${\mathcal{Q}}\hookrightarrow{\mathcal{S}}$ is null-homotopic.

Then from 5.0 and 5.1 we can derive the following version of Quillen’s fiber
theorem which we will use repeatedly.

######  Theorem 5.0 (Quillen’s fiber theorem).

Let $f:{\mathcal{T}}\to{\mathcal{S}}$ be a map between posets of finite
height. Let $m\geq-1$, and suppose that for all $x\in{\mathcal{S}}$,
$f^{-1}({\mathcal{S}}_{\leq x})*{\mathcal{S}}_{>x}$ is $m$-connected. Then $f$
is an $(m+1)$-equivalence.

We will also use the non-Hausdorff mapping cylinder (see [BM]) of a map
$f:{\mathcal{T}}\to{\mathcal{S}}$ between posets. This is the poset
$M_{f}={\mathcal{T}}\coprod{\mathcal{S}}$ such that its ordering $\leq$
restricts to the given ordering in ${\mathcal{T}}$ and ${\mathcal{S}}$, and
for $x\in{\mathcal{T}}$ and $y\in{\mathcal{S}}$ we have $x<y$ if $f(x)\leq y$.
The adjective non-Hausdorff is motivated by the non-Hausdorffness of the
construction when considering finite posets as finite topological spaces. We
refer to [BM] for a more elaborate discussion.

######  Lemma 5.2.

Let $f:{\mathcal{T}}\to{\mathcal{S}}$ be a map of posets of finite height,
where ${\mathcal{S}}$ is bounded. If
$M:=\\{x\in{\mathcal{T}}~{}|~{}f(x)=1_{{\mathcal{S}}}\\}$ is an antichain then
$M_{f}\setminus\\{1_{{\mathcal{S}}}\\}$ is homotopy equivalent to

$\bigvee_{x\in M}\Sigma{\mathcal{T}}_{<x}.$

###### Proof.

Let ${\mathcal{X}}=M_{f}\setminus\\{1_{{\mathcal{S}}}\\}$. Then
${\mathcal{X}}\setminus M$ is contractible since the map
$r:{\mathcal{X}}\setminus M\to{\mathcal{S}}\setminus\\{1_{{\mathcal{S}}}\\}$,
defined by $r(x)=f(x)$ if $x\in{\mathcal{T}}$ or $r(x)=x$ if
$x\in{\mathcal{S}}$, is a homotopy equivalence with inverse given by the
natural inclusion of ${\mathcal{S}}\setminus\\{1_{{\mathcal{S}}}\\}$ in
${\mathcal{X}}$. Therefore,
$|{\mathcal{X}}|\simeq|{\mathcal{X}}|/|{\mathcal{X}}\setminus M|$, and since
$M$ is an antichain, we conclude that $|{\mathcal{X}}|/|{\mathcal{X}}\setminus
M|$ is homotopy equivalent to the wedge of the suspension of the links in
$|{\mathcal{X}}|$ of $x\in M$. The fact that
$\operatorname{Lk}_{|{\mathcal{X}}|}(x)=|{\mathcal{T}}_{<x}|$ then concludes
the proof. ∎

In general, we also have the following formula relating the Euler
characteristic of posets connected by an order-preserving map (see e.g.
Corollary 3.2 of [Wa1]).

######  Lemma 5.3.

Let $f:{\mathcal{T}}\to{\mathcal{S}}$ be a map of finite posets. Then

$\tilde{\chi}({\mathcal{T}})=\tilde{\chi}({\mathcal{S}})-\sum_{x\in{\mathcal{S}}}\tilde{\chi}(f^{-1}({\mathcal{S}}_{\leq
x}))\tilde{\chi}({\mathcal{S}}_{>x}).$

###  5.2. Homotopy formulas and results for posets from 2 \- 4

Our first homotopy equivalence relates the strictly partial decomposition with
the subposet of $(\sqcup,h)$-complemented subobjects. Recall that
$\Phi(\sigma)=\bigvee_{x\in\sigma}x$ for a decomposition
$\sigma\in{\mathcal{D}}(X,\sqcup)$.

######  Proposition 5.0.

The map
$\Phi:\operatorname{\mathcal{PD}}(X,\sqcup)^{*}\setminus{\mathcal{D}}(X,\sqcup)\to\operatorname{{\mathcal{S}}}_{h}(X,\sqcup)^{*}$
is a homotopy equivalence with inverse
$i:\operatorname{{\mathcal{S}}}_{h}(X,\sqcup)^{*}\to\operatorname{\mathcal{PD}}(X,\sqcup)^{*}\setminus{\mathcal{D}}(X,\sqcup)$
given by $i(x)=\\{x\\}$. In particular,
$\operatorname{{\mathcal{S}}}_{h}(X,\sqcup)^{*}$ is a deformation retract of
$\operatorname{\mathcal{PD}}(X,\sqcup)^{*}\setminus{\mathcal{D}}(X,\sqcup)$.

Moreover,
$\mathcal{O}\operatorname{\mathcal{PD}}(X,\sqcup)^{*}\setminus\mathcal{O}{\mathcal{D}}(X,\sqcup)$
is also homotopy equivalent to
$\operatorname{{\mathcal{S}}}_{h}(X,\sqcup)^{*}$.

###### Proof.

This follows from 3.1, the definition of
$\operatorname{{\mathcal{S}}}_{h}(X,\sqcup)$ and the relations $\Phi i(x)=x$
and $i\Phi(\sigma)\geq\sigma$. ∎

Now we apply the wedge-decomposition result to relate the ordered and
unordered versions of the posets
${\mathcal{D}},\operatorname{\mathcal{PD}},{\mathcal{F}},\operatorname{\mathcal{PF}},\mathcal{B},\mathcal{PB}$.
Recall the definition of $\mathcal{O}K$ for a simplicial complex $K$ from 4.0.

######  Theorem 5.0.

If $\operatorname{{\mathcal{S}}}(X,\sqcup)$ has finite height, the following
hold:

  1. (1)

$\mathcal{O}{\mathcal{D}}(X,\sqcup)^{\circ}\simeq{\mathcal{D}}(X,\sqcup)^{\circ}\vee\bigvee_{\sigma\in{\mathcal{D}}(X,\sqcup)^{\circ}}S^{|\sigma|-2}*{\mathcal{D}}(X,\sqcup)^{\circ}_{<\sigma}.$

Hence $\mathcal{O}{\mathcal{D}}(X,\sqcup)$ is Cohen-Macaulay if
${\mathcal{D}}(X,\sqcup)$ is.

  2. (2)

$\mathcal{O}\operatorname{\mathcal{PD}}(X,\sqcup)^{*}\simeq\operatorname{\mathcal{PD}}(X,\sqcup)^{*}\vee\bigvee_{\sigma\in{\mathcal{D}}(X,\sqcup)^{\circ}}S^{|\sigma|-2}*\operatorname{\mathcal{PD}}(X,\sqcup)^{*}_{<\sigma}.$

Hence $\mathcal{O}\operatorname{\mathcal{PD}}(X,\sqcup)$ is Cohen-Macaulay if
$\operatorname{\mathcal{PD}}(X,\sqcup)$ is.

  3. (3)

If $K$ is a finite-dimensional simplicial complex then

$\mathcal{O}K\simeq K\vee\bigvee_{\sigma\in
K}\bigg{(}\bigvee_{D(|\sigma|)}S^{|\sigma|-1}\bigg{)}*\operatorname{Lk}_{K}(\sigma),$

where $D(n)=\sum_{i=0}^{n}(-1)^{i}\frac{n!}{i!}$. Moreover, $K$ is Cohen-
Macaulay (resp. Cohen-Macaulay over a ring $R$) if and only if $\mathcal{O}K$
is Cohen-Macaulay (resp. Cohen-Macaulay over $R$). In particular, this applies
to the complexes ${\mathcal{F}},\operatorname{\mathcal{PF}},\mathcal{B}$ or
$\mathcal{PB}$.

###### Proof.

These conclusions follow by the description of the fibers in 4.3 in
conjunction with 5.1 and 5.0. Note that, when $K$ is a simplicial complex
regarded as a poset, then $K_{>\sigma}$ is isomorphic to
$\operatorname{Lk}_{K}(\sigma)$ for any simplex $\sigma\in K$. The Cohen-
Macaulay implication in items (1) and (2) follows by 4.3(1) and (2)
respectively, jointly with Corollary 9.7 of [Q].

