# Projection algebras and free projection- and idempotent-generated regular
$*$-semigroups00footnotetext: This work was supported by the following grants:
Future Fellowship FT190100632 of the Australian Research Council;
EP/V032003/1, EP/S020616/1 and EP/V003224/1 of the Engineering and Physical
Sciences Research Council. The second author thanks the Sydney Mathematical
Research Institute, the University of Sydney and Western Sydney University for
partially funding his visit to Sydney in 2023 during which part of this
research was undertaken.

James East, Robert D. Gray, P.A. Azeef Muhammed, Nik Ruškuc

###### Abstract

The purpose of this paper is to introduce a new family of semigroups—the free
projection-generated regular $*$-semigroups—and initiate their systematic
study. Such a semigroup $\operatorname{\textup{{PG}}}(P)$ is constructed from
a projection algebra $P$, using the recent groupoid approach to regular
$*$-semigroups. The assignment $P\mapsto\operatorname{\textup{{PG}}}(P)$ is a
left adjoint to the forgetful functor that maps a regular $*$-semigroup $S$ to
its projection algebra ${\bf P}(S)$. In fact, the category of projection
algebras is coreflective in the category of regular $*$-semigroups. The
algebra ${\bf P}(S)$ uniquely determines the biordered structure of the
idempotents ${\bf E}(S)$, up to isomorphism, and this leads to a category
equivalence between projection algebras and regular $*$-biordered sets. As a
consequence, $\operatorname{\textup{{PG}}}(P)$ can be viewed as a quotient of
the classical free idempotent-generated (regular) semigroups
$\operatorname{\textup{{IG}}}(E)$ and $\operatorname{\textup{{RIG}}}(E)$,
where $E={\bf E}(\operatorname{\textup{{PG}}}(P))$; this is witnessed by a
number of presentations in terms of generators and defining relations. The
semigroup $\operatorname{\textup{{PG}}}(P)$ can also be viewed as the
fundamental groupoid of a simplicial complex explicitly constructed from $P$.
The theory is then illustrated on a number of examples. In one direction, the
free construction applied to the projection algebras of adjacency semigroups
yields a new family of graph-based path semigroups. In another, it turns out
that, remarkably, the Temperley–Lieb monoid $\mathcal{T}\\!\mathcal{L}_{n}$ is
the free regular $*$-semigroup over its own projection algebra ${\bf
P}(\mathcal{T}\\!\mathcal{L}_{n})$.

_Keywords_ : Regular $*$-semigroup, projection algebra, chained projection
groupoid, free projection-generated regular $*$-semigroup, forgetful functor,
adjoint, coreflectivity, regular semigroup, biordered set, free idempotent-
generated semigroup, presentation, fundamental groupoid, Temperley–Lieb
monoid.

MSC: 20M50 (primary), 18B40, 20M05, 20M10, 20M17, 20M20.

###### Contents

  1. 1 Introduction
  2. 2 Regular $*$-semigroups
    1. 2.1 Definitions and basic properties
    2. 2.2 Adjacency semigroups
    3. 2.3 Diagram monoids
  3. 3 Projection algebras and chained projection groupoids
    1. 3.1 Preliminaries on small categories
    2. 3.2 Projection algebras
    3. 3.3 Path categories and chain groupoids
    4. 3.4 Chained projection groupoids
    5. 3.5 A category isomorphism
  4. 4 Construction of the chain semigroup
    1. 4.1 Linked pairs of projections
    2. 4.2 The reduced chain groupoid
    3. 4.3 The chain semigroup
  5. 5 Freeness of the chain semigroup
  6. 6 Projection algebras and biordered sets
    1. 6.1 Preliminaries on biordered sets
    2. 6.2 From projection algebras to $*$-biordered sets
    3. 6.3 From $*$-biordered sets to projection algebras
    4. 6.4 A category equivalence, and more on freeness
  7. 7 Presentations
    1. 7.1 Preliminaries on presentations
    2. 7.2 Presentation over $P$
    3. 7.3 Presentation as a quotient of $\operatorname{\textup{{IG}}}(E)$
    4. 7.4 Presentation as a quotient of $\operatorname{\textup{{RIG}}}(E)$
  8. 8 Adjacency semigroups and bridging path semigroups
  9. 9 Temperley–Lieb monoids
  10. 10 Topological interpretation
    1. 10.1 Special and degenerate linked pairs
    2. 10.2 Graphs and complexes

##  1 Introduction

Set-based free objects exist for many classes of algebras, such as groups,
monoids, lattices, rings and modules. These are typically built from a base
set, and defined in terms of a universal mapping property. Formally, the
existence of such free algebras in a category ${\bf C}$ amounts to the
forgetful functor ${\bf C}\to{\bf Set}$ (which maps an algebra to its
underlying set) having a left adjoint ${\bf Set}\to{\bf C}$; full definitions
will be recalled later in the paper. Such an adjoint exists for example if
${\bf C}$ forms a variety [9, 8, 13], but this is not the case in general,
even for some ‘classical’ algebras such as fields [8, Exercise 2.3]. From the
early days of semigroup theory, it was recognised that the idempotents form a
useful ‘skeleton’ of a semigroup. As the theory developed, it emerged that
this phenomenon was governed by the existence of forgetful functors from
various classes of semigroups into categories of idempotent-like structures
such as semilattices, biordered sets and others. Adjoints of some of these
functors led to important classes of free semigroups, which will be discussed
more below.

The current paper is concerned with _regular $*$-semigroups_. These were
introduced in [58] as an intermediate class between inverse semigroups and
regular semigroups. They have attained prominence recently, as the so-called
diagram monoids come equipped with a natural regular $*$-structure [30, 50,
29, 6, 5, 10]. These monoids are the building blocks of diagram algebras, such
as the Brauer, Temperley–Lieb and partition algebras, among others, which in
turn have important applications in theoretical physics, low-dimensional
topology, representation theory, and many other parts of mathematics and
science [37, 11, 60, 49, 44, 45]. Every regular $*$-semigroup $S$ contains a
set ${\bf P}(S)$ of distinguished idempotents known as projections, which can
be given the structure of a _projection algebra_ [41]. The category ${\bf PA}$
of such algebras played a key role in the groupoid representation of regular
$*$-semigroups in the recent paper [28]. The assignment $S\mapsto{\bf P}(S)$
is a forgetful functor from the category ${\bf RSS}$ of regular $*$-semigroups
to ${\bf PA}$. It turns out that this functor has a left adjoint ${\bf
PA}\to{\bf RSS}$, and this leads to the notion of a _free (projection-
generated) regular $*$-semigroup_ $\operatorname{\textup{{PG}}}(P)$ over a
projection algebra $P$. The purpose of this paper is to show how to construct
these free semigroups, and initiate their systematic study.

The link between regular $*$-semigroups and their projection algebras has many
parallels with the link between arbitrary semigroups and their biordered sets
of idempotents; the latter form the cornerstone of Nambooripad’s theory of
regular semigroups [54]. In that situation we have a forgetful functor ${\bf
E}:{\bf Sgp}\to{\bf BS}$, which maps a semigroup $S$ to its biordered set

$E={\bf E}(S)=\\{e\in S:e^{2}=e\\}.$

The latter has the structure of a partial algebra, where a product $ef$ (for
$e,f\in E$) is only defined if at least one of $ef$ or $fe$ is equal to $e$ or
$f$. A deep result of Easdown and Nambooripad states that ${\bf E}$ has a left
adjoint ${\bf BS}\to{\bf Sgp}$. This formulation can be found in [55, Theorem
6.10], but has its basis in [26, Theorem 3.3]. The adjoint of ${\bf E}$ maps
an (abstract) biordered set $E$ to the _free (idempotent-generated) semigroup_
$\operatorname{\textup{{IG}}}(E)$, which is defined by the presentation

$\operatorname{\textup{{IG}}}(E)=\langle X_{E}:x_{e}x_{f}=x_{ef}\ \text{if
$ef$ is defined in $E$}\rangle,$

where here $X_{E}=\\{x_{e}:e\in E\\}$ is an alphabet in one-one correspondence
with $E$. The key point is that the biordered set of the semigroup
$\operatorname{\textup{{IG}}}(E)$ is precisely $E$, when one identifies $e\in
E$ with the equivalence class of the one-letter word $x_{e}$.

The biorder approach has additional power in the case of regular semigroups.
The restriction of the above forgetful functor ${\bf E}:{\bf Sgp}\to{\bf BS}$
to regular semigroups maps into the category of regular biordered sets, which
were axiomatised by Nambooripad [54], but the adjoint ${\bf BS}\to{\bf Sgp}$
maps regular biordered sets to non-regular semigroups in general. Instead, the
restriction has a different adjoint, mapping $E$ to the _free regular
(idempotent-generated) semigroup_ $\operatorname{\textup{{RIG}}}(E)$; see [54]
for a combinatorial/topological definition, and [59] for a presentation.

The free semigroups $\operatorname{\textup{{IG}}}(E)$ and
$\operatorname{\textup{{RIG}}}(E)$ turn out to have very intricate structure,
and have therefore become a subject of broad interest in their own right [54,
56, 26, 35, 36, 24, 23, 19, 17, 12, 59, 51, 21, 18, 20, 34, 25]. As one strand
of research, it was shown in [12] that maximal subgroups of
$\operatorname{\textup{{IG}}}(E)$ and $\operatorname{\textup{{RIG}}}(E)$ are
(isomorphic to) fundamental groups of certain natural complexes associated to
$E$. The main motivation for this result was to address a folklore conjecture
that such maximal subgroups were always free, and the topological viewpoint
led to the discovery of a biordered set inducing the non-free subgroup
$\mathbb{Z}\times\mathbb{Z}$; see [12, Section 5]. Soon after, the conjecture
was turned upside down, when it was shown in [35] that _every_ group is
isomorphic to a maximal subgroup of some $\operatorname{\textup{{IG}}}(E)$.
Since then, many substantial studies have emerged exploring the structure of
$\operatorname{\textup{{IG}}}(E)$ and $\operatorname{\textup{{RIG}}}(E)$ in
the case that $E={\bf E}(S)$ is the biordered set of some important semigroup
$S$ [36, 23, 17, 21].

We note in passing that biordered sets are not the only structures that have
been used to model the idempotent ‘skeleton’ of semigroups. For example, early
work focussed on the so-called _warp_ of a semigroup $S$ [59, 14, 15, 62],
which was again a partial algebra with underlying set $E={\bf E}(S)$, but
which retained _all_ products $ef$ for which $e,f,ef\in E$. Similarly, the
biordered set can sometimes be given additional structure, as is indeed the
case with regular $*$-semigroups, whose biordered sets have involutions [57].

We now return to our main topic, regular $*$-semigroups and projection
algebras, linked by the forgetful functor ${\bf P}:{\bf RSS}\to{\bf PA}$. The
latter maps a regular $*$-semigroup $S$ to its set

$P={\bf P}(S)=\\{p\in S:p^{2}=p=p^{*}\\}$

of _projections_ , which is then given the structure of a _projection algebra_
, as originally defined (under a different name) by Imaoka [41]. This algebra
has a unary operation $\theta_{p}$ for each $p\in P$, which is defined by
$q\theta_{p}=pqp$ for $q\in P$. Such algebras were axiomatised in [41], where
it was shown that they are the appropriate vehicle for transformation
representations of _fundamental_ regular $*$-semigroups, building on work of
Munn in the inverse case [53]; see also [43].