The “only if” implication of the Cohen-Macaulay part of item (3) follows by an
application of Theorem 5.1 of [BWW] by noting that if $\sigma\in K$, then
$F_{K}^{-1}(K_{\leq\sigma})$ is Cohen-Macaulay. That is,
$F_{K}^{-1}(K_{\leq\sigma})$ is the poset of injective words on the alphabet
$\sigma$, and this was proved to be shellable (and hence Cohen-Macaulay) by
[BW].

For the “if” implication, it remains to prove that if $\mathcal{O}K$ is Cohen-
Macaulay then the links $\operatorname{Lk}_{K}(\sigma)$ are spherical. It is
not hard to show that they have the correct dimension. Now fix $\sigma\in K$,
and let $z\in F_{K}^{-1}(\sigma)$. Let $(v_{i})_{i\in I}$ be a total ordering
of the vertices of $\operatorname{Lk}_{K}(\sigma)$. Consider the map
$H:\operatorname{Lk}_{K}(\sigma)\to\mathcal{O}K_{>z}$ given by
$H(\tau)=(z,v_{i_{1}},\ldots,v_{i_{k}})$ if
$\tau=\\{v_{i_{1}}<\ldots<v_{i_{k}}\\}$. Clearly, $H$ is a well-defined order-
preserving poset map. Let
$F^{\prime}:\mathcal{O}K_{>z}\to\operatorname{Lk}_{K}(\sigma)$ be the map that
forgets the ordering and deletes the vertices in $\sigma$, that is
$F^{\prime}(w)=F_{K}(w)\setminus\sigma$. Thus $F^{\prime}H(\tau)=\tau$ is the
identity map, so $F^{\prime}$ induces a surjective map on the homotopy (and
homology) groups. Since $\mathcal{O}K_{>z}$ is spherical of the same dimension
than $\operatorname{Lk}_{K}(\sigma)$, we conclude that
$\operatorname{Lk}_{K}(\sigma)$ is spherical. This concludes the proof. ∎

We study now natural maps between our posets that might be useful to compute
the homotopy type relations between them. We begin with the relation between
the complex of partial bases $\mathcal{B}(X,\sqcup,P)$ and the frame complex.
Since these are inflations in the sense of [BWW], by their Theorem 6.2 adapted
to the infinite situation, we conclude:

######  Proposition 5.0.

For a collection $P=(P_{x})_{x\in\mathcal{A}}$ of non-empty sets $P_{x}$
indexed by the set of vertices $\mathcal{A}$ of
$\operatorname{\mathcal{PF}}(X,\sqcup)$, we have the following homotopy
equivalences arising from the projection map
$p:\mathcal{PB}(X,\sqcup,P)\to\operatorname{\mathcal{PF}}(X,\sqcup)$:

$\mathcal{PB}(X,\sqcup,P)\simeq\operatorname{\mathcal{PF}}(X,\sqcup)\vee\bigvee_{\sigma\in\operatorname{\mathcal{PF}}(X,\sqcup)}\bigg{(}\bigvee_{\prod_{x\in\sigma}|P_{x}|-1}S^{|\sigma|-1}\bigg{)}*\operatorname{\mathcal{PF}}(X,\sqcup)_{>\sigma},$
$\mathcal{B}(X,\sqcup,P)\simeq{\mathcal{F}}(X,\sqcup)\vee\bigvee_{\sigma\in{\mathcal{F}}(X,\sqcup)}\bigg{(}\bigvee_{\prod_{x\in\sigma}|P_{x}|-1}S^{|\sigma|-1}\bigg{)}*{\mathcal{F}}(X,\sqcup)_{>\sigma}.$

Moreover,
$p^{-1}(\operatorname{\mathcal{PF}}(X,\sqcup)_{\leq\sigma})=P_{x_{1}}*\ldots*P_{x_{r}}$
if $\sigma=\\{x_{1},\ldots,x_{r}\\}$.

In particular, if $\operatorname{\mathcal{PF}}(X,\sqcup)$ (resp.
${\mathcal{F}}(X,\sqcup)$) is homotopically Cohen-Macaulay then
$\mathcal{PB}(X,\sqcup,P)$ (resp. $\mathcal{B}(X,\sqcup,P)$) is homotopy
equivalent to a wedge of spheres of dimension
$\dim(\operatorname{\mathcal{PF}}(X,\sqcup))$ (resp.
$\dim({\mathcal{F}}(X,\sqcup))$).

###### Proof.

The homotopy equivalence is a consequence of [BWW, Theorem 6.2] adapted to the
infinite case. Alternatively, this follows from 5.0 since
$p^{-1}(\operatorname{\mathcal{PF}}(X,\sqcup)_{\leq\sigma})=P_{x_{1}}*\ldots*P_{x_{r}}$
for $\sigma=\\{x_{1},\ldots,x_{r}\\}\in\operatorname{\mathcal{PF}}(X,\sqcup)$.
This argument also holds for the restricted map
$p:\mathcal{B}(X,\sqcup,P)\to{\mathcal{F}}(X,\sqcup)$ since
${\mathcal{F}}(X,\sqcup)$ and $\mathcal{B}(X,\sqcup,P)$ are full subcomplexes
of $\operatorname{\mathcal{PF}}(X,\sqcup)$ and $\mathcal{PB}(X,\sqcup,P)$
respectively.

Then by [Q, Corollary 9.7], since the lower fibers
$p^{-1}(\operatorname{\mathcal{PF}}(X,\sqcup)_{\leq\sigma})$ are homotopy CM
of the correct dimension, if $\operatorname{\mathcal{PF}}(X,\sqcup)$ is
homotopy CM then $\mathcal{PB}(X,\sqcup,P)$ is homotopy CM as well. The same
conclusions hold for ${\mathcal{F}}(X,\sqcup)$ and $\mathcal{B}(X,\sqcup,P)$.
∎

For inflation complexes, we can easily establish the reciprocal to Cohen-
Macaulayness.

######  Corollary 5.0.

If $\mathcal{PB}(X,\sqcup,P)$ is homotopy CM, then
$\operatorname{\mathcal{PF}}(X,\sqcup)$ is homotopy CM. The same conclusions
hold for $\mathcal{B}(X,\sqcup,P)$ and ${\mathcal{F}}(X,\sqcup)$.

###### Proof.

Suppose that $\mathcal{PB}(X,\sqcup,P)$ is homotopy CM. By 5.0,
$\operatorname{\mathcal{PF}}(X,\sqcup)$ is spherical. It remains to prove that
the links of its simplices are spherical of the correct dimension. Let
$\sigma\in\operatorname{\mathcal{PF}}(X,\sqcup)$ and consider a partial basis
$B\in\mathcal{PB}(X,\sqcup,P)$ with $p(B)=\sigma$. Then
$\operatorname{Lk}_{\mathcal{PB}(X,\sqcup,P)}(B)$ is isomorphic to the
inflation of the complex
$\operatorname{Lk}_{\operatorname{\mathcal{PF}}(X,\sqcup)}(\sigma)$ over the
collection $P^{B}:=(P_{x})_{x\in\mathcal{A}_{\sigma}}$, where
$\mathcal{A}_{\sigma}$ is the set of vertices of
$\operatorname{Lk}_{\operatorname{\mathcal{PF}}(X,\sqcup)}(\sigma)$. Note that
both links have the correct dimension. By Theorem 6.2 of [BWW], since
$\operatorname{Lk}_{\mathcal{PB}(X,\sqcup,P)}(B)$ is spherical, we conclude
that $\operatorname{Lk}_{\operatorname{\mathcal{PF}}(X,\sqcup)}(\sigma)$ is
spherical.

The proof for $\mathcal{B}(X,\sqcup,P)$ and ${\mathcal{F}}(X,\sqcup)$ is
analogous. ∎

We now derive some results on the augmented Bergman complex.

######  Lemma 5.4.

$\Delta(X,\sqcup)$ is homotopy equivalent to the subposet
$M_{\Phi}\setminus\\{1\\}$ of the non-Hausdorff mapping cylinder $M_{\Phi}$ of
the map
$\Phi:{\mathcal{F}}(X,\sqcup)\to\operatorname{{\mathcal{S}}}(X,\sqcup)$.