Projection algebras took on a new level of significance in [28], where they
became the (structured) object sets of so-called _chained projection
groupoids_. The main result [28, Theorem 8.1] is a category isomorphism ${\bf
RSS}\cong{\bf CPG}$, where the latter is the category of all such groupoids.
These groupoids are in fact triples $(P,\mathcal{G},\varepsilon)$, where $P$
is an (abstract) projection algebra, $\mathcal{G}$ is an ordered groupoid
whose structure has tight algebraic and order-theoretic links to $P$, and
$\varepsilon:\mathscr{C}\to\mathcal{G}$ is a functor from a natural _chain
groupoid_ $\mathscr{C}=\mathscr{C}(P)$ built from $P$. Seen through the
groupoid lens, the forgetful functor ${\bf CPG}\to{\bf PA}$ maps
$(P,\mathcal{G},\varepsilon)\mapsto P$. The key construction in the current
paper is a left adjoint ${\bf PA}\to{\bf CPG}$. This maps
$P\mapsto(P,\overline{\mathscr{C}},\nu)$, where $\overline{\mathscr{C}}$ is an
appropriate quotient of $\mathscr{C}$, and
$\nu:\mathscr{C}\to\overline{\mathscr{C}}$ is the quotient map. Topologically,
$\overline{\mathscr{C}}$ is the fundamental groupoid of a natural complex
built from $P$. Applying the isomorphism ${\bf S}:{\bf CPG}\to{\bf RSS}$ from
[28] yields an adjoint ${\bf PA}\to{\bf RSS}$ to the forgetful functor ${\bf
P}:{\bf RSS}\to{\bf PA}$, and hence establishes the existence of the _free
(projection-generated) regular $*$-semigroups_
$\operatorname{\textup{{PG}}}(P)={\bf S}(P,\overline{\mathscr{C}},\nu)$. In
fact, we show that the adjoint is a right inverse of ${\bf P}$, and this has
the consequence that ${\bf PA}$ is _coreflective_ in ${\bf RSS}$ (and in ${\bf
CPG}$). We also remark that the existence of the adjoint answers a question
not settled by the isomorphism ${\bf RSS}\cong{\bf CPG}$ from [28], in that it
shows that every projection algebra $P$ can be realised as $P={\bf P}(S)$ for
some regular $*$-semigroup $S$. This fact was first established by Imaoka in
his above-mentioned work on fundamental semigroups [41]; see also [43].

At this point, the semigroups $\operatorname{\textup{{PG}}}(P)$ become the
subject of the rest of the paper. There are a number of ways to understand
these semigroups, starting from their original definition as _chain
semigroups_ $\operatorname{\textup{{PG}}}(P)={\bf
S}(P,\overline{\mathscr{C}},\nu)$. It is immediate from our construction that
$\operatorname{\textup{{PG}}}(P)$ is generated by $P$. Building on this,
another of our main results establishes a presentation

$\operatorname{\textup{{PG}}}(P)\cong\langle X_{P}:x_{p}^{2}=x_{p},\
(x_{p}x_{q})^{2}=x_{p}x_{q},\ x_{p}x_{q}x_{p}=x_{q\theta_{p}}\ \text{for
}p,q\in P\rangle,$

where $X_{P}=\\{x_{p}:p\in P\\}$ is an alphabet in one-one correspondence with
$P$. Two further presentations allow us to understand the explicit
relationship between the new regular $*$-semigroup
$\operatorname{\textup{{PG}}}(P)$ and the classical free idempotent-generated
semigroups $\operatorname{\textup{{IG}}}(E)$ and
$\operatorname{\textup{{RIG}}}(E)$, where $E={\bf
E}(\operatorname{\textup{{PG}}}(P))$.

As was the case with these classical semigroups, the free regular
$*$-semigroups $\operatorname{\textup{{PG}}}(P)$ are very interesting in their
own right. For example, when $P={\bf P}(A_{\Gamma})$ is the projection algebra
of an adjacency semigroup (in the sense of Jackson and Volkov [42]), the free
semigroup $\operatorname{\textup{{PG}}}(P)$ is an apparently-new graph-based
path semigroup. As another example, it turns out that a finite _Temperley–Lieb
monoid_ $\mathcal{T}\\!\mathcal{L}_{n}$ is a free regular $*$-semigroup over
its own projection algebra ${\bf P}(\mathcal{T}\\!\mathcal{L}_{n})$, giving
yet another fundamental way to understand this important diagram monoid. The
situation for other diagram monoids is more delicate, and is taken up for the
partition monoid in the forthcoming paper [31].

We now give a brief overview of the structure of the paper; the introduction
to each section contains a fuller summary of its contents.

  * •

Sections 2 and 3 contain the preliminary material we need. The former covers
the basics on regular $*$-semigroups, and provides some key examples, namely
adjacency semigroups and diagram monoids. The latter gives an overview of the
relevant constructions and results from [28], concerning regular
$*$-semigroups, projection algebras and chained projection groupoids, leading
up to the isomorphism ${\bf RSS}\cong{\bf CPG}$.

  * •

Sections 4 and 5 are central for this paper: they introduce the semigroups
$\operatorname{\textup{{PG}}}(P)$, and demonstrate their freeness. The former
is the content of Definition 4.21 and Theorem 4.22. The latter is achieved in
Theorem 5.1, which shows that the functor ${{\bf PA}\to{\bf
RSS}:P\mapsto\operatorname{\textup{{PG}}}(P)}$ is indeed a left adjoint to the
forgetful functor ${{\bf RSS}\to{\bf PA}:S\mapsto{\bf P}(S)}$, and also
establishes the coreflectivity of ${\bf PA}$ in ${\bf RSS}$. A number of
semigroup-theoretic consequences are given in Theorems 5.8 and 5.9.

  * •

Section 6 explores the connection between projection algebras and regular
$*$-biordered sets, as defined in [57]. The main result here is Theorem 6.19,
which establishes an equivalence between the categories ${\bf PA}$ and ${\bf
RSBS}$ of all such structures. As a consequence, Theorem 6.20 shows that the
semigroups $\operatorname{\textup{{PG}}}(P)$ are also free with respect to the
forgetful functor ${{\bf RSS}\to{\bf RSBS}:S\mapsto{\bf E}(S)}$. En route, we
show in Proposition 6.10 that the projection algebra of a regular
$*$-semigroup uniquely determines its biordered set, up to isomorphism.

  * •

In Section 7 we give three presentations (by generators and defining
relations) for the semigroups $\operatorname{\textup{{PG}}}(P)$. The first, in
Theorem 7.2, involves $P$ as a generating set. The other two, in Theorems 7.9
and 7.11, utilise the generating set $E={\bf
E}(\operatorname{\textup{{PG}}}(P))$, and exhibit
$\operatorname{\textup{{PG}}}(P)$ as an explicit quotient of
$\operatorname{\textup{{IG}}}(E)$ and of $\operatorname{\textup{{RIG}}}(E)$,
respectively.

  * •

Sections 8 and 9 illustrate our theory on some important examples. In the
former, we will see that the free construction applied to an adjacency
semigroup results in an apparently-new graph-based _bridging path semigroup_ ,
which we believe is worthy of further study. In the latter, we return to
diagram monoids, showing in Theorem 9.1 that the free regular $*$-semigroup
associated to the projection algebra of a finite Temperley–Lieb monoid
$\mathcal{T}\\!\mathcal{L}_{n}$ is, somewhat remarkably, isomorphic to
$\mathcal{T}\\!\mathcal{L}_{n}$ itself.

  * •

Finally, Section 10 provides a topological interpretation of the free regular
$*$-semigroups, with Theorems 10.11 and 10.12 establishing a link with the
fundamental group(oid)s of certain natural graphs and complexes associated to
projection algebras. We conclude the paper by recasting our earlier examples
using this topological framework, and exploring some further ones. As an
intriguing consequence, the semigroup-theoretic structure of the
Temperley–Lieb monoid $\mathcal{T}\\!\mathcal{L}_{n}$ allows us to immediately
deduce that the components of its associated complex are simply connected;
this is not obvious, a priori, as these complexes are highly intricate.

A feature of the work presented here is that it establishes tight connections
between different types of mathematical objects. This has certainly caused
some presentational and notational challenges for the authors. In the hope of
somewhat easing such challenges for the reader, we have collected the key
notation in Tables 1–4. The information presented there might not make much
sense at this stage of reading the paper, but we hope that the reader can use
the tables as a ready reference throughout.

Notation | Specification | Notes | Reference
---|---|---|---
${\bf BS}$ | biordered sets (bosets) |  | Subsection 6.1
${\bf CPG}$ | chained projection groupoids |  isomorphic to ${\bf RSS}$ via ${\bf G}$ and ${\bf S}$ |  Subsection 3.4, Theorem 3.9
${\bf OG}$ | ordered groupoids |  | Subsection 3.3
${\bf PA}$ | projection algebras |  coreflective in ${\bf RSS}$ and in ${\bf CPG}$ |  Subsection 3.2, Theorems 5.1 and 5.2
${\bf RBS}$ | regular bosets |  | Subsection 6.1
${\bf RSBS}$ | regular $*$-bosets |  equivalent to ${\bf PA}$ via ${\bf E}$ and ${\bf P}$ |  Subsection 6.1, Theorem 6.19
${\bf RSS}$ | regular $*$-semigroups |  isomorphic to ${\bf CPG}$ via ${\bf G}$ and ${\bf S}$ |  Subsection 2.1, Theorem 3.9
${\bf Set}$ | sets |  |  Section 1, Remark 5.10
${\bf Sgp}$ | semigroups |  |  Section 1, Remark 5.10
Table 1: Large categories. Notation | Meaning | Notes | Reference
---|---|---|---
$\mathscr{C}:{\bf PA}\rightarrow{\bf OG}$ |  $\mathscr{C}(P)$ – chain groupoid associated with projection algebra $P$ |  | Subsection 3.3
${\bf E}:{\bf RSS}\rightarrow{\bf RSBS}$ |  ${\bf E}(S)$ – boset associated with regular $*$-semigroup $S$ | forgetful | Subsection 6.1
${\bf E}:{\bf PA}\rightarrow{\bf RSBS}$ |  ${\bf E}(P)$ – boset associated with projection algebra $P$ | equivalence | Subsection 6.2
${\bf F}:{\bf PA}\rightarrow{\bf CPG}$ |  ${\bf F}(P)$ – free chained projection groupoid associated with projection algebra $P$ | adjoint | Subsection 4.2
${\bf G}:{\bf RSS}\rightarrow{\bf CPG}$ |  ${\bf G}(S)$ – chained projection groupoid associated with regular $*$-semigroup $S$ | isomorphism | Subsection 3.5
${\bf P}:{\bf RSS}\rightarrow{\bf PA}$ |  ${\bf P}(S)$ – projection algebra of regular $*$-semigroup $S$ | forgetful | Subsection 3.2
${\bf P}:{\bf RSBS}\rightarrow{\bf PA}$ |  ${\bf P}(E)$ – projection algebra of regular $*$-boset $E$ | equivalence | Subsection 3.2
${\bf S}:{\bf CPG}\rightarrow{\bf RSS}$ |  ${\bf S}(P,\mathcal{G},\varepsilon)$ – regular $*$-semigroup associated with chained projection groupoid $(P,\mathcal{G},\varepsilon)$ | isomorphism | Subsection 3.5
Table 2: Functors. Notation | Meaning | Reference
---|---|---
$\mathcal{C}_{P}$ | chain semigroup of $P$, $\mathcal{C}_{P}=\operatorname{\textup{{PG}}}(P)$ | Subsection 4.3
$\mathscr{C}(P)$ | chain groupoid of $P$ | Subsection 3.3
$\overline{\mathscr{C}}(P)$ | reduced chain groupoid of $P$ | Subsection 4.2
$G_{P}$ | graph associated with $P$ | Subsection 10.2
$\operatorname{\textup{{IG}}}(E)$ | free idempotent-generated semigroup over $E$ | Subsection 7.3
$K_{P},K_{P}^{\prime}$ | two complexes associated with $P$ | Subsection 10.2
$\operatorname{\textup{{PG}}}(P)$ |  free projection-generated regular $*$-semigroup over $P$, $\operatorname{\textup{{PG}}}(P)=\mathcal{C}_{P}$ | Section 5
$\mathscr{P}(P)$ | path category of $P$ | Subsection 3.3
$\operatorname{\textup{{RIG}}}(E)$ | free idempotent-generated regular semigroup over $E$ | Subsection 7.4
Table 3: Semigroups, small categories and other structures associated with a projection algebra $P$ or boset $E$. Notation | Name | Reference
---|---|---
$A_{\Gamma}$ | adjacency semigroup of digraph $\Gamma$ | Subsection 2.2
$B_{\Gamma}$ | bridging path semigroup of digraph $\Gamma$ | Section 8
$\mathcal{B}_{n}$ | Brauer monoid | Subsection 2.3
$\mathcal{M}_{n}$ | Motzkin monoid | Subsection 2.3
$\mathcal{P}_{n}$ | partition monoid | Subsection 2.3
$\mathcal{P}\mathcal{B}_{n}$ | partial Brauer monoid | Subsection 2.3
$\mathcal{T}\\!\mathcal{L}_{n}$ | Temperley–Lieb monoid | Subsection 2.3
Table 4: Concrete semigroups.