###### Proof.

Let ${\mathcal{M}}=M_{\Phi}\setminus\\{1\\}$, and consider the map
$\varphi:\Delta(X,\sqcup)\to{\mathcal{M}}$ given by

$\varphi(\sigma\cup\\{x_{1},\ldots,x_{r}\\})=\begin{cases}\sigma&r=0,\\\
x_{r}&r>0.\end{cases}$

Clearly, $\varphi$ is a well-defined order-preserving map. To prove it is a
homotopy equivalence, we show that the lower homotopy fibers are contractible.
Indeed, for $\sigma\in{\mathcal{F}}(X,\sqcup)$ we have

$\varphi^{-1}\big{(}{\mathcal{M}}_{\leq\sigma}\big{)}=\Delta(X,\sqcup)_{\leq\sigma}\simeq*.$

On the other hand, for
$x\in\operatorname{{\mathcal{S}}}(X,\sqcup)\setminus\\{1\\}$ we get

$\sigma\cup\\{x_{1},\ldots,x_{r}\\}\in\varphi^{-1}\big{(}{\mathcal{M}}_{\leq
x}\big{)}\,\Longleftrightarrow\,\Phi(\sigma)\leq x\text{ and }x_{r}\leq x.$

When $r>0$, the condition $\Phi(\sigma)\leq x$ follows immediately from
$x_{r}\leq x$. In this way, we can construct a contraction homotopy inside
this fiber:

$\sigma\cup\\{x_{1},\ldots,x_{r}\\}\leq\sigma\cup\\{x_{1},\ldots,x_{r},x\\}\geq\\{x\\}.$

Note that all these terms lie in
$\varphi^{-1}\big{(}{\mathcal{M}}_{\leq\sigma}\big{)}$.

Since the lower homotopy fibers are contractible, by Quillen’s fiber Theorem
we conclude that $\varphi$ is a homotopy equivalence. ∎

Since the elements of the poset ${\mathcal{F}}(X,\sqcup)$ whose join is the
unique maximal element of $\operatorname{{\mathcal{S}}}(X,\sqcup)$ is exactly
the set of full frames, they form an antichain. In conjunction with 5.2 this
immediately implies the following result which is well known in the matroid
case (see [BKRRSSSZ]).

######  Proposition 5.0.

Let $M$ be the set of full frames of $\operatorname{{\mathcal{S}}}(X,\sqcup)$.
Then $\Delta(X,\sqcup)$ is homotopy equivalent to

$\bigvee_{\sigma\in M}S^{|\sigma|-1}.$

In particular, $\Delta(X,\sqcup)$ is spherical, and if $M$ is empty (i.e.
there are no full frames) then $\Delta(X,\sqcup)$ is contractible.

We have the following partial result on the connectivity of
$\operatorname{\mathcal{PD}}(X,\sqcup)^{*}$ if have information on the lower
intervals of both $\operatorname{{\mathcal{S}}}_{h}(X,\sqcup)$ and the partial
decompositions.

######  Theorem 5.0.

Let $n$ denote the height of $\operatorname{{\mathcal{S}}}(X,\sqcup)$. Assume
that (LIP) holds and that for all
$y\in\operatorname{{\mathcal{S}}}_{h}(X,\sqcup)$, the poset
$\operatorname{{\mathcal{S}}}_{h}(Y,\sqcup)^{*}$ is spherical of dimension
$h(y)-2$. Then $\operatorname{\mathcal{PD}}(X,\sqcup)^{*}$ is
$(n-3)$-connected. Moreover, the inclusion
$\operatorname{{\mathcal{S}}}_{h}(X,\sqcup)^{*}\hookrightarrow\operatorname{\mathcal{PD}}(X,\sqcup)^{*}$
is an $(n-2)$-equivalence.

###### Proof.

Note that $\operatorname{{\mathcal{S}}}_{h}(X,\sqcup)^{*}$ is spherical of
dimension $n-2$ by hypothesis, and that the inclusion
$\operatorname{{\mathcal{S}}}_{h}(X,\sqcup)^{*}\hookrightarrow\operatorname{\mathcal{PD}}(X,\sqcup)^{*}\setminus{\mathcal{D}}(X,\sqcup)$
is a homotopy equivalence. Therefore, it is enough to show that the inclusion
$i:\operatorname{\mathcal{PD}}(X,\sqcup)^{*}\setminus{\mathcal{D}}(X,\sqcup)\hookrightarrow\operatorname{\mathcal{PD}}(X,\sqcup)^{*}$
is an $(n-2)$-equivalence.

We analyze the lower fibers of full decompositions for the map $i$ to apply
later Quillen’s fiber Theorem. Let
$\sigma_{0}\in{\mathcal{D}}(X,\sqcup)^{\circ}$. By (LIP), the lower fiber
$i^{-1}({\mathcal{D}}(X,\sqcup)^{\circ})$ can be identified with the poset

$F_{\sigma_{0}}:=\big{(}\prod_{y\in\sigma}\operatorname{\mathcal{PD}}(Y,\sqcup)\big{)}^{*}\setminus\prod_{y\in\sigma}{\mathcal{D}}(Y,\sqcup).$

Write $\sigma_{0}=\\{y_{1},\ldots,y_{r}\\}$, and
$\operatorname{\mathcal{PD}}(y_{i}):=\operatorname{\mathcal{PD}}(Y_{i},\sqcup)$,
${\mathcal{D}}(y_{i}):={\mathcal{D}}(Y_{i},\sqcup)$ and
$\operatorname{{\mathcal{S}}}_{h}(y_{i}):=\operatorname{{\mathcal{S}}}_{h}(Y_{i},\sqcup)$.
We claim that $F_{\sigma_{0}}$ is homotopy equivalent to
${\mathcal{T}}:=\big{(}\prod_{i=1}^{r}\operatorname{{\mathcal{S}}}_{h}(y_{i})\big{)}^{*}$.

Let $\Psi:F_{\sigma_{0}}\to{\mathcal{T}}$ be the map such that

$\Psi(\tau_{1},\ldots,\tau_{r})=(\Phi(\tau_{1}),\ldots,\Phi(\tau_{r})).$

Clearly, $\Psi$ is well-defined and order-preserving.

On the other hand, we have a map $\Gamma:{\mathcal{T}}\to F_{\sigma_{0}}$
which is just the canonical inclusion
$\Gamma(x_{1},\ldots,x_{r})=(\\{x_{1}\\}\setminus\\{0\\},\ldots,\\{x_{r}\\}\setminus\\{0\\})$.
Then $\Psi\Gamma$ is the identity map on ${\mathcal{T}}$, and

$\Gamma\Psi(\tau_{1},\ldots,\tau_{r})=(\\{\Phi(\tau_{1})\\}\setminus\\{0\\},\ldots,\\{\Phi(\tau_{r})\\}\setminus\\{0\\})\geq(\tau_{1},\ldots,\tau_{r}).$

Now, ${\mathcal{T}}$ is the proper part of the direct product of bounded
posets, and hence it is a standard fact that the geometric realization of
${\mathcal{T}}$ is isomorphic to an iterated suspension of a join of the
spaces $|\operatorname{{\mathcal{S}}}_{h}(y_{i})^{*}|$:

$|{\mathcal{T}}|\cong\Sigma(\ldots\Sigma\big{(}\Sigma\big{(}|\operatorname{{\mathcal{S}}}_{h}(y_{1})^{*}|*|\operatorname{{\mathcal{S}}}_{h}(y_{2})^{*}|\big{)}*|\operatorname{{\mathcal{S}}}_{h}(y_{3})^{*}|\big{)}\ldots*|\operatorname{{\mathcal{S}}}_{h}(y_{r})^{*}|\big{)}$

See [Wa2]. Since $\operatorname{{\mathcal{S}}}_{h}(y_{i})^{*}$ is
$(h(y_{i})-3)$-connected by hypothesis, we conclude that ${\mathcal{T}}$ is

$3(r-1)+\sum_{i}(h(y_{i})-3)=3(r-1)+n-3r=n-3$

connected. On the other hand, we know that
${\mathcal{D}}(X,\sqcup)^{\circ}_{>\sigma_{0}}$ is the proper part of the
partition lattice on $\sigma_{0}$, so it is spherical of dimension $r-3$, and
hence $(r-4)$-connected (the empty space is $(-2)$-connected by convention).
Thus $F_{\sigma_{0}}*{\mathcal{D}}(X,\sqcup)^{\circ}_{>\sigma_{0}}$ is

$(n-3)+(r-4)+2=n+r-5$

connected. In particular, since $r\geq 2$, these are $(n-3)$-connected. By
Quillen’s fiber Theorem we conclude that $i$ is an $(n-2)$-equivalence, and
therefore $\operatorname{\mathcal{PD}}(X,\sqcup)^{*}$ is $(n-3)$-connected. ∎

We finally state a result on the top-homology groups.

######  Proposition 5.0.

Let $n$ be the height of $\operatorname{{\mathcal{S}}}(X,\sqcup)$ and suppose
$\sigma\in{\mathcal{F}}(X,\sqcup)$ is a full frame (i.e. a decomposition of
size $n$). Then the Coxeter complex $S_{\sigma}$ associated to $\sigma$ gives
a non-zero class
$[S_{\sigma}]\in\widetilde{H}_{n-2}(\operatorname{{\mathcal{S}}}_{h}(X,\sqcup)^{*},{\mathbb{Z}})$.

Moreover, $S_{\sigma}$ is null-homotopic when embedded in
$\operatorname{\mathcal{PD}}(X)^{*}$ via the canonical inclusion
$\operatorname{{\mathcal{S}}}_{h}(X,\sqcup)^{*}\hookrightarrow\operatorname{\mathcal{PD}}(X,\sqcup)^{*}$.

###### Proof.

There is an $(n-2)$-dimensional sphere embedded in the order complex of
$\operatorname{{\mathcal{S}}}_{h}(X,\sqcup)$ which is obtained by taking the
usual apartment of a basis, here regarded as a frame. Concretely, this sphere
is a Coxeter complex obtained as the order complex of the subposet

$S_{\sigma}:=\big{\\{}\bigvee_{x\in\tau}x\mathrel{{\>:\>}}\emptyset\neq\tau\subsetneq\sigma\big{\\}}.$

Since $S_{\sigma}$ is an $(n-2)$-dimensional sphere, its class gives a non-
zero cycle for the $(n-2)$-homology group of
$\operatorname{{\mathcal{S}}}_{h}(X,\sqcup)^{*}$.

On the other hand, note that the image of $S_{\sigma}$ under the canonical map
$\operatorname{{\mathcal{S}}}_{h}(X,\sqcup)^{*}\to\operatorname{\mathcal{PD}}(X,\sqcup)^{*}$
is contained in the subposet

$T_{\sigma}=\\{\tau\mathrel{{\>:\>}}\emptyset\neq\tau\subseteq\tau^{\prime}\in{\mathcal{D}}(X,\sqcup)^{\circ}_{\geq\sigma}\\}.$

Then $T_{\sigma}$ is the proper part of the partial decomposition poset of the
Boolean lattice on $\sigma$. In particular, by 5.0, the poset $T_{\sigma}$ is
contractible. Hence the inclusion of the sphere
$S_{\sigma}\hookrightarrow\operatorname{\mathcal{PD}}(X,\sqcup)^{*}$ is null-
homotopic (and thus trivial in homology). ∎

######  Corollary 5.0.

Let $n$ be the height of $\operatorname{{\mathcal{S}}}(X,\sqcup)$, and assume
the following conditions hold:

  * •

(LIP);

  * •

for all $y\in\operatorname{{\mathcal{S}}}_{h}(X,\sqcup)$, the poset
$\operatorname{{\mathcal{S}}}_{h}(Y,\sqcup)^{*}$ is spherical of dimension
$h(y)-2$;

  * •

the spheres $S_{\sigma}$, for $\sigma$ a full frame, generate the $(n-2)$th
homotopy group of $\operatorname{{\mathcal{S}}}_{h}(X,\sqcup)^{*}$.

Then
$\operatorname{{\mathcal{S}}}_{h}(X,\sqcup)^{*}\hookrightarrow\operatorname{\mathcal{PD}}(X,\sqcup)^{*}$
is null-homotopic. In particular, $\operatorname{\mathcal{PD}}(X,\sqcup)^{*}$
is $(n-2)$-connected.

###### Proof.

By 5.0, it is enough to show that
$\operatorname{{\mathcal{S}}}_{h}(X,\sqcup)^{*}\hookrightarrow\operatorname{\mathcal{PD}}(X,\sqcup)^{*}$
maps a generators of the $(n-2)$th homotopy group to the trivial element. By
hypothesis, we can take the generators given by the classes of the spheres
obtained as the ordered complexes of $S_{\sigma}$ for the full frames
$\sigma$. By 5.0, these spheres are null-homotopic in
$\operatorname{\mathcal{PD}}(X,\sqcup)^{*}$. The result follows. ∎

###  5.3. Unique complementation

Now we explore some consequences on the relations between the different posets
we defined when we additionally have unique downward
$(\sqcup,h)$-complementation and (EP) holds.

######  Lemma 5.5.

If $\operatorname{{\mathcal{S}}}(X,\sqcup)$ is uniquely downward
$(\sqcup,h)$-complemented and the map $G$ defined in 4.5 is surjective, then
$G$ is an isomorphism between $\mathcal{O}{\mathcal{D}}(X,\sqcup)^{\circ}$ and
the opposite poset of $\Delta\operatorname{{\mathcal{S}}}(X,\sqcup)^{*}$.

###### Proof.

Since $G$ is a surjective poset map by hypothesis, it is enough to prove that
$G(\tau)\supseteq G(\sigma)$ implies that $\tau\leq\sigma$.

Assume then that $G(\tau)\supseteq G(\sigma)$. Write
$\tau=(z_{1},\ldots,z_{r})$ and $\sigma=(w_{1},\ldots,w_{s})$. For $1\leq
j\leq s-1$, let $1\leq\beta(j)\leq r$ be such that

$y_{j}:=w_{1}\vee\ldots\vee w_{j}=z_{1}\vee\ldots\vee z_{\beta(j)}.$

We claim that

$w_{j+1}=z_{\beta(j)+1}\vee\ldots\vee z_{\beta(j+1)}.$

Indeed, $w_{j+1}$ is the unique $(\sqcup,h)$-complement of

$y_{j}=z_{1}\vee\ldots\vee z_{\beta(j)}$

in

$y_{j}\vee w_{j+1}=y_{j}\vee z_{\beta(j)+1}\vee\ldots\vee z_{\beta(j)}.$

But also $z_{\beta(j)+1}\vee\ldots\vee z_{\beta(j)}$ is the unique
$(\sqcup,h)$-complement of $y_{j}$ in $y_{j}\vee w_{j+1}$. Thus they must
coincide. This shows that $\tau\leq\sigma$. ∎

######  Lemma 5.6.

If $\operatorname{{\mathcal{S}}}(X,\sqcup)$ is uniquely downward
$(\sqcup,h)$-complemented and satisfies (EP), then the map $G$ is surjective.
In particular, $\mathcal{O}{\mathcal{D}}(X,\sqcup)^{\circ}$ is isomorphic to
$(\Delta\operatorname{{\mathcal{S}}}(X,\sqcup)^{*})^{\mathrm{op}}$.

###### Proof.

We prove this by induction on the size of a chain. Let
$c=\\{z_{0}<z_{1}\ldots<z_{r}\\}\in\Delta\operatorname{{\mathcal{S}}}(X,\sqcup)^{*}$
be a chain. If $r=0$, let $w_{0}$ be the unique $(\sqcup,h)$-complement of
$z_{0}$. Thus $\\{z_{0},w_{0}\\}$ is a $\sqcup$-decomposition and
$G(z_{0},w_{0})=(z_{0})$. Assume then that $r>0$, and let
$c^{\prime}=c\setminus\\{z_{r}\\}$. Hence there exists an ordered
$\sqcup$-decomposition $(x_{0},\ldots,x_{r})$ with $z_{i}=x_{0}\vee\ldots\vee
x_{i}$, for $0\leq i\leq r-1$. By uniqueness, we have
$x_{r}=z_{r-1}^{\perp}>z_{r}^{\perp}$. Let $u$ be the unique
$(\sqcup,h)$-complement of $z_{r}^{\perp}$ in $x_{r}$. Then
$\\{u,z_{r}^{\perp}\\}\in{\mathcal{D}}(\operatorname{{\mathcal{S}}}(X,\sqcup)_{\leq
x_{r}})$ and it is $\sqcup$-compatible in
$\operatorname{{\mathcal{S}}}(X,\sqcup)$. By (EP), we see that
$\\{x_{0},\ldots,x_{r-1},u,z_{r}^{\perp}\\}\in{\mathcal{D}}(X,\sqcup)$ and
also $z_{r}=z_{r-1}\vee u$. Hence
$G(x_{0},\ldots,x_{r-1},u,z_{r}^{\perp})=(z_{0},\ldots,z_{r-1},z_{r})$. ∎

######  Corollary 5.0.

Suppose that ${\mathcal{D}}(X,\sqcup)$ has a unique minimum element
$\sigma_{0}$. Then ${\mathcal{D}}(X,\sqcup)$ is the partition lattice on
$\sigma_{0}$, $\operatorname{{\mathcal{S}}}_{h}(X,\sqcup)$ is a Boolean
lattice on the set $\sigma_{0}$ and
$\operatorname{\mathcal{PD}}(X,\sqcup)^{*}$ is contractible. Moreover,
$\operatorname{{\mathcal{S}}}_{h}(X,\sqcup)$, ${\mathcal{D}}(X,\sqcup)$,
$\operatorname{\mathcal{PD}}(X,\sqcup)$ are Cohen-Macaulay.