##  2 Regular $*$-semigroups

###  2.1 Definitions and basic properties

Here we gather the background on regular $*$-semigroups that we will need in
the rest of the paper. For proofs of the various assertions, see for example
[58, 41, 28]. For more on semigroups in general we refer to [39, 16].

A _regular $*$-semigroup_ is a semigroup $S$ with a unary operation
${{}^{*}:S\to S:a\mapsto a^{*}}$ satisfying

$(a^{*})^{*}=a=aa^{*}a\qquad\text{and}\qquad(ab)^{*}=b^{*}a^{*}\qquad\text{for
all $a,b\in S$.}$

From the identity $a=aa^{*}a$, it is clear that a regular $*$-semigroup is
(von Neumann) regular. The identities $(a^{*})^{*}=a$ and
$(ab)^{*}=b^{*}a^{*}$ say that ∗ is an involution.

We write ${\bf RSS}$ for the category of regular $*$-semigroups with
$*$-morphisms, i.e. maps ${\phi:S\to S^{\prime}}$ satisfying

$(ab)\phi=(a\phi)(b\phi)\qquad\text{and}\qquad(a^{*})\phi=(a\phi)^{*}\qquad\text{for
all $a,b\in S$.}$

Given a regular $*$-semigroup $S$, we write

${\bf P}(S)=\\{p\in S:p^{2}=p=p^{*}\\}\qquad\text{and}\qquad{\bf E}(S)=\\{e\in
S:e^{2}=e\\}$

for the sets of all _projections_ and _idempotents_ of $S$, respectively.
Important properties of these elements include the following:

  1. (RS1)

The projections are precisely the elements of the form $aa^{*}$, for $a\in S$.

  2. (RS2)

The product of two projections is an idempotent, but need not be a projection.

  3. (RS3)

Any idempotent $e$ is the product of two projections, namely
$e=(ee^{*})(e^{*}e)$.

  4. (RS4)

The product of two idempotents need not be an idempotent.

  5. (RS5)

For all $p,q\in{\bf P}(S)$ we have $pqp\in{\bf P}(S)$.

  6. (RS6)

More generally, we have $aqa^{*}\in{\bf P}(S)$ for all $a\in S$ and $q\in{\bf
P}(S)$.

Our seventh item is an elaboration on (RS3). For projections $p,q\in{\bf
P}(S)$, we write $p\mathrel{\mathscr{F}}q$ if $p=pqp$ and $q=qpq$, and say
that $p$ and $q$ are _friendly_. It is easy to check that
$ee^{*}\mathrel{\mathscr{F}}e^{*}e$ for any idempotent $e\in{\bf E}(S)$. Thus,
(RS3) says that every idempotent is a product of a pair of
$\mathrel{\mathscr{F}}$-related projections. As noted on [28, p. 20], such
expressions are unique:

  1. (RS7)

$pq=rs\ \Leftrightarrow\ [p=r\text{ and }q=s]$ for all
$(p,q),(r,s)\in{\mathrel{\mathscr{F}}}$.

More generally, any product of idempotents in a regular $*$-semigroup is equal
to a product $p_{1}\cdots p_{k}$ of projections satisfying
$p_{1}\mathrel{\mathscr{F}}\cdots\mathrel{\mathscr{F}}p_{k}$, though such
expressions need not be unique; see [28, Proposition 3.16].

Because of (RS5), each projection $p\in P={\bf P}(S)$ induces a map

$\theta_{p}:P\to P\qquad\text{given by}\qquad q\theta_{p}=pqp\qquad\text{for
$q\in P$.}$ (2.1)

Taking these maps as unary operations gives $P$ the structure of a so-called
_projection algebra_ ; we will discuss these more formally in Section 3, and
will use them extensively in the rest of the paper.

We now describe some examples that we will use to illustrate the ideas
developed in the paper. For more examples, see [28] or [57], but note that
regular $*$-semigroups were called ‘special $*$-semigroups’ in the latter.

###  2.2 Adjacency semigroups

Adjacency semigroups were introduced in [42], and were discussed as a key
example in [28]. They can be viewed in several ways: as combinatorial
completely $0$-simple regular $*$-semigroups, or as symmetrical square
$0$-bands. Here we take the graph theoretic approach of [42]. General
completely $0$-simple regular $*$-semigroups were discussed in detail in [28,
Section 3.3].

Let $\Gamma=(P,E)$ be a symmetric, reflexive digraph, with vertex set $P$, and
edge set $E\subseteq P\times P$. The _adjacency semigroup_ $A_{\Gamma}$ is the
regular $*$-semigroup with:

  * •

underlying set $A_{\Gamma}=(P\times P)\cup\\{0\\}$,

  * •

involution $0^{*}=0$ and $(p,q)^{*}=(q,p)$, and

  * •

product $0^{2}=0=0(p,q)=(p,q)0$ and $(p,q)(r,s)=\begin{cases}(p,s)&\text{if
$(q,r)\in E$}\\\ 0&\text{otherwise.}\end{cases}$

We identify a vertex $p\in P$ with the pair $(p,p)\in A_{\Gamma}$. In this
way, the projections and idempotents of $A_{\Gamma}$ are the sets
$P_{0}=P\cup\\{0\\}$ and $E_{0}=E\cup\\{0\\}$, and we have $(p,q)=pq$ for all
$(p,q)\in E$. The projection algebra operations are given, for $p,q\in P_{0}$,
by

$q\theta_{p}=\begin{cases}p&\text{if $(p,q)\in E$}\\\
0&\text{otherwise.}\end{cases}$ (2.2)

In particular, $\theta_{0}$ is the constant map with image $0$. It also
follows that ${\mathrel{\mathscr{F}}}=E\cup\\{(0,0)\\}$.

###  2.3 Diagram monoids

A key class of examples of regular $*$-semigroups come from the so-called
_diagram monoids_. This class includes important families such as partition
monoids $\mathcal{P}_{n}$, Brauer monoids $\mathcal{B}_{n}$, Temperley–Lieb
monoids $\mathcal{T}\\!\mathcal{L}_{n}$, partial Brauer monoids
$\mathcal{P}\mathcal{B}_{n}$ and Motzkin monoids $\mathcal{M}_{n}$. The
elements of $\mathcal{P}_{n}$ are the set partitions of
$\\{1,\ldots,n\\}\cup\\{1^{\prime},\ldots,n^{\prime}\\}$. These partitions are
represented and multiplied diagrammatically, as in Figure 1; for formal
definitions and an extended discussion see [28]. The involution
$\alpha\mapsto\alpha^{*}$ corresponds to a vertical reflection, as also shown
in Figure 1. The elements of $\mathcal{B}_{n}$ (resp.
$\mathcal{P}\mathcal{B}_{n}$) are the partitions whose blocks have size $2$
(resp. $\leq 2$), while $\mathcal{T}\\!\mathcal{L}_{n}$ (resp.
$\mathcal{M}_{n}$) consists of partitions from $\mathcal{B}_{n}$ (resp.
$\mathcal{P}\mathcal{B}_{n}$) that can be drawn in planar fashion within the
rectangle bounded by the vertices. Thus, in Figure 1 we have
$\beta\in\mathcal{T}\\!\mathcal{L}_{6}$ and $\alpha\beta\in\mathcal{M}_{6}$.

$\alpha=$$\beta=$$=\alpha\beta$

$\alpha=$$=\alpha^{*}$

Figure 1: Diagrammatic representation, multiplication and involution in
$\mathcal{P}_{6}$.

The Temperley–Lieb monoid $\mathcal{T}\\!\mathcal{L}_{n}$ will be considered
in Section 9 as a major example. To study it we will require the following
well-known presentation by generators and relations; see [10, 27] for proofs.

######  Theorem 2.3.

The Temperley–Lieb monoid has monoid presentation

$\mathcal{T}\\!\mathcal{L}_{n}\cong\langle X_{T}:R_{T}\rangle,$

where $X_{T}=\\{t_{1},\ldots,t_{n-1}\\}$, and where $R_{T}$ is the set of
relations

$\displaystyle t_{i}^{2}$ $\displaystyle=t_{i}$ for all $i$ (T1)
$\displaystyle t_{i}t_{j}$ $\displaystyle=t_{j}t_{i}$ if $|i-j|>1$ (T2)
$\displaystyle t_{i}t_{j}t_{i}$ $\displaystyle=t_{i}$ if $|i-j|=1$. (T3)

In the above presentation, the generator $t_{i}$ corresponds to the diagram

$\tau_{i}=$$1$$i$$i\\!\\!+\\!\\!1$$n$. (2.4)

##  3 Projection algebras and chained projection groupoids

In this section we give an overview of the constructions and results we will
need from [28], building up to the isomorphism between the categories of
regular $*$-semigroups and chained projection groupoids; see Theorem 3.9. The
presentation here is necessarily streamlined; for more details, and for proofs
of the various assertions, see [28].

###  3.1 Preliminaries on small categories

Categories considered in the paper come in two kinds: large categories whose
objects and morphisms are algebraic structures and structure-preserving
mappings; and small categories, which are thought of as algebraic objects in
their own right. The former are treated in the usual way; see for example [7,
48]. The current section explains how we view the latter, following [28].

A small category $\mathcal{C}$ will be identified with its morphism set. The
objects of $\mathcal{C}$ are identified with the identities, the set of which
is denoted $v\mathcal{C}$.111The choice of notation alludes to the graph-
theoretic interpretation of objects and morphisms as vertices and edges in a
digraph. This viewpoint will become prominent in the final section of the
paper. The domain and range maps are denoted ${\bf d},{\bf r}:\mathcal{C}\to
v\mathcal{C}$, and we compose morphisms left-to-right, so that $a\circ b$ is
defined when ${\bf r}(a)={\bf d}(b)$, and then ${\bf d}(a\circ b)={\bf d}(a)$
and ${\bf r}(a\circ b)={\bf r}(b)$. For $p,q\in v\mathcal{C}$ we write

$\mathcal{C}(p,q)=\\{a\in\mathcal{C}:{\bf d}(a)=p,\ {\bf r}(a)=q\\}$

for the set of all morphisms $p\to q$.

A _$*$ -category_ is a small category $\mathcal{C}$ with an involution, i.e. a
map ${\mathcal{C}\to\mathcal{C}:a\mapsto a^{*}}$ satisfying the following, for
all $a,b\in\mathcal{C}$:

  * •

${\bf d}(a^{*})={\bf r}(a)$, ${\bf r}(a^{*})={\bf d}(a)$ and $(a^{*})^{*}=a$.

  * •

If ${\bf r}(a)={\bf d}(b)$, then $(a\circ b)^{*}=b^{*}\circ a^{*}$.

A _groupoid_ is a $*$-category for which we additionally have $a\circ
a^{*}={\bf d}(a)$ (and hence also $a^{*}\circ a={\bf r}(a)$) for all
$a\in\mathcal{C}$. In a groupoid, we typically write $a^{*}=a^{-1}$ for
$a\in\mathcal{C}$.

An _ordered $*$-category_ (respectively, _ordered groupoid_) is a $*$-category
(respectively, groupoid) $\mathcal{C}$ equipped with a partial order $\leq$
satisfying the following, for all $a,b,c,d\in\mathcal{C}$ and $p\in
v\mathcal{C}$:

  * •

If $a\leq b$, then ${\bf d}(a)\leq{\bf d}(b)$, ${\bf r}(a)\leq{\bf r}(b)$ and
$a^{*}\leq b^{*}$.

  * •

If $a\leq b$ and $c\leq d$, and if ${\bf r}(a)={\bf d}(c)$ and ${\bf
r}(b)={\bf d}(d)$, then $a\circ c\leq b\circ d$.

  * •

For all $p\leq{\bf d}(a)$ and $q\leq{\bf r}(a)$, there exist unique $u,v\leq
a$ with ${\bf d}(u)=p$ and ${\bf r}(v)=q$. These elements are denoted
$u={}_{p}{\downharpoonleft}a$ and $v=a{\downharpoonright}_{q}$, and are called
the _left_ and _right restrictions_ of $a$ to $p$ and $q$, respectively.

A _congruence_ on a small category $\mathcal{C}$ is an equivalence relation
$\approx$ on $\mathcal{C}$ satisfying the following, for all
$a,b,u,v\in\mathcal{C}$:

  * •

$a\approx b\ \Rightarrow\ [{\bf d}(a)={\bf d}(b)$ and ${\bf r}(a)={\bf
r}(b)]$,

  * •

$a\approx b\ \Rightarrow\ [u\circ a\approx u\circ b$ and $a\circ v\approx
b\circ v]$, whenever the stated compositions are defined.

For a subset $\Omega\subseteq\mathcal{C}\times\mathcal{C}$ with ${\bf
d}(a)={\bf d}(b)$ and ${\bf r}(a)={\bf r}(b)$ for all $(a,b)\in\Omega$, we
write $\Omega^{\sharp}$ for the congruence on $\mathcal{C}$ generated by
$\Omega$.

An _ordered $*$-congruence_ on an ordered $*$-category $\mathcal{C}$ is a
congruence $\approx$ satisfying the following, for all $a,b\in\mathcal{C}$ and
$p\in v\mathcal{C}$:

  * •

$a\approx b\ \Rightarrow\ a^{*}\approx b^{*}$,

  * •

$[a\approx b$ and $p\leq{\bf d}(a)]\ \Rightarrow\
{}_{p}{\downharpoonleft}a\approx{}_{p}{\downharpoonleft}b$.

When ${\approx}=\Omega^{\sharp}$, these two conditions can be verified by
showing that they hold for all $(a,b)\in\Omega$; see [28, Lemma 2.7].

Given an (ordered $*$-) congruence $\approx$ on an (ordered $*$-) category
$\mathcal{C}$, the quotient $\mathcal{C}/{\approx}$ is an (ordered $*$-)
category. Identifying an object $p\in v\mathcal{C}$ with its $\approx$-class,
we have ${v(\mathcal{C}/{\approx})=v\mathcal{C}}$.

###  3.2 Projection algebras

The properties of projections of regular $*$-semigroups are formalised in what
are now known as projection algebras, going back to Imaoka [41], who called
them ‘$P$-groupoids’. Here we recall the definition of these algebras, and
list some known results that will be used later in the paper.

A _projection algebra_ is a unary algebra $P$, with set of operations
$\\{\theta_{p}:p\in P\\}$ in one-one correspondence with $P$, satisfying the
following axioms, for all $p,q\in P$:

  1.   2. (P1)

$p\theta_{p}=p$,

  3. (P2)

$\theta_{p}\theta_{p}=\theta_{p}$,

  4. (P3)

$p\theta_{q}\theta_{p}=q\theta_{p}$,

  5. (P4)

$\theta_{p}\theta_{q}\theta_{p}=\theta_{q\theta_{p}}$,

  6. (P5)

$\theta_{p}\theta_{q}\theta_{p}\theta_{q}=\theta_{p}\theta_{q}$.

The elements of a projection algebra are called _projections_. We write ${\bf
PA}$ for the category of projection algebras with projection algebra
morphisms, defined as maps $\phi:P\to P^{\prime}$ satisfying

$(p\theta_{q})\phi=(p\phi)\theta_{q\phi}\qquad\text{for all $p,q\in P$.}$
(3.1)

Projection algebras can also be thought of as binary algebras, with a single
operation $\diamond$ given by $q\diamond p=q\theta_{p}$. In this formulation,
projection algebra morphisms are simply maps $\phi:P\to P^{\prime}$ satisfying
$(q\diamond p)\phi=(q\phi)\diamond(p\phi)$ for $p,q\in P$. The binary approach
was used in [43], and compared in detail to the unary approach in [28, Remark
4.2].

Given a regular $*$-semigroup $S$, the set of projections

$P={\bf P}(S)=\\{p\in S:p^{2}=p=p^{*}\\}$

becomes a projection algebra, with unary operations $\theta_{p}$ as in (2.1).
Conversely, any projection algebra is the projection algebra of some regular
$*$-semigroup. This latter fact was proved in [41, 43, 28], and it also
follows from Theorem 4.22 below; see also [57]. The assignment $S\mapsto{\bf
P}(S)$ is the object part of a (forgetful) functor

${\bf P}:{\bf RSS}\to{\bf PA}.$ (3.2)

Given a $*$-morphism $\phi:S\to S^{\prime}$, the projection algebra morphism
${\bf P}(\phi):{\bf P}(S)\to{\bf P}(S^{\prime})$ is simply the restriction
${\bf P}(\phi)=\phi|_{{\bf P}(S)}$. It is easy to check that ${\bf P}(\phi)$
is indeed a morphism as defined in (3.1).

A projection algebra $P$ has three associated relations, $\leq$,
$\leq_{\mathrel{\mathscr{F}}}$ and $\mathrel{\mathscr{F}}$, defined for
$p,q\in P$ by

$p\leq q\ \Leftrightarrow\ p=p\theta_{q},\qquad
p\leq_{\mathrel{\mathscr{F}}}q\ \Leftrightarrow\
p=q\theta_{p}\qquad\text{and}\qquad p\mathrel{\mathscr{F}}q\ \Leftrightarrow\
[p\leq_{\mathrel{\mathscr{F}}}q\text{ and }q\leq_{\mathrel{\mathscr{F}}}p].$
(3.3)

The relation $\leq$ is a partial order, and we have $p\leq q\ \Leftrightarrow\
p=r\theta_{q}$ for some $r\in P$. The relation $\leq_{\mathrel{\mathscr{F}}}$
is reflexive, and $\mathrel{\mathscr{F}}$ is reflexive and symmetric; neither
$\leq_{\mathrel{\mathscr{F}}}$ nor $\mathrel{\mathscr{F}}$ is transitive in
general.

We now list some important properties of projection algebras proved in [28,
Section 4]. Specifically, for any projection algebra $P$, and for any
$p,q,r\in P$:

  1. (PA1)

$p\theta_{q}\mathrel{\mathscr{F}}q\theta_{p}$,

  2. (PA2)

$[p\leq q\leq_{\mathrel{\mathscr{F}}}r$ or $p\leq_{\mathrel{\mathscr{F}}}q\leq
r]\ \Rightarrow\ p\leq_{\mathrel{\mathscr{F}}}r$,

  3. (PA3)

$p\leq q\ \Rightarrow\ p\leq_{\mathrel{\mathscr{F}}}q$,

  4. (PA4)

$p\leq q\ \Rightarrow\ \theta_{p}=\theta_{p}\theta_{q}=\theta_{q}\theta_{p}$,

  5. (PA5)

$p\leq_{\mathrel{\mathscr{F}}}q\ \Rightarrow\
\theta_{p}=\theta_{p}\theta_{q}\theta_{p}$.

We will also need the following simple result:

######  Lemma 3.4.

For any $p_{1},\ldots,p_{k},q,r\in P$ we have

$\theta_{q\theta_{p_{1}}\cdots\theta_{p_{k}}}=\theta_{p_{k}}\cdots\theta_{p_{1}}\theta_{q}\theta_{p_{1}}\cdots\theta_{p_{k}}\qquad\text{and}\qquad
r\theta_{p_{k}}\cdots\theta_{p_{1}}\theta_{q}\theta_{p_{1}}\cdots\theta_{p_{k}}\theta_{r}=q\theta_{p_{1}}\cdots\theta_{p_{k}}\theta_{r}.$

###### Proof.

The first claim follows by iterating (P4). The second follows by applying the
first, and then (P3):

$r\theta_{p_{k}}\cdots\theta_{p_{1}}\theta_{q}\theta_{p_{1}}\cdots\theta_{p_{k}}\theta_{r}=r\theta_{q\theta_{p_{1}}\cdots\theta_{p_{k}}}\theta_{r}=q\theta_{p_{1}}\cdots\theta_{p_{k}}\theta_{r}.\qed$

###  3.3 Path categories and chain groupoids

Let $P$ be a projection algebra. A _( $P$-)path_ is a path in the graph of the
$\mathrel{\mathscr{F}}$-relation, i.e. a tuple

$\mathfrak{p}=(p_{1},p_{2},\ldots,p_{k})\in P^{k}\qquad\text{for some $k\geq
1$, such that
$p_{1}\mathrel{\mathscr{F}}p_{2}\mathrel{\mathscr{F}}\cdots\mathrel{\mathscr{F}}p_{k}$.}$

We write ${\bf d}(\mathfrak{p})=p_{1}$ and ${\bf r}(\mathfrak{p})=p_{k}$. We
identify each $p\in P$ with the path $(p)$ of length $1$. The _path category_
of $P$ is the ordered $*$-category $\mathscr{P}=\mathscr{P}(P)$ of all
$P$-paths, with:

  * •

object set $v\mathscr{P}=P$,

  * •

composition
$(p_{1},\ldots,p_{k})\circ(p_{k},\ldots,p_{l})=(p_{1},\ldots,p_{k},\ldots,p_{l})$,

  * •

involution $(p_{1},\ldots,p_{k})^{\textup{rev}}=(p_{k},\ldots,p_{1})$,

  * •

restrictions
${}_{q}{\downharpoonleft}(p_{1},\ldots,p_{k})=(q_{1},\ldots,q_{k})$ and
$(p_{1},\ldots,p_{k}){\downharpoonright}_{r}=(r_{1},\ldots,r_{k})$, for $q\leq
p_{1}$ and $r\leq p_{k}$, where

$q_{i}=q\theta_{p_{2}}\cdots\theta_{p_{i}}=q\theta_{p_{1}}\cdots\theta_{p_{i}}\qquad\text{and}\qquad
r_{i}=r\theta_{p_{k-1}}\cdots\theta_{p_{i}}=r\theta_{p_{k}}\cdots\theta_{p_{i}}\qquad\text{for
$1\leq i\leq k$.}$ (3.5)

Given a projection algebra $P$, we write $\Omega=\Omega(P)$ for the set of all
pairs $(\mathfrak{s},\mathfrak{t})\in\mathscr{P}\times\mathscr{P}$ of the
following two forms:

  1. ($\mathsf{\Omega}$1)

$\mathfrak{s}=(p,p)$ and $\mathfrak{t}=(p)$, for some $p\in P$,

  2. ($\mathsf{\Omega}$2)

$\mathfrak{s}=(p,q,p)$ and $\mathfrak{t}=(p)$, for some
$(p,q)\in{\mathrel{\mathscr{F}}}$,

and we write ${\approx}=\Omega^{\sharp}$ for the congruence on $\mathscr{P}$
generated by $\Omega$. This is an ordered $*$-congruence, and the quotient is
a groupoid, the _chain groupoid_ of $P$:

$\mathscr{C}=\mathscr{C}(P)=\mathscr{P}/{\approx}.$

The elements of $\mathscr{C}$ are called _( $P$-)chains_, and we denote by
$[\mathfrak{p}]$ the chain containing the path ${\mathfrak{p}\in\mathscr{P}}$.