###### Proof.

By 3.0, ${\mathcal{D}}(X,\sqcup)$ is the partition lattice on $\sigma_{0}$. On
the other hand, any element of
$x\in\operatorname{{\mathcal{S}}}_{h}(X,\sqcup)$ is part of a
$\sqcup$-decomposition, so there exists a unique set $\tau\subseteq\sigma_{0}$
such that $x$ is the join of the elements of $\tau$. Therefore
$\operatorname{{\mathcal{S}}}_{h}(X,\sqcup)$ is the Boolean lattice on the set
$\sigma_{0}$. In particular $\operatorname{{\mathcal{S}}}_{h}(X,\sqcup)$ and
${\mathcal{D}}(X,\sqcup)$ are Cohen-Macaulay.

Now we prove that $\operatorname{\mathcal{PD}}(X,\sqcup)^{*}$ is contractible.
In fact, we show that if
$\sigma\in\operatorname{\mathcal{PD}}(X,\sqcup)\setminus{\mathcal{D}}(X,\sqcup)$,
then $\operatorname{\mathcal{PD}}(X,\sqcup)_{>\sigma}\setminus\\{\\{1\\}\\}$
is contractible (so $\operatorname{\mathcal{PD}}(X,\sqcup)^{*}$ is
contractible by taking $\sigma=\\{\\}$).

Let
$\sigma\in\operatorname{\mathcal{PD}}(X,\sqcup)\setminus{\mathcal{D}}(X,\sqcup)$.
Then $\sigma$ is contained in some $\sqcup$-decomposition (i.e. partition)
$\tau$ and every element $y\in\sigma$ can be obtained by joining elements of
$\sigma_{0}$. Thus, the set

$f(\sigma):=\sigma\cup\\{x\in\sigma_{0}\mathrel{{\>:\>}}x\nleq y\text{ for all
}y\in\sigma\\}$

is a partition of $\sigma_{0}$. Now we fix $\sigma$, and let
$\sigma^{\prime}>\sigma$. Then $f(\sigma^{\prime})\geq\sigma^{\prime}>\sigma$.
Since $\sigma\notin{\mathcal{D}}(X,\sqcup)$ and
$f(\sigma)\in{\mathcal{D}}(X,\sqcup)$, we see that $f(\sigma)>\sigma$. Thus we
have the following inequalities in
$\operatorname{\mathcal{PD}}(X,\sqcup)_{>\sigma}\setminus\\{\\{1\\}\\}$:

$\sigma^{\prime}\leq f(\sigma^{\prime})\geq f(\sigma).$

This proves that
$\operatorname{\mathcal{PD}}(X,\sqcup)_{>\sigma}\setminus\\{\\{1\\}\\}$ is
contractible. It is also not hard to show that this interval has the correct
dimension.

Since $f(\emptyset)=\sigma_{0}$, the homotopy above is

$\sigma^{\prime}\leq f(\sigma^{\prime})\geq\sigma_{0}.$

Hence $\operatorname{\mathcal{PD}}(X,\sqcup)^{*}$ is contractible.

On the other hand, if $\sigma\in{\mathcal{D}}(X,\sqcup)$ then

$\operatorname{\mathcal{PD}}(X,\sqcup)_{>\sigma}={\mathcal{D}}(X,\sqcup)_{>\sigma}\cong\Pi(\sigma)_{>\sigma},$

which is a Cohen-Macaulay poset. Therefore, upper intervals in the poset
$\operatorname{\mathcal{PD}}(X,\sqcup)^{*}$ are spherical of the correct
dimension.

Finally, since lower intervals in $\operatorname{\mathcal{PD}}(X,\sqcup)$ are
products of partial decomposition posets, which are partial decompositions of
Boolean lattices, we conclude that $\operatorname{\mathcal{PD}}(X,\sqcup)$ is
a Cohen-Macaulay poset. ∎

######  Proposition 5.0.

If $\operatorname{{\mathcal{S}}}(X,\sqcup)$ is uniquely downward
$(\sqcup,h)$-complemented and satisfies (EP) then
$\operatorname{\mathcal{PD}}(X,\sqcup)^{*}$ deformation retracts onto
${\mathcal{D}}(X,\sqcup)^{\circ}$.

###### Proof.

We prove that for any
$\tau\in\operatorname{\mathcal{PD}}(X,\sqcup)^{*}\setminus{\mathcal{D}}(X,\sqcup)$,
the interval
${\mathcal{D}}_{\tau}:={\mathcal{D}}(X,\sqcup)^{\circ}_{\geq\tau}$ is
contractible. Let $\sigma\in{\mathcal{D}}_{\tau}$. For $y\in\sigma$, let
$\tau_{y}:=\\{x\in\tau\mathrel{{\>:\>}}x\leq y\\}$ and
$\alpha(y):=\bigvee_{x\in\tau_{y}}x$. Then $\alpha(y)\leq y$. Let
$\beta(y)\leq y$ be the unique $(\sqcup,h)$-complement for $\alpha(y)$ in $y$.
Hence
$\\{\alpha(y),\beta(y)\\}\setminus\\{0\\}\in{\mathcal{D}}(\operatorname{{\mathcal{S}}}(X,\sqcup)_{\leq
y})$ and it is $\sqcup$-compatible in
$\operatorname{{\mathcal{S}}}(X,\sqcup)$. By the (EP) property,
$\sigma^{\prime}_{y}:=(\sigma\setminus\\{y\\})\cup\\{\alpha(y),\beta(y)\\}\setminus\\{0\\}$
is a $\sqcup$-decomposition of $\operatorname{{\mathcal{S}}}(X,\sqcup)$. We
can continue in this way until producing a $\sqcup$-decomposition

$\sigma^{\prime}=\\{\alpha(y),\beta(y):y\in\sigma\\}\setminus\\{0\\}\in{\mathcal{D}}(X,\sqcup).$

Note that $\sigma^{\prime}\leq\sigma$ and that $\tau\leq\sigma^{\prime}$. Now,
let $x$ be the join of the elements of $\tau$ and $x^{\prime}$ be the
$(\sqcup,h)$-complement of $x$ in $\operatorname{{\mathcal{S}}}(X,\sqcup)$. We
claim that $\beta(y)\leq x^{\prime}$. Since $x=\bigvee_{y\in\sigma}\alpha(y)$,
if $x^{\prime\prime}=\bigvee_{y\in\sigma}\beta(y)$ then
$\\{x,x^{\prime\prime}\\}\in{\mathcal{D}}(X,\sqcup)$, so $x,x^{\prime\prime}$
are $(\sqcup,h)$-complements of each other since $\sigma^{\prime}$ is a
$\sqcup$-decomposition. By uniqueness, $x^{\prime\prime}=x^{\prime}$, and this
implies that $\beta(y)\leq x^{\prime}$. Hence we have the following zigzag
whose elements lie in ${\mathcal{D}}_{\tau}$:

$\sigma\geq\sigma^{\prime}\leq\\{x,x^{\prime}\\}.$

It remains to prove that the map $\sigma\mapsto\sigma^{\prime}$ is order-
preserving. For simplicity, we prove it for covering relations, so assume that
$\sigma\leq\rho$, where $\rho$ is obtained by joining two elements
$y_{1},y_{2}$ of $\sigma$. Then we have the $\sqcup$-decompositions by
$(\sqcup,h)$-complements $y_{1}=\alpha(y_{1})\vee\beta(y_{1})$ and
$y_{2}=\alpha(y_{2})\vee\beta(y_{2})$. Clearly $\alpha(y_{1}\vee
y_{2})=\alpha(y_{1})\vee\alpha(y_{2})$. Let $z=\beta(y_{1})\vee\beta(y_{2})$,
which exists since $\sigma^{\prime}$ is a $\sqcup$-decomposition. This also
implies that $h(z)=h(\beta(y_{1}))+h(\beta(y_{2}))$, and that
$z\wedge(\alpha(y_{1})\vee\alpha(y_{2}))=0$. We also have that
$z\vee\alpha(y_{1})\vee\alpha(y_{2})=y_{1}\vee y_{2}$. Therefore $z$ is an
$(\sqcup,h)$-complement for $\alpha(y_{1})\vee\alpha(y_{2})$ in $y_{1}\vee
y_{2}$. By uniqueness, $z=\beta(y_{1}\vee y_{2})$. Hence $\rho^{\prime}$ is
obtained from $\sigma^{\prime}$ after taking the joins
$\alpha(y_{1})\vee\alpha(y_{2})$ and $\beta(y_{1})\vee\beta(y_{2})$. Thus
$\sigma^{\prime}\leq\rho^{\prime}$. ∎

######  Corollary 5.0.

Suppose that $\operatorname{{\mathcal{S}}}(X,\sqcup)$ is uniquely downward
$(\sqcup,h)$-complemented, and let
$\sigma\in\operatorname{\mathcal{PF}}(X,\sqcup)$ be a simplex of size $n-1$.
Then the following hold:

  1. (1)

$\sigma$ is contained in a unique maximal simplex.

  2. (2)

${\mathcal{F}}(X,\sqcup)$, viewed as a poset, deformation retracts onto the
subposet $\widehat{{\mathcal{F}}}(X,\sqcup)$ of frames of size $\neq n-1$.

  3. (3)

If (EP) holds then we have a poset inclusion
$\widehat{{\mathcal{F}}}(X,\sqcup)^{\mathrm{op}}\hookrightarrow{\mathcal{D}}(X,\sqcup)^{\circ}$
given by $\tau\mapsto\tau\cup\\{\big{(}\bigvee_{x\in\tau}x\big{)}^{\perp}\\}$.

###### Proof.

Let $\sigma\in\operatorname{\mathcal{PF}}(X,\sqcup)$ be a frame of size $n-1$,
and recall that $\Phi(\sigma)$ is the join of the elements of $\sigma$. Since
$\sigma$ is a partial decomposition,
$\sigma\cup\\{\Phi(\sigma)^{\perp}\\}\in{\mathcal{D}}(X,\sqcup)$. Now,
$|\sigma|=n-1$ implies that $h(\Phi(\sigma))=n-1$ and
$h(\Phi(\sigma)^{\perp})=1$. Therefore $\sigma\cup\\{\Phi(\sigma)^{\perp}\\}$
is a frame of size $n$, and clearly
$\operatorname{\mathcal{PF}}(X,\sqcup)_{>\sigma}=\\{\sigma\cup\\{\Phi(\sigma)^{\perp}\\}\\}$
by uniqueness of the complement. This proves item (1).

Item (2) is an easy consequence of (1). We write down the deformation retract.
Let $\widehat{{\mathcal{F}}}(X,\sqcup)$ be the subposet of
${\mathcal{F}}(X,\sqcup)$ consisting of frames of size $\neq n-1$. By the
previous paragraph,
${\mathcal{F}}(X,\sqcup)_{>\sigma}=\\{\sigma\cup\\{\Phi(\sigma)^{\perp}\\}\\}$
for any frame $\sigma$ of size $n-1$. Therefore the retraction
$r:{\mathcal{F}}(X,\sqcup)\to\widehat{{\mathcal{F}}}(X,\sqcup)$ that takes a
frame of size $n-1$ to the unique maximal frame of size $n$ containing it is
an order-preserving map such that $r(\sigma)\geq\sigma$ for all
$\sigma\in{\mathcal{F}}(X,\sqcup)$ and it is the identity on
$\widehat{{\mathcal{F}}}(X,\sqcup)$. Hence
${\mathcal{F}}(X,\sqcup)\simeq\widehat{{\mathcal{F}}}(X,\sqcup)$.

The function
$\widehat{{\mathcal{F}}}(X,\sqcup)^{\mathrm{op}}\hookrightarrow{\mathcal{D}}(X,\sqcup)^{\circ}$
defined in item (3) is an inclusion of posets by 3.0. ∎

######  Corollary 5.0.

Assume the following hold for $\operatorname{{\mathcal{S}}}(X,\sqcup)$:

  1. (1)

$\operatorname{{\mathcal{S}}}(X,\sqcup)$ is finite and uniquely downward
$(\sqcup,h)$-complemented.

  2. (2)

For $y\in\operatorname{{\mathcal{S}}}(X,\sqcup)$,
$\operatorname{{\mathcal{S}}}(X,\sqcup)_{\leq
y}=\operatorname{{\mathcal{S}}}(Y,\sqcup)$ for a representative $y=[(Y,i)]$.

  3. (3)

(EP) holds.

  4. (4)

The lower-intervals in ${\mathcal{D}}(X,\sqcup)$ and
$\operatorname{\mathcal{PD}}(X,\sqcup)$ decompose as in 4.4(2).

Then $\tilde{\chi}(\mathcal{O}\operatorname{\mathcal{PD}}(X,\sqcup))=-1$.

###### Proof.

We compute $\tilde{\chi}(\mathcal{O}\operatorname{\mathcal{PD}}(X,\sqcup))$ by
using the wedge decomposition given in 5.0(2) and the following formula for
the Euler characteristic of a finite poset:

(2)
$\tilde{\chi}\big{(}{{\mathcal{D}}(X,\sqcup)^{\circ}}\big{)}=-1-\sum_{\sigma\in{\mathcal{D}}(X,\sqcup)^{\circ}}\tilde{\chi}({\mathcal{D}}(X,\sqcup)^{\circ}_{<\sigma}).$

First, note that $\operatorname{{\mathcal{S}}}(Y,\sqcup)$ is also uniquely
downward $(\sqcup,h)$-complemented since it coincides with
$\operatorname{{\mathcal{S}}}(X,\sqcup)_{\leq y}$ for $y=[(Y,i)]$. From this
it is also not hard to see that (EP) holds for
$\operatorname{{\mathcal{S}}}(Y,\sqcup)$. Thus, by 5.0,
$\operatorname{\mathcal{PD}}(Y,\sqcup)^{*}\simeq{\mathcal{D}}(Y,\sqcup)^{\circ}$
for any $y=[(Y,i)]\in\operatorname{{\mathcal{S}}}(X,\sqcup)$.

Now let $\sigma=\\{y_{1},\ldots,y_{r}\\}\in{\mathcal{D}}(X,\sqcup)^{\circ}$,
with representatives $y_{i}=[(Y_{i},j_{i})]$. We have

$\displaystyle\tilde{\chi}(\operatorname{\mathcal{PD}}(X,\sqcup)^{*}_{<\sigma})$
$\displaystyle=\tilde{\chi}\bigg{(}\big{(}\prod_{i}\operatorname{\mathcal{PD}}(Y_{i},\sqcup)\big{)}^{*}\bigg{)}=\prod_{i}\tilde{\chi}\big{(}\operatorname{\mathcal{PD}}(Y_{i},\sqcup)^{*}\big{)}=\prod_{i}\tilde{\chi}\big{(}{\mathcal{D}}(Y_{i},\sqcup)^{\circ}\big{)},$

and

$\displaystyle\tilde{\chi}({\mathcal{D}}(X,\sqcup)^{\circ}_{<\sigma})$
$\displaystyle=\tilde{\chi}\bigg{(}\big{(}\prod_{i}{\mathcal{D}}(Y_{i},\sqcup)\big{)}^{\circ}\bigg{)}=\tilde{\chi}\bigg{(}{\mathcal{D}}(Y_{1},\sqcup)^{\circ}*\ldots*{\mathcal{D}}(Y_{r},\sqcup)^{\circ}\bigg{)}$
$\displaystyle=(-1)^{|\sigma|-1}\prod_{i}\tilde{\chi}({\mathcal{D}}(Y_{i},\sqcup)^{\circ})$
$\displaystyle=(-1)^{|\sigma|-1}\tilde{\chi}(\operatorname{\mathcal{PD}}(X,\sqcup)^{*}_{<\sigma}).$