######  Remark 3.6.

Any chain $\mathfrak{c}=[\mathfrak{p}]\in\mathscr{C}$ can be uniquely
represented in the form $[p_{1},\ldots,p_{k}]$, where each $p_{i}$ is distinct
from $p_{i+1}$ (if $i\leq k-1$) and from $p_{i+2}$ (if $i\leq k-2$). This
‘reduced form’ can be found by successively reducing $\mathfrak{p}$, using the
rules

$(p,p)\to(p)\ \ \text{for $p\in P$}\qquad\text{and}\qquad(p,q,p)\to(p)\ \
\text{for $(p,q)\in{\mathrel{\mathscr{F}}}$}.$

One way to establish uniqueness is to show that the rewriting system
$(\mathscr{P},{\to})$ is (locally) confluent and Noetherian, in the sense of
[40].

For any projection algebra morphism $\phi:P\to P^{\prime}$, there is a well-
defined ordered groupoid morphism

$\mathscr{C}(\phi):\mathscr{C}(P)\to\mathscr{C}(P^{\prime})\qquad\text{given
by}\qquad[p_{1},\ldots,p_{k}]\mathscr{C}(\phi)=[p_{1}\phi,\ldots,p_{k}\phi].$
(3.7)

In this way, $\mathscr{C}$ can be viewed as a functor from the category ${\bf
PA}$ of projection algebras to the category ${\bf OG}$ of ordered groupoids.

###  3.4 Chained projection groupoids

A _weak projection groupoid_ is a pair $(P,\mathcal{G})$, consisting of an
ordered groupoid $\mathcal{G}$ whose object set $P$ has the structure of a
projection algebra, and for which the restriction to $P$ of the order on
$\mathcal{G}$ coincides with the projection algebra order $\leq$ from (3.3).
For any $a\in\mathcal{G}$, we have a pair of maps

$\vartheta_{a}:{\bf d}(a)^{\downarrow}\to{\bf r}(a)^{\downarrow}:p\mapsto{\bf
r}({}_{p}{\downharpoonleft}a)\qquad\text{and}\qquad\Theta_{a}=\theta_{{\bf
d}(a)}\vartheta_{a}:P\to{\bf r}(a)^{\downarrow},$

where here $q^{\downarrow}=\\{p\in P:p\leq q\\}$ is the down-set of $q\in P$.
A _projection groupoid_ is a weak projection groupoid $(P,\mathcal{G})$ for
which:

  1. (G1)

$\theta_{p\Theta_{a}}=\Theta_{a^{-1}}\theta_{p}\Theta_{a}$ for all $p\in P$
and $a\in\mathcal{G}$.

Let $(P,\mathcal{G})$ be a projection groupoid. An _evaluation map_ is an
ordered $v$-functor $\varepsilon:\mathscr{C}(P)\to\mathcal{G}$, meaning that
the following hold:

$\varepsilon(p)=p\text{ for $p\in
P$},\quad\varepsilon(\mathfrak{c}\circ\mathfrak{d})=\varepsilon(\mathfrak{c})\circ\varepsilon(\mathfrak{d})\text{
if ${\bf r}(\mathfrak{c})={\bf
d}(\mathfrak{d})$}\quad\text{and}\quad\varepsilon({}_{p}{\downharpoonleft}\mathfrak{c})={}_{p}{\downharpoonleft}(\varepsilon(\mathfrak{c}))\text{
for $p\leq{\bf d}(\mathfrak{c})$.}$

We note in passing that evaluation maps are written to the left of their
arguments, as we feel they are easier to read this way (and never need to be
composed). Since $\mathscr{C}(P)$ is generated by chains of length $2$, the
functor $\varepsilon$ is completely determined by the elements
$\varepsilon[p,q]$ for $(p,q)\in{\mathrel{\mathscr{F}}}$. These elements
feature in the remaining assumptions and constructions involving evaluation
maps.

Given a morphism $b\in\mathcal{G}$, a pair of projections $(e,f)\in P\times P$
is said to be _$b$ -linked_ if

$f=e\Theta_{b}\theta_{f}\qquad\text{and}\qquad e=f\Theta_{b^{-1}}\theta_{e}.$

Given such a $b$-linked pair $(e,f)$, and writing $q={\bf d}(b)$ and $r={\bf
r}(b)$, we define further projections

$e_{1}=e\theta_{q},\qquad e_{2}=f\Theta_{b^{-1}},\qquad
f_{1}=e\Theta_{b}\qquad\text{and}\qquad f_{2}=f\theta_{r}.$

The groupoid $\mathcal{G}$ then contains two well-defined morphisms:

$\displaystyle\lambda(e,b,f)$
$\displaystyle=\varepsilon[e,e_{1}]\circ{}_{e_{1}}{\downharpoonleft}b\circ\varepsilon[f_{1},f]$
and $\displaystyle\rho(e,b,f)$
$\displaystyle=\varepsilon[e,e_{2}]\circ{}_{e_{2}}{\downharpoonleft}b\circ\varepsilon[f_{2},f]$
$\displaystyle=\varepsilon[e,e_{1}]\circ
b{\downharpoonright}_{f_{1}}\circ\varepsilon[f_{1},f]$
$\displaystyle=\varepsilon[e,e_{2}]\circ
b{\downharpoonright}_{f_{2}}\circ\varepsilon[f_{2},f].$

A _chained projection groupoid_ is a triple $(P,\mathcal{G},\varepsilon)$,
where $(P,\mathcal{G})$ is a projection groupoid, and
$\varepsilon:\mathscr{C}(P)\to\mathcal{G}$ is an evaluation map for which:

  1. (G2)

$\lambda(e,b,f)=\rho(e,b,f)$ for every $b\in\mathcal{G}$, and every $b$-linked
pair $(e,f)$.

We write ${\bf CPG}$ for the category of all chained projection groupoids with
_chained projection functors_ as morphisms. A chained projection functor
$(P,\mathcal{G},\varepsilon)\to(P^{\prime},\mathcal{G}^{\prime},\varepsilon^{\prime})$
is an ordered groupoid functor $\phi:\mathcal{G}\to\mathcal{G}^{\prime}$ such
that

  * •

$v\phi=\phi|_{P}:P\to P^{\prime}$ is a projection algebra morphism, and

  * •

$\phi$ respects evaluation maps, in the sense that the following diagram
commutes:

${\mathscr{C}(P)}$ ${\mathscr{C}(P^{\prime})}$ ${\mathcal{G}}$
${\mathcal{G}^{\prime},}$$\scriptstyle{\mathscr{C}(v\phi)}$$\scriptstyle{\varepsilon}$$\scriptstyle{\varepsilon^{\prime}}$$\scriptstyle{\phi}$

where $\mathscr{C}(v\phi)$ is constructed from $v\phi$ as in (3.7).
Explicitly, this is to say that

$(\varepsilon[p_{1},\ldots,p_{k}])\phi=\varepsilon^{\prime}[p_{1}\phi,\ldots,p_{k}\phi]$

whenever $p_{1},\ldots,p_{k}\in P$ and
$p_{1}\mathrel{\mathscr{F}}\cdots\mathrel{\mathscr{F}}p_{k}$.

###  3.5 A category isomorphism

The main result of [28] is that the categories ${\bf RSS}$ and ${\bf CPG}$, of
regular $*$-semigroups and chained projection groupoids, are isomorphic. The
proof involves two functors, ${\bf G}$ and ${\bf S}$, between the two
categories, which operate as follows.

A regular $*$-semigroup $S$ determines a chained projection groupoid ${\bf
G}(S)=(P,\mathcal{G},\varepsilon)$, where:

  * •

$P$ is the projection algebra of $S$.

  * •

$\mathcal{G}$ is an ordered groupoid built from $S$ as follows. The morphisms
are the elements of $S$, and the objects/identities are the projections, with
${\bf d}(a)=aa^{*}$ and ${\bf r}(a)=a^{*}a$ for $a\in S$. The composition and
involution are given by $a\circ b=ab$ when ${\bf r}(a)={\bf d}(b)$, and
$a^{-1}=a^{*}$. Restrictions are given by ${}_{p}{\downharpoonleft}a=pa$ and
$a{\downharpoonright}_{q}=aq$ for $p\leq{\bf d}(a)$ and $q\leq{\bf r}(a)$.

  * •

$\varepsilon:\mathscr{C}(P)\to\mathcal{G}$ is the evaluation map given by
$\varepsilon[p_{1},\ldots,p_{k}]=p_{1}\cdots p_{k}$, where this product is
taken in $S$.

Any $*$-morphism $S\to S^{\prime}$ is also a chained projection functor ${\bf
G}(S)\to{\bf G}(S^{\prime})$.

Conversely, any chained projection groupoid $(P,\mathcal{G},\varepsilon)$
gives rise to a regular $*$-semigroup ${\bf S}(P,\mathcal{G},\varepsilon)$,
with underlying set $\mathcal{G}$, and:

  * •

involution given by $a^{*}=a^{-1}$,

  * •

product defined, for $a,b\in\mathcal{G}$ with ${\bf r}(a)=p$ and ${\bf
d}(b)=q$, by

$a\bullet
b=a{\downharpoonright}_{p^{\prime}}\circ\varepsilon[p^{\prime},q^{\prime}]\circ{}_{q^{\prime}}{\downharpoonleft}b,\qquad\text{where}\qquad
p^{\prime}=q\theta_{p}\quad\text{and}\quad q^{\prime}=p\theta_{q}.$ (3.8)

Any chained projection functor
$(P,\mathcal{G},\varepsilon)\to(P^{\prime},\mathcal{G}^{\prime},\varepsilon^{\prime})$
is also a $*$-morphism ${{\bf S}(P,\mathcal{G},\varepsilon)\to{\bf
S}(P^{\prime},\mathcal{G}^{\prime},\varepsilon^{\prime})}$.

######  Theorem 3.9 (see [28, Theorem 8.1]).

${\bf G}$ and ${\bf S}$ are mutually inverse isomorphisms between the
categories ${\bf RSS}$ and ${\bf CPG}$. ∎

##  4 Construction of the chain semigroup

We now come to the main focus of our study: the chain semigroup
$\mathcal{C}_{P}$ associated to a projection algebra $P$. This semigroup is
built in Subsection 4.3 by first constructing a chained projection groupoid
$(P,\overline{\mathscr{C}},\nu)$, and then applying the functor ${{\bf S}:{\bf
CPG}\to{\bf RSS}}$ from Theorem 3.9. Here $\overline{\mathscr{C}}$ is a
homomorphic image of the chain groupoid $\mathscr{C}$ (see Subsection 4.2),
and $\nu$ is the quotient map $\mathscr{C}\to\overline{\mathscr{C}}$. The
definition of $\overline{\mathscr{C}}$ involves the notion of _linked pairs_
of projections, which are introduced in Subsection 4.1.

###  4.1 Linked pairs of projections

######  Definition 4.1.

Let $P$ be a projection algebra, and let $p\in P$. A pair of projections
$(e,f)\in P\times P$ is said to be _$p$ -linked_ if

$f=e\theta_{p}\theta_{f}\qquad\text{and}\qquad e=f\theta_{p}\theta_{e}.$ (4.2)

Associated to such a $p$-linked pair $(e,f)$ we define the tuples

$\lambda(e,p,f)=(e,e\theta_{p},f)\qquad\text{and}\qquad\rho(e,p,f)=(e,f\theta_{p},f).$

The next two results gather some important basic properties of $p$-linked
pairs. Recall that $\mathscr{P}=\mathscr{P}(P)$ denotes the path category of
$P$.

######  Lemma 4.3.

If $(e,f)$ is $p$-linked, then

  1. (i)

$(f,e)$ is also $p$-linked, and we have
$\lambda(e,p,f)^{\textup{rev}}=\rho(f,p,e)$ and
$\rho(e,p,f)^{\textup{rev}}=\lambda(f,p,e)$,

  2. (ii)

$e,f\leq_{\mathrel{\mathscr{F}}}p$,

  3. (iii)

Each of $e$ and $f$ is $\mathrel{\mathscr{F}}$-related to each of
$e\theta_{p}$ and $f\theta_{p}$; consequently both $\lambda(e,p,f)$ and
$\rho(e,p,f)$ belong to $\mathscr{P}(e,f)$.

###### Proof.

(i). This follows directly by inspecting Definition 4.1.

(ii). By the symmetry afforded by part (i), it suffices to show that
$e\leq_{\mathrel{\mathscr{F}}}p$. For this we use (4.2), Lemma 3.4 and (P3) to
calculate

$p\theta_{e}=p\theta_{f\theta_{p}\theta_{e}}=p\theta_{e}\theta_{p}\theta_{f}\theta_{p}\theta_{e}=e\theta_{p}\theta_{f}\theta_{p}\theta_{e}=f\theta_{p}\theta_{e}=e.$

(iii). By symmetry, it suffices to show that
$e\mathrel{\mathscr{F}}e\theta_{p}\mathrel{\mathscr{F}}f$. Combining
${e\leq_{\mathrel{\mathscr{F}}}p}$ with (PA1), it follows that
$e=p\theta_{e}\mathrel{\mathscr{F}}e\theta_{p}$. We obtain
$f\leq_{\mathrel{\mathscr{F}}}e\theta_{p}$ directly from (4.2). Using (P4) and
(4.2) we calculate
$f\theta_{e\theta_{p}}=f\theta_{p}\theta_{e}\theta_{p}=e\theta_{p}$, so that
$e\theta_{p}\leq_{\mathrel{\mathscr{F}}}f$. ∎

######  Remark 4.4.

Lemma 4.3(iii) says that $(e,f)$ being $p$-linked implies
$e,f\mathrel{\mathscr{F}}e\theta_{p},f\theta_{p}$. The converse of this holds
as well, as (4.2) says that $f\leq_{\mathrel{\mathscr{F}}}e\theta_{p}$ and
$e\leq_{\mathrel{\mathscr{F}}}f\theta_{p}$. Thus, we could take
${e,f\mathrel{\mathscr{F}}e\theta_{p},f\theta_{p}}$ as an equivalent
definition for $(e,f)$ to be $p$-linked.

######  Remark 4.5.

Consider a projection $p\in P$, and a $p$-linked pair $(e,f)$. By Lemma
4.3(ii) we have $e,f\leq_{\mathrel{\mathscr{F}}}p$, and of course we also have
$e\theta_{p},f\theta_{p}\leq p$. These relationships are all shown in Figure
2. In the diagram, each arrow $s\to t$ stands for the $P$-path
$(s,t)\in\mathscr{P}$. Thus, the upper and lower paths from $e$ to $f$
correspond to $\lambda(e,p,f)$ and $\rho(e,p,f)$, respectively.

$e$$p$$e\theta_{p}$$f\theta_{p}$$f$ Figure 2: A projection $p\in P$, and a
$p$-linked pair $(e,f)$, as in Definition 4.1. Dotted and dashed lines
indicate $\leq_{\mathrel{\mathscr{F}}}$ and $\leq$ relationships,
respectively. Lines with arrows indicate the paths $\lambda(e,p,f)$ and
$\rho(e,p,f)$. Of course, by Lemma 4.3(i), the dual diagram in which all four
arrows are reversed is also valid. See Remark 4.5 for more details.

######  Lemma 4.6.

If $(e,f)$ is $p$-linked, and if $e^{\prime}\leq e$, then
$(e^{\prime},f^{\prime})$ is $p$-linked, where
$f^{\prime}=e^{\prime}\theta_{p}\theta_{f}$, and we have

${}_{e^{\prime}}{\downharpoonleft}\lambda(e,p,f)=\lambda(e^{\prime},p,f^{\prime})\qquad\text{and}\qquad{}_{e^{\prime}}{\downharpoonleft}\rho(e,p,f)=\rho(e^{\prime},p,f^{\prime}).$

###### Proof.

To show that $(e^{\prime},f^{\prime})$ is $p$-linked, we must show that
$f^{\prime}=e^{\prime}\theta_{p}\theta_{f^{\prime}}$ and
$e^{\prime}=f^{\prime}\theta_{p}\theta_{e^{\prime}}$. For the first we use the
definition of $f^{\prime}$ and Lemma 3.4 several times to calculate

$e^{\prime}\theta_{p}\theta_{f^{\prime}}=e^{\prime}\theta_{p}\theta_{e^{\prime}\theta_{p}\theta_{f}}=e^{\prime}\theta_{p}\theta_{f}\theta_{p}\theta_{e^{\prime}}\theta_{p}\theta_{f}=f\theta_{p}\theta_{e^{\prime}}\theta_{p}\theta_{f}={e^{\prime}}\theta_{p}\theta_{f}=f^{\prime}.$

For the second we have

$\displaystyle
f^{\prime}\theta_{p}\theta_{e^{\prime}}=e^{\prime}\theta_{p}\theta_{f}\theta_{p}\theta_{e^{\prime}}$
$\displaystyle=f\theta_{p}\theta_{e^{\prime}}$ by definition, and by Lemma 3.4
$\displaystyle=f\theta_{p}\theta_{e}\theta_{e^{\prime}}$ by (PA4), as
$e^{\prime}\leq e$ $\displaystyle=e\theta_{e^{\prime}}$ by (4.2)
$\displaystyle=e^{\prime}$ as $e^{\prime}\leq_{\mathrel{\mathscr{F}}}e$ by
(PA3).

We now show that
${}_{e^{\prime}}{\downharpoonleft}\lambda(e,p,f)=\lambda(e^{\prime},p,f^{\prime})$,
the proof that
${}_{e^{\prime}}{\downharpoonleft}\rho(e,p,f)=\rho(e^{\prime},p,f^{\prime})$
being analogous. Using (3.5) and Definition 4.1, we have

${}_{e^{\prime}}{\downharpoonleft}\lambda(e,p,f)=(e^{\prime},e^{\prime}\theta_{e\theta_{p}},e^{\prime}\theta_{e\theta_{p}}\theta_{f})\qquad\text{and}\qquad\lambda(e^{\prime},p,f^{\prime})=(e^{\prime},e^{\prime}\theta_{p},f^{\prime}).$

Both have first component $e^{\prime}$. We deduce equality of the second and
third components from the following calculations:

  * •

$e^{\prime}\theta_{e\theta_{p}}=e^{\prime}\theta_{p}\theta_{e}\theta_{p}=e^{\prime}\theta_{e}\theta_{p}\theta_{e}\theta_{p}=e^{\prime}\theta_{e}\theta_{p}=e^{\prime}\theta_{p}$,
using $e^{\prime}\leq e$, (P4) and (P5),

  * •

$(e^{\prime}\theta_{e\theta_{p}})\theta_{f}=e^{\prime}\theta_{p}\theta_{f}=f^{\prime}$,
using the previous calculation, and the definition of $f^{\prime}$. ∎

###  4.2 The reduced chain groupoid

We now use linked pairs to define the reduced chain groupoid of a projection
algebra.

######  Definition 4.7.

Let $P$ be a projection algebra, and let $\Xi=\Xi(P)$ be the set of all pairs
${(\mathfrak{s},\mathfrak{t})\in\mathscr{P}\times\mathscr{P}}$ of the forms
($\mathsf{\Omega}$1) and ($\mathsf{\Omega}$2), as well as:

  1. ($\mathsf{\Omega}$3)

$\mathfrak{s}=\lambda(e,p,f)$ and $\mathfrak{t}=\rho(e,p,f)$, for some $p\in
P$, and some $p$-linked pair $(e,f)$.

Let ${\mathrel{\raise
1.07639pt\hbox{\oalign{$\approx$\crcr\kern-3.61664pt\cr$\approx$}}}}=\Xi^{\sharp}$
be the congruence on $\mathscr{P}$ generated by $\Xi$, and define the quotient

$\overline{\mathscr{C}}=\overline{\mathscr{C}}(P)=\mathscr{P}/{\mathrel{\raise
1.07639pt\hbox{\oalign{$\approx$\crcr\kern-3.61664pt\cr$\approx$}}}}.$

As $\Omega\subseteq\Xi$, we have ${\approx}\subseteq{\mathrel{\raise
1.07639pt\hbox{\oalign{$\approx$\crcr\kern-3.61664pt\cr$\approx$}}}}$, and so
$\overline{\mathscr{C}}$ is a quotient of the chain groupoid
$\mathscr{C}=\mathscr{C}(P)=\mathscr{P}/{\approx}$. As such
$\overline{\mathscr{C}}$ is itself a groupoid, which we call the _reduced
chain groupoid of $P$_. The elements of $\overline{\mathscr{C}}$ are called
_reduced ( $P$-)chains_, and we denote by $[[\mathfrak{p}]]$ the reduced chain
containing the path ${\mathfrak{p}\in\mathscr{P}}$. We denote the quotient map
$\mathscr{C}\to\overline{\mathscr{C}}$ by

$\nu:\mathscr{C}\to\overline{\mathscr{C}},\qquad\text{which is given
by}\qquad\nu[\mathfrak{p}]=[[\mathfrak{p}]]\qquad\text{for
$\mathfrak{p}\in\mathscr{P}$.}$

######  Remark 4.8.

We will see in Subsection 10.1 that there exists a natural (and generally
smaller) subset of $\Xi$ that also generates the congruence $\mathrel{\raise
1.07639pt\hbox{\oalign{$\approx$\crcr\kern-3.61664pt\cr$\approx$}}}$. For now
it is more convenient to use $\Xi$, as its definition is more symmetrical.

In Proposition 4.13 we show that $(P,\overline{\mathscr{C}},\nu)$ is a chained
projection groupoid. We build towards this with a number of lemmas.

######  Lemma 4.9.

$\mathrel{\raise
1.07639pt\hbox{\oalign{$\approx$\crcr\kern-3.61664pt\cr$\approx$}}}$ is an
ordered $*$-congruence, and $\overline{\mathscr{C}}$ is an ordered groupoid.

###### Proof.

We have already observed that $\overline{\mathscr{C}}$ is a groupoid. As
explained in Section 3.1, we can show that ${\mathrel{\raise
1.07639pt\hbox{\oalign{$\approx$\crcr\kern-3.61664pt\cr$\approx$}}}}=\Xi^{\sharp}$
is an ordered $*$-congruence by showing that

$\mathfrak{s}^{\textup{rev}}\mathrel{\raise
1.07639pt\hbox{\oalign{$\approx$\crcr\kern-3.61664pt\cr$\approx$}}}\mathfrak{t}^{\textup{rev}}\qquad\text{and}\qquad{}_{p}{\downharpoonleft}\mathfrak{s}\mathrel{\raise
1.07639pt\hbox{\oalign{$\approx$\crcr\kern-3.61664pt\cr$\approx$}}}{}_{p}{\downharpoonleft}\mathfrak{t}\qquad\text{for
all $(\mathfrak{s},\mathfrak{t})\in\Xi$, and all $p\leq{\bf
d}(\mathfrak{s})$.}$

When $(\mathfrak{s},\mathfrak{t})\in\Xi$ has type ($\mathsf{\Omega}$1) or
($\mathsf{\Omega}$2), this was done in [28, Lemma 5.16]. For pairs of type
($\mathsf{\Omega}$3) we apply Lemmas 4.3 and 4.6. ∎

It is clear that $(P,\overline{\mathscr{C}})$ is a weak projection groupoid.
To show that it is a projection groupoid we need to verify (G1). To do so, we
need to understand the maps $\Theta_{\mathfrak{c}}=\theta_{{\bf
d}(\mathfrak{c})}\vartheta_{\mathfrak{c}}$.