By 5.0(2) and the above equalities, we get

$\displaystyle\tilde{\chi}(\mathcal{O}\operatorname{\mathcal{PD}}(X,\sqcup)^{*})$
$\displaystyle=\tilde{\chi}({\mathcal{D}}(X,\sqcup)^{\circ})-\sum_{\sigma\in{\mathcal{D}}(X,\sqcup)^{\circ}}(-1)^{|\sigma|-2}\tilde{\chi}(\operatorname{\mathcal{PD}}(X,\sqcup)^{*}_{<\sigma})$
$\displaystyle=\tilde{\chi}({\mathcal{D}}(X,\sqcup)^{\circ})+\sum_{\sigma\in{\mathcal{D}}(X,\sqcup)^{\circ}}\tilde{\chi}({\mathcal{D}}(X,\sqcup)^{\circ}_{<\sigma})$
$\displaystyle=\tilde{\chi}({\mathcal{D}}(X,\sqcup)^{\circ})+(-\tilde{\chi}({\mathcal{D}}(X,\sqcup)^{\circ})-1)$
$\displaystyle=-1.$

∎

##  6\. Application to classical examples

This section shows how our constructions give rise to well-studied objects.

###  6.1. Finitely generated free groups

Let ${\mathcal{C}}$ be the category of groups. We take $\sqcup$ to be the
coproduct of this category, which is the free product of groups, denoted by
$*$. Then, for a group $X$, we have that $\operatorname{{\mathcal{S}}}(X)$ is
the lattice of subgroups. If $X$ is a free group of finite rank $n$,
$\operatorname{{\mathcal{S}}}(X,*)=\operatorname{FC}(X)=\operatorname{FC}_{n}$
is the poset of free factors of $X$. Hatcher and Vogtmann studied this poset
in several articles (see for example [HV98a, HV98b, HV22]), and they proved it
is Cohen-Macaulay of dimension $n$. Note that this poset is ranked, and the
poset-rank of an element is exactly its rank as a free group. Moreover,
$\operatorname{FC}(X)$ is indeed a lattice by the Kurosh subgroup theorem, and
for $H\in\operatorname{FC}(X)$ we have $\operatorname{FC}(X)_{\leq
H}=\operatorname{FC}(H)$.

On the other hand, if $H_{1}*H_{2}*\ldots*H_{r}\in\operatorname{FC}(X)$, then
$\\{H_{1},\ldots,H_{r}\\}$ spans a Boolean lattice such that the map
$I\subseteq\\{1,\ldots,r\\}\mapsto\langle H_{i}\mathrel{{\>:\>}}i\in
I\rangle\in\operatorname{FC}(X)$ is a lattice embedding. Therefore,
${\mathcal{D}}(X,*)$ is the poset of decompositions into free factors, and we
may just denote it by ${\mathcal{D}}(X)$. That is, its elements are sets of
free factors $\\{H_{1},\ldots,H_{r}\\}$ such that $H_{1}*\ldots*H_{r}=X$. This
coincides with the poset defined by Hatcher and Vogtmann in [HV22]. It is
proved in [HV22] that ${\mathcal{D}}(X)^{\circ}$ is spherical of dimension
$n-2$.

The poset $\operatorname{FC}(X)$ is (non-uniquely) downward
$(\sqcup,h)$-complemented, it satisfies (EP) and (CP). We naturally have
$\operatorname{\mathcal{PD}}(X)_{\leq\sigma}=\prod_{H\in\sigma}\operatorname{\mathcal{PD}}(H)$
for any $\sigma\in\operatorname{\mathcal{PD}}(X)$, so (LIP) also holds.
Moreover, upper intervals in ${\mathcal{D}}(X)$ are partition lattices by 2.2,
and lower intervals are products of smaller decomposition posets. Thus we
conclude that ${\mathcal{D}}(X)$ is Cohen-Macaulay.

######  Theorem 6.0 (Hatcher-Vogtmann).

Let $X$ be a free group of finite rank $n$. Then the posets
$\operatorname{FC}(X)$ and ${\mathcal{D}}(X)$ are Cohen-Macaulay of dimension
$n$ and $n-1$ respectively.

By 5.0 we obtain the following corollary.

######  Corollary 6.0.

Let $X$ be a free group of finite rank $n$. Then the poset
$\mathcal{O}{\mathcal{D}}(X)$ is Cohen-Macaulay of dimension $n-1$.

Next, note that every partial frame $\\{H_{1},\ldots,H_{r}\\}$ of $X$ extends
to a full frame. Hence $\operatorname{\mathcal{PF}}(X,*)={\mathcal{F}}(X,*)$
and the set $\mathcal{A}$ of vertices of ${\mathcal{F}}(X,*)$ is exactly the
set of free factors of $X$ of rank $1$. We denote this complex simply by
${\mathcal{F}}(X)$.

For the partial basis complex, we choose the family
$P=(P_{H})_{H\in\mathcal{A}}$, where $P_{H}=\\{x\in
H\mathrel{{\>:\>}}\left\langle x\right\rangle=H\\}\cong\\{\pm 1\\}$. Then
$\mathcal{B}(X,*)$ is exactly the complex of partial bases of a free group,
that is, the maximal simplices are bases of $X$. This complex is also denoted
by $\operatorname{PB}(X)=\operatorname{PB}_{n}$, and it was shown to be Cohen-
Macaulay of dimension $n-1$ by Sadofschi Costa [SC]. In particular, by 5.0, we
conclude that ${\mathcal{F}}(X)$ is Cohen-Macaulay.

######  Theorem 6.0 (Sadofschi Costa).

Let $X$ be a free group of finite rank $n$. Then the posets ${\mathcal{F}}(X)$
and $\operatorname{PB}(X)$ are Cohen-Macaulay of dimension $n-1$.

Again 5.0 yields the following corollary.

######  Corollary 6.0.

Let $X$ be a free group of finite rank $n$. Then the posets
$\mathcal{O}{\mathcal{F}}(X)$ and $\mathcal{O}\operatorname{PB}(X)$ are Cohen-
Macaulay of dimension $n-1$.

However, little is known about the homotopy type of the poset of partial
decompositions
$\operatorname{\mathcal{PD}}(X):=\operatorname{\mathcal{PD}}(X,*)$. Thus we
propose the following question.

######  Question 6.0.

Let $X$ be a free group of rank $n$. Are the proper parts of
$\operatorname{\mathcal{PD}}(X)$ and $\mathcal{O}{\mathcal{P}}(X)$ spherical
(of dimension $2n-3$)? If so, are they Cohen-Macaulay?

By 5.0, 6.0 and (LIP), we see that $\operatorname{\mathcal{PD}}(X)^{*}$ is at
least $(n-3)$-connected.

If $n=2$ then $\operatorname{\mathcal{PD}}(X)^{*}={\mathcal{F}}(X)$, which is
connected by 6.0. Thus $\operatorname{\mathcal{PD}}(X)^{*}$ is connected for
all $n\geq 2$. Moreover, $\operatorname{\mathcal{PD}}(X)^{*}$ is simply
connected for $n\geq 3$:

######  Proposition 6.0.

If $n\geq 3$ then $\operatorname{\mathcal{PD}}(X)^{*}$ is simply connected.