######  Lemma 4.10.

For $\mathfrak{c}=[[p_{1},\ldots,p_{k}]]\in\overline{\mathscr{C}}$, we have
$\Theta_{\mathfrak{c}}=\theta_{p_{1}}\cdots\theta_{p_{k}}$.

###### Proof.

Using (3.5), we first calculate

$q\vartheta_{\mathfrak{c}}={\bf
r}({}_{q}{\downharpoonleft}\mathfrak{c})=q\theta_{p_{2}}\cdots\theta_{p_{k}}\qquad\text{for
$q\leq p_{1}={\bf d}(\mathfrak{c})$.}$

It follows from this that $t\Theta_{\mathfrak{c}}=t\theta_{{\bf
d}(\mathfrak{c})}\vartheta_{\mathfrak{c}}=t\theta_{p_{1}}\theta_{p_{2}}\cdots\theta_{p_{k}}$
for arbitrary $t\in P$. ∎

######  Lemma 4.11.

If $P$ is a projection algebra, then $(P,\overline{\mathscr{C}})$ is a
projection groupoid.

###### Proof.

To verify (G1), consider a reduced chain
$\mathfrak{c}=[[p_{1},\ldots,p_{k}]]\in\overline{\mathscr{C}}$, and let $q\in
P$. Then by Lemmas 4.10 and 3.4 we have

$\theta_{q\Theta_{\mathfrak{c}}}=\theta_{q\theta_{p_{1}}\cdots\theta_{p_{k}}}=\theta_{p_{k}}\cdots\theta_{p_{1}}\theta_{q}\theta_{p_{1}}\cdots\theta_{p_{k}}=\Theta_{\mathfrak{c}^{-1}}\theta_{q}\Theta_{\mathfrak{c}}.\qed$

We now bring in the quotient map $\nu:\mathscr{C}\to\overline{\mathscr{C}}$
from Definition 4.7.

######  Lemma 4.12.

If $P$ is a projection algebra, then $\nu$ is an evaluation map.

###### Proof.

Clearly $\nu$ is a $v$-functor. To see that it is ordered, consider a path
$\mathfrak{p}=(p_{1},\ldots,p_{k})\in\mathscr{P}$, and let $q\leq p_{1}$. Then
with the $q_{i}$ as in (3.5), we have

$\nu({}_{q}{\downharpoonleft}[\mathfrak{p}])=\nu[q_{1},\ldots,q_{k}]=[[q_{1},\ldots,q_{k}]]={}_{q}{\downharpoonleft}[[p_{1},\ldots,p_{k}]]={}_{q}{\downharpoonleft}(\nu[\mathfrak{p}]).\qed$

######  Proposition 4.13.

If $P$ is a projection algebra, then $(P,\overline{\mathscr{C}},\nu)$ is a
chained projection groupoid.

###### Proof.

It remains to verify (G2). To do so, fix a $\mathfrak{c}$-linked pair $(e,f)$,
where $\mathfrak{c}=[[p_{1},\ldots,p_{k}]]\in\overline{\mathscr{C}}$. So

$f=e\Theta_{\mathfrak{c}}\theta_{f}\qquad\text{and}\qquad
e=f\Theta_{\mathfrak{c}^{-1}}\theta_{e},$ (4.14)

and we must show that $\lambda(e,\mathfrak{c},f)=\rho(e,\mathfrak{c},f)$.
Keeping $\varepsilon=\nu$ in mind, these morphisms are defined by

$\lambda(e,\mathfrak{c},f)=[[e,e_{1}]]\circ{}_{e_{1}}{\downharpoonleft}\mathfrak{c}\circ[[f_{1},f]]\qquad\text{and}\qquad\rho(e,\mathfrak{c},f)=[[e,e_{2}]]\circ\mathfrak{c}{\downharpoonright}_{f_{2}}\circ[[f_{2},f]],$
(4.15)

in terms of the projections

$e_{1}=e\theta_{p_{1}},\qquad e_{2}=f\Theta_{\mathfrak{c}^{-1}},\qquad
f_{1}=e\Theta_{\mathfrak{c}}\qquad\text{and}\qquad f_{2}=f\theta_{p_{k}}.$

For convenience, we will write

${}_{e_{1}}{\downharpoonleft}\mathfrak{c}=[[u_{1},\ldots,u_{k}]]\qquad\text{and}\qquad\mathfrak{c}{\downharpoonright}_{f_{2}}=[[v_{1},\ldots,v_{k}]].$

Using (3.5), and $e_{1}=e\theta_{p_{1}}$ and $f_{2}=f\theta_{p_{k}}$, we have

$u_{i}=e_{1}\theta_{p_{2}}\cdots\theta_{p_{i}}=e\theta_{p_{1}}\cdots\theta_{p_{i}}\qquad\text{and}\qquad
v_{i}=f_{2}\theta_{p_{k-1}}\cdots\theta_{p_{i}}=f\theta_{p_{k}}\cdots\theta_{p_{i}},$

for each $i$. Keeping in mind that the compositions in (4.15) exist, we have

$\lambda(e,\mathfrak{c},f)=[[e,u_{1},\ldots,u_{k},f]]\qquad\text{and}\qquad\rho(e,\mathfrak{c},f)=[[e,v_{1},\ldots,v_{k},f]],$

and we must show that these are equal. It will also be convenient to
additionally write $u_{0}=e$ and $v_{k+1}=f$. To assist with understanding the
coming arguments, these projections are shown in Figure 3 (in the case $k=4$).
Our task is essentially to show that the large ‘rectangle’ at the bottom of
the diagram commutes, modulo $\mathrel{\raise
1.07639pt\hbox{\oalign{$\approx$\crcr\kern-3.61664pt\cr$\approx$}}}$.

We recall the notion of $p$-linked pairs (Definition 4.1) and claim that:

$(u_{i-1},v_{i+1})\text{ is }p_{i}\text{-linked for each }1\leq i\leq k.$
(4.16)

To prove this, we must show that

$v_{i+1}=u_{i-1}\theta_{p_{i}}\theta_{v_{i+1}}\qquad\text{and}\qquad
u_{i-1}=v_{i+1}\theta_{p_{i}}\theta_{u_{i-1}}.$

First we note that (4.14) and Lemma 4.10 give

$f=e\theta_{p_{1}}\cdots\theta_{p_{k}}\theta_{f}\qquad\text{and}\qquad
e=f\theta_{p_{k}}\cdots\theta_{p_{1}}\theta_{e}.$

Combining this with Lemma 3.4, we obtain

$\displaystyle u_{i-1}\theta_{p_{i}}\theta_{v_{i+1}}$
$\displaystyle=e\theta_{p_{1}}\cdots\theta_{p_{i-1}}\cdot\theta_{p_{i}}\cdot\theta_{f\theta_{p_{k}}\cdots\theta_{p_{i+1}}}$
$\displaystyle=e\theta_{p_{1}}\cdots\theta_{p_{i-1}}\cdot\theta_{p_{i}}\cdot\theta_{p_{i+1}}\cdots\theta_{p_{k}}\theta_{f}\theta_{p_{k}}\cdots\theta_{p_{i+1}}=f\theta_{p_{k}}\cdots\theta_{p_{i+1}}=v_{i+1}.$

The proof that $u_{i-1}=v_{i+1}\theta_{p_{i}}\theta_{u_{i-1}}$ is analogous,
and (4.16) is proved. Now, the linked pairs in (4.16) lead to the paths

$\displaystyle\lambda(u_{i-1},p_{i},v_{i+1})$
$\displaystyle=(u_{i-1},u_{i-1}\theta_{p_{i}},v_{i+1})$ and
$\displaystyle\rho(u_{i-1},p_{i},v_{i+1})$
$\displaystyle=(u_{i-1},v_{i+1}\theta_{p_{i}},v_{i+1})$
$\displaystyle=(u_{i-1},u_{i},v_{i+1})$
$\displaystyle=(u_{i-1},v_{i},v_{i+1}),$

as in Definition 4.1. Since
$(\lambda(u_{i-1},p_{i},v_{i+1}),\rho(u_{i-1},p_{i},v_{i+1}))$ is a pair of
the form ($\mathsf{\Omega}$3), we have

$[[u_{i-1},v_{i},v_{i+1}]]=[[u_{i-1},u_{i},v_{i+1}]]\qquad\text{for each
$1\leq i\leq k$.}$ (4.17)

In other words, each of the small rectangles at the bottom of Figure 3
commutes, and then it follows that the large rectangle commutes. Formally, we
repeatedly use (4.17) in the indicated places, as follows:

$\displaystyle\rho(e,\mathfrak{c},f)=[[e,v_{1},\ldots,v_{k},f]]$
$\displaystyle=[[\underline{u_{0},v_{1},v_{2}},v_{3},v_{4},\ldots,v_{k},v_{k+1}]]$
$\displaystyle=[[u_{0},\underline{u_{1},v_{2},v_{3}},v_{4},\ldots,v_{k},v_{k+1}]]$
$\displaystyle=[[u_{0},u_{1},\underline{u_{2},v_{3},v_{4}},\ldots,v_{k},v_{k+1}]]$
$\displaystyle\hskip 3.69885pt\vdots$
$\displaystyle=[[u_{0},u_{1},u_{2},u_{3},u_{4},\ldots,u_{k},v_{k+1}]]=[[e,u_{1},\ldots,u_{k},f]]=\lambda(e,\mathfrak{c},f),$

and the proof is complete. ∎

${}_{\phantom{0}}e=$$u_{0}$$v_{5}$$=f_{\phantom{0}}$$p_{1}$$u_{1}$$v_{1}$$p_{2}$$u_{2}$$v_{2}$$p_{3}$$u_{3}$$v_{3}$$p_{4}$$u_{4}$$v_{4}$
Figure 3: The projections $e,f,p_{i},u_{i},v_{i}$ from the proof of
Proposition 4.13, shown here in the case $k=4$. Dashed lines indicate $\leq$
relationships. Each arrow $s\to t$ represents the $P$-path
$(s,t)\in\mathscr{P}$, so the upper and lower paths $e\to f$ represent
$\lambda(e,\mathfrak{c},f)=(e,u_{1},\ldots,u_{k},f)$ and
$\rho(e,\mathfrak{c},f)=(e,v_{1},\ldots,v_{k},f)$, respectively.

For a projection algebra $P$, we write ${\bf
F}(P)=(P,\overline{\mathscr{C}},\nu)$ for the chained projection groupoid from
Proposition 4.13. The assignment $P\mapsto{\bf F}(P)$ can be thought of as an
object map ${{\bf PA}\to{\bf CPG}}$. We can extend this to morphisms as well.
Indeed, fix a projection algebra morphism $\phi:P\to P^{\prime}$. In what
follows we use the standard abbreviations for the constructions associated to
$P$, and use dashes to distinguish those for $P^{\prime}$; e.g.
$\mathscr{P}=\mathscr{P}(P)$ and
$\mathscr{C}^{\prime}=\mathscr{C}(P^{\prime})$. We first define a mapping

$\varphi:\mathscr{P}\to\overline{\mathscr{C}}^{\prime}\qquad\text{by}\qquad\mathfrak{p}\varphi=\nu^{\prime}([\mathfrak{p}]\mathscr{C}(\phi)),\qquad\text{i.e.}\qquad(p_{1},\ldots,p_{k})\varphi=[[p_{1}\phi,\ldots,p_{k}\phi]].$
(4.18)

Note that $\varphi$ is the composite of three ordered $*$-functors, namely the
quotient map $\mathscr{P}\to\mathscr{C}$, followed by
${\mathscr{C}(\phi):\mathscr{C}\to\mathscr{C}^{\prime}}$, and then the
evaluation
$\nu^{\prime}:\mathscr{C}^{\prime}\to\overline{\mathscr{C}}^{\prime}$. Hence
$\varphi$ is itself an ordered $*$-functor. It is straightforward to verify
that $\Xi\subseteq\ker(\varphi)$, meaning that
$\mathfrak{s}\varphi=\mathfrak{t}\varphi$ for all
$(\mathfrak{s},\mathfrak{t})\in\Xi$. Therefore there is a well-defined ordered
groupoid functor

$\displaystyle{\bf
F}(\phi):\overline{\mathscr{C}}\to\overline{\mathscr{C}}^{\prime}\qquad\text{given
by}\qquad$ $\displaystyle[[\mathfrak{p}]]{\bf
F}(\phi)=\mathfrak{p}\varphi=\nu^{\prime}([\mathfrak{p}]\mathscr{C}(\phi)),$
i.e. $\displaystyle[[p_{1},\ldots,p_{k}]]{\bf
F}(\phi)=[[p_{1}\phi,\ldots,p_{k}\phi]].$

######  Proposition 4.19.

${\bf F}$ is a functor ${\bf PA}\to{\bf CPG}$.