###### Proof.

We only need to analyze the case $n=3$. By 5.0, we show that classes of the
standard apartments generate the fundamental group of
$\operatorname{FC}(X)^{*}$. This follows by Proposition 5.8 of [SC] applied to
the empty partial basis there. ∎

###  6.2. Vector spaces

Let ${\mathcal{C}}$ be the category of vector spaces over a field
${\mathbb{K}}$, and let $X$ be a ${\mathbb{K}}$-vector space of finite
dimension $n$. Here we take $\sqcup$ to be the coproduct of the category, that
is, the direct sum $\oplus$ of vector spaces. Then
$\operatorname{{\mathcal{S}}}(X)=\operatorname{{\mathcal{S}}}(X,\oplus)$ is
the poset of subspaces of $X$ since every subspace has a direct sum
complement. The order complex of $\operatorname{{\mathcal{S}}}(X)^{*}$ is the
Tits building of $\operatorname{SL}(X)$, which is spherical by the famous
Solomon-Tits theorem [So]. Intervals in $\operatorname{{\mathcal{S}}}(X)$ are
subspace posets again, so $\operatorname{{\mathcal{S}}}(X)$ is indeed Cohen-
Macaulay of dimension $n$. This poset is also ranked, and the rank of an
element coincides with its dimension as a ${\mathbb{K}}$-vector space. Note
that $\\{V_{1},\ldots,V_{r}\\}\subseteq\operatorname{{\mathcal{S}}}(X)$ are
subspaces in direct sum if and only if
$\sum_{i}\dim(V_{i})=\dim(V_{1}+\ldots+V_{r})$. Hence
${\mathcal{D}}(X,\oplus)$ is the poset of sets of non-zero subspaces which are
in direct sum and span $X$. We denote it simply by ${\mathcal{D}}(X)$. This
poset was shown to be shellable for finite fields in [We1], and in particular
Cohen-Macaulay. In the case $X=\mathbb{F}_{q}^{n}$, the reduced Euler
characteristic of ${\mathcal{D}}(X)$ is

(3) $\frac{(-1)^{n}}{n}\cdot\prod_{i=1}^{n-1}(q^{i}-1)\cdot f_{n}(q),$

where $f_{n}(q)$ is a monic polynomial of degree $\binom{n}{2}$ and positive
integer coefficients related to a $q$-analog of Catalan-numbers. The
coefficient sequences of these polynomials seem to be always unimodal (see
[We1, pp.241-242]).

On the other hand, for arbitrary fields, we show later that
${\mathcal{D}}(X)^{*}$ is spherical by invoking the results from [C] on the
Charney poset $\mathcal{G}(X,\oplus)$ (see 4.0).

Similar to the case of the free factor poset,
${\mathcal{F}}(X):={\mathcal{F}}(X,\oplus)=\operatorname{\mathcal{PF}}(X,\oplus)$
and we take $\operatorname{PB}(X):=\mathcal{B}(X,\oplus)$ to be the complex of
partial bases of $X$ (i.e. linearly independent sets of vectors). For finite
vector spaces $X$, the complexes ${\mathcal{F}}(X)$ and $\operatorname{PB}(X)$
are matroid complexes, which are shellable and in particular Cohen-Macaulay of
dimension $n-1$ (see [Bjo2] and 5.0). For infinite fields, this follows by
Theorem 2.6 of [vdK] since fields satisfy the Bass’ stable range condition
$\mathrm{SR}_{2}$. By item (i) of such theorem, we see that
$\mathcal{O}\operatorname{PB}(X)$ is spherical of dimension $n-1$, and by (ii)
there also the intervals $\mathcal{O}\operatorname{PB}(X)_{>z}$, for
$z\in\mathcal{O}\operatorname{PB}(X)$, are spherical. This implies that
$\mathcal{O}\operatorname{PB}(X)$ is Cohen-Macaulay of dimension $n-1$. Then
by 5.0, $\operatorname{PB}(X)$ is Cohen-Macaulay of dimension $n-1$. By 5.0,
${\mathcal{F}}(X)$ is Cohen-Macaulay of dimension $n-1$. Again, by 5.0, we see
that $\mathcal{O}{\mathcal{F}}(X)$ is Cohen-Macaulay of dimension $n-1$.

Finally,
$\operatorname{\mathcal{PD}}(X):=\operatorname{\mathcal{PD}}(X,\oplus)$ is the
poset of partial direct sum decompositions of $X$, that is, its elements are
sets of non-zero subspaces of $X$ which are in direct sum. An unpublished work
by Hanlon, Hersh and Shareshian [HHS] proves that
$\operatorname{\mathcal{PD}}(X)^{*}$ is Cohen-Macaulay of dimension $2n-3$,
and in particular spherical, if $X=\mathbb{F}_{q}^{n}$. Moreover, they show
that the reduced Euler characteristic of this poset is

(4) $-\frac{1}{n}\cdot q^{\binom{n}{2}}\cdot\prod_{i=1}^{n-1}(q^{i}-1).$

Note that this formula is closely related to the Euler characteristic of the
join of the poset of proper non-zero subspaces and ${\mathcal{D}}(X)$. The
former has Euler characteristic $(-1)^{n}q^{\binom{n}{2}}$, and the Euler
characteristic of the latter is given in 3. The join of two posets has as its
Euler characteristic the negative of the product of the two Euler
characteristics, so in 4 the polynomial $f_{n}(q)$ is missing.

For finite-dimensional vector spaces $X$ over arbitrary fields, the poset
$\operatorname{\mathcal{PD}}(X)$ is homotopy equivalent to
$\operatorname{CB}(X)$ (see [BPW]), which was proved to be $(2n-4)$-connected
in [MPW]. Hence, $\operatorname{\mathcal{PD}}(X)$ is spherical of dimension
$2n-3$. By 6.0 and 5.0, we can also conclude that
$\mathcal{O}\operatorname{\mathcal{PD}}(X)^{*}$ is spherical.

In order to conclude Cohen-Macaulayness for $\operatorname{\mathcal{PD}}(X)$,
we must show that intervals are spherical. Note that (LIP) holds, so if
$\sigma\in\operatorname{\mathcal{PD}}(X)$, then
$\operatorname{\mathcal{PD}}(X)_{\leq\sigma}\cong\prod_{S\in\sigma}\operatorname{\mathcal{PD}}(S)$
is spherical by induction. However, upper intervals do not have a clear
description, unless we have full decomposition. That is, if
$\sigma\in{\mathcal{D}}(X)$ then
$\operatorname{\mathcal{PD}}(X)_{\geq\sigma}={\mathcal{D}}(X)_{\geq\sigma}$ is
the partition lattice on $\sigma$, whose proper part is spherical.

Finally, note $\operatorname{{\mathcal{S}}}(X)$ is downward
$(\sqcup,h)$-complemented for $\sqcup=\oplus$, and it satisfies (EP) and (CP).
Hence by 4.0, the Charney complex $\Delta\mathcal{G}(X,\oplus)$ is isomorphic
to the poset $(\mathcal{O}{\mathcal{D}}(X,\oplus)^{\circ})^{\mathrm{op}}$.
Since the $\mathcal{G}(X,\oplus)$ is spherical by [C], we conclude that
$\mathcal{O}{\mathcal{D}}(X)^{\circ}=\mathcal{O}{\mathcal{D}}(X,\oplus)^{\circ}$
is spherical of dimension $n-2$. Thus ${\mathcal{D}}(X)^{\circ}$ is spherical
of dimension $n-2$ by 5.0.

We conclude the following theorem whose pieces follow from the references and
arguments in the preceding paragraphs.

######  Theorem 6.0.

Let $X$ be a vector space of finite dimension $n$ over a field ${\mathbb{K}}$.
Then the posets $\operatorname{PB}(X)$, $\mathcal{O}\operatorname{PB}(X)$,
${\mathcal{F}}(X)$, $\mathcal{O}{\mathcal{F}}(X)$,
$\operatorname{{\mathcal{S}}}(X)$, ${\mathcal{D}}(X)$ and
$\mathcal{O}{\mathcal{D}}(X)$ are Cohen-Macaulay.

The posets $\operatorname{\mathcal{PD}}(X)^{*}$ and
$\mathcal{O}\operatorname{\mathcal{PD}}(X)^{*}$ are spherical of dimension
$2n-3$. If ${\mathbb{K}}$ is a finite field then
$\operatorname{\mathcal{PD}}(X)$ and
$\mathcal{O}\operatorname{\mathcal{PD}}(X)$ are Cohen-Macaulay.

As we mentioned above, the upper intervals in the partial decomposition posets
are harder to compute. We leave open then the following question.

######  Question 6.0.

Are the posets $\operatorname{\mathcal{PD}}(X)$ and
$\mathcal{O}\operatorname{\mathcal{PD}}(X)$ Cohen-Macaulay, for any finite-
dimensional vector space $X$?

We close this section with a brief discussion on the Euler characteristic of
frames, partial decompositions and some of the ordered versions.

For $X=\mathbb{F}_{q}^{n}$, it is easily checked that the Euler characteristic
of $\mathcal{O}{\mathcal{F}}(X)$ is a $q$-analog of the derangements numbers.
3 provides a list for small $n$, and 4 contains the values for the unordered
version. These polynomials do not seem to be well-behaved and we do not know
of any other proposed $q$-analog of derangement numbers coinciding with them.