###### Proof.

We first verify that ${\bf F}(\phi)$, as above, is a chained projection
functor from ${{\bf F}(P)=(P,\overline{\mathscr{C}},\nu)}$ to ${\bf
F}(P^{\prime})=(P^{\prime},\overline{\mathscr{C}}^{\prime},\nu^{\prime})$. The
restriction of ${\bf F}(\phi)$ to $P=v\overline{\mathscr{C}}$ is $\phi$, which
is a projection algebra morphism by assumption. To show that ${\bf F}(\phi)$
preserves evaluation maps we need to check that the following diagram
commutes:

${\mathscr{C}(P)}$ ${\mathscr{C}(P^{\prime})}$ ${\overline{\mathscr{C}}(P)}$
${\overline{\mathscr{C}}(P^{\prime}),}$$\scriptstyle{\mathscr{C}(\phi)}$$\scriptstyle{\nu}$$\scriptstyle{\nu^{\prime}}$$\scriptstyle{{\bf
F}(\phi)}$

and this is routine. It is also routine to check that ${\bf
F}(\phi\circ\phi^{\prime})={\bf F}(\phi)\circ{\bf F}(\phi^{\prime})$ for
composable morphisms $\phi$ and $\phi^{\prime}$, and that ${\bf
F}(\operatorname{id}_{P})=\operatorname{id}_{{\bf F}(P)}$ for all $P$. ∎

###  4.3 The chain semigroup

As a result of Proposition 4.13 and Theorem 3.9, we have a regular
$*$-semigroup ${\bf S}(P,\overline{\mathscr{C}},\nu)$, which we will denote by
$\mathcal{C}_{P}$, and call the _chain semigroup of $P$_. In line with
Subsection 3.5, we denote the product in $\mathcal{C}_{P}$ by $\bullet$. To
give an explicit description of $\bullet$, consider an arbitrary pair of
reduced chains $\mathfrak{c}=[[p_{1},\ldots,p_{k}]]$ and
$\mathfrak{d}=[[q_{1},\ldots,q_{l}]]$, and write $p={\bf
r}(\mathfrak{c})=p_{k}$ and $q={\bf d}(\mathfrak{d})=q_{1}$. As in (3.8), and
remembering that the evaluation map is $\nu$, we have

$\mathfrak{c}\bullet\mathfrak{d}=\mathfrak{c}{\downharpoonright}_{p^{\prime}}\circ[[p^{\prime},q^{\prime}]]\circ{}_{q^{\prime}}{\downharpoonleft}\mathfrak{d}\qquad\text{where}\qquad
p^{\prime}=q\theta_{p}\qquad\text{and}\qquad q^{\prime}=p\theta_{q}.$

Using (3.5), we have
$\mathfrak{c}{\downharpoonright}_{p^{\prime}}=[[p_{1}^{\prime},\ldots,p_{k}^{\prime}]]$
and
${}_{q^{\prime}}{\downharpoonleft}\mathfrak{d}=[[q_{1}^{\prime},\ldots,q_{l}^{\prime}]]$,
where

$p_{i}^{\prime}=p^{\prime}\theta_{p_{k}}\cdots\theta_{p_{i}}\qquad\text{and}\qquad
q_{j}^{\prime}=q^{\prime}\theta_{q_{1}}\cdots\theta_{q_{j}}\qquad\text{for
$1\leq i\leq k$ and $1\leq j\leq l$,}$

and where $p^{\prime}=p_{k}^{\prime}$ and $q^{\prime}=q_{1}^{\prime}$. It
follows that

$\mathfrak{c}\bullet\mathfrak{d}=\mathfrak{c}{\downharpoonright}_{p^{\prime}}\circ[[p^{\prime},q^{\prime}]]\circ{}_{q^{\prime}}{\downharpoonleft}\mathfrak{d}=[[p_{1}^{\prime},\ldots,p_{k}^{\prime}]]\circ[[p_{k}^{\prime},q_{1}^{\prime}]]\circ[[q_{1}^{\prime},\ldots,q_{l}^{\prime}]]=[[p_{1}^{\prime},\ldots,p_{k}^{\prime},q_{1}^{\prime},\ldots,q_{l}^{\prime}]]$

is simply the _concatenation_ of
$\mathfrak{c}{\downharpoonright}_{p^{\prime}}$ and
${}_{q^{\prime}}{\downharpoonleft}\mathfrak{d}$, which we denote by
$\mathfrak{c}{\downharpoonright}_{p^{\prime}}\oplus{}_{q^{\prime}}{\downharpoonleft}\mathfrak{d}$.
As special cases we have

$\mathfrak{c}\bullet\mathfrak{d}=\mathfrak{c}\oplus\mathfrak{d}$ if ${\bf
r}(\mathfrak{c})\mathrel{\mathscr{F}}{\bf d}(\mathfrak{d})$ and
$\mathfrak{c}\bullet\mathfrak{d}=\mathfrak{c}\circ\mathfrak{d}$ if ${\bf
r}(\mathfrak{c})={\bf d}(\mathfrak{d})$. (4.20)

######  Definition 4.21.

The _chain semigroup_ $\mathcal{C}_{P}$ of a projection algebra $P$, is the
regular $*$-semigroup defined as follows.

  1. (CP1)

The elements of $\mathcal{C}_{P}$ are the reduced ($P$-)chains,
$[[p_{1},\ldots,p_{k}]]$, as in Definition 4.7.

  2. (CP2)

The product $\bullet$ in $\mathcal{C}_{P}$ is defined, for
$\mathfrak{c},\mathfrak{d}\in\mathcal{C}_{P}$ with $p={\bf r}(\mathfrak{c})$
and $q={\bf d}(\mathfrak{d})$, by

$\mathfrak{c}\bullet\mathfrak{d}=\mathfrak{c}{\downharpoonright}_{p^{\prime}}\oplus{}_{q^{\prime}}{\downharpoonleft}\mathfrak{d},\qquad\text{where}\qquad
p^{\prime}=q\theta_{p}\quad\text{and}\quad q^{\prime}=p\theta_{q},$

and where $\oplus$ denotes concatenation, as above.

  3. (CP3)

The involution in $\mathcal{C}_{P}$ is given by
$[[p_{1},\ldots,p_{k}]]^{*}=[[p_{k},\ldots,p_{1}]]$.

  4. (CP4)

The projections of $\mathcal{C}_{P}$ have the form $[[p]]=p$, for $p\in P$,
and consequently ${\bf P}(\mathcal{C}_{P})=P$. Moreover, these projection
algebras have the same operations, as

$p\bullet q\bullet p=q\theta_{p}\qquad\text{for all $p,q\in P$.}$

  5. (CP5)

The idempotents of $\mathcal{C}_{P}$ have the form $p\bullet q=[[p,q]]$, for
$(p,q)\in{\mathrel{\mathscr{F}}}$.

We have proved the following.

######  Theorem 4.22.

If $P$ is a projection algebra, then its chain semigroup $\mathcal{C}_{P}$ is
a projection-generated (equivalently, idempotent-generated) regular
$*$-semigroup whose projection algebra is precisely $P$. ∎

We have now introduced the main concept of our study, the chain semigroup
associated to a projection algebra, and the remaining sections of the paper
investigate these semigroups from a number of different angles:

  * •

their incarnation as free objects in the category of regular $*$-semigroups
(Section 5),

  * •

their idempotent/biordered structure (Section 6),

  * •

their presentations by generators and defining relations (Section 7),

  * •

their relationship to free (regular) idempotent-generated semigroups (Section
7), and

  * •

their topological structure (Section 10).

In Sections 8 and 9 we present a number of natural examples.

##  5 Freeness of the chain semigroup

In the previous section we showed how to construct the chain semigroup
$\mathcal{C}_{P}={\bf S}(P,\overline{\mathscr{C}},\nu)$ from a projection
algebra $P$. Now we will explain how $\mathcal{C}_{P}$ is rightfully thought
of as ‘the free regular $*$-semigroup with projection algebra $P$’. In
categorical language, this is to say that the chain semigroups are the objects
in the image of a left adjoint to the forgetful functor ${\bf P}:{\bf
RSS}\to{\bf PA}$ from (3.2). (The precise meanings of these terms are given
below.) The forgetful functor in question maps a regular $*$-semigroup $S$ to
its projection algebra ${\bf P}(S)$. It follows from Proposition 4.19 (and the
isomorphism ${\bf RSS}\cong{\bf CPG}$) that the assignment
$P\mapsto\mathcal{C}_{P}$ is the object part of a functor ${\bf PA}\to{\bf
RSS}$. Our main goal here is to prove the following result (where again
definitions are given below).

######  Theorem 5.1.

The functor ${\bf PA}\to{\bf RSS}:P\mapsto\mathcal{C}_{P}$ is a left adjoint
to the forgetful functor ${\bf RSS}\to{\bf PA}:S\mapsto{\bf P}(S)$, and ${\bf
PA}$ is coreflective in ${\bf RSS}$.

In fact, it will be more convenient to prove the following groupoid version of
Theorem 5.1; the two theorems are equivalent via the isomorphism ${\bf
RSS}\cong{\bf CPG}$.

######  Theorem 5.2.

The functor ${\bf PA}\to{\bf CPG}:P\mapsto(P,\overline{\mathscr{C}},\nu)$ is a
left adjoint to the forgetful functor ${\bf CPG}\to{\bf
PA}:(P,\mathcal{G},\varepsilon)\mapsto P$, and ${\bf PA}$ is coreflective in
${\bf CPG}$.

We now give the (standard) definitions of the terms appearing in the above
results; for more details see [7, 48].

######  Definition 5.3.

Consider two categories ${\bf C}$ and ${\bf D}$, and a pair of functors ${{\bf
F},{\bf G}:{\bf C}\to{\bf D}}$. A _natural transformation_ $\eta:{\bf
F}\to{\bf G}$ is a family $\eta=(\eta_{C})_{C\in v{\bf C}}$, where each
$\eta_{C}:{\bf F}(C)\to{\bf G}(C)$ is a morphism in ${\bf D}$, and such that
the following condition holds:

  * •

For every pair of objects $C,C^{\prime}\in v{\bf C}$, and for every morphism
$\phi:C\to C^{\prime}$ in ${\bf C}$, the following diagram commutes:

${{\bf F}(C)}$ ${{\bf F}(C^{\prime})}$ ${{\bf G}(C)}$ ${{\bf
G}(C^{\prime}).}$$\scriptstyle{{\bf
F}(\phi)}$$\scriptstyle{\eta_{C}}$$\scriptstyle{\eta_{C^{\prime}}}$$\scriptstyle{{\bf
G}(\phi)}$

We call $\eta$ a _natural isomorphism_ if each $\eta_{C}$ is an isomorphism
(in ${\bf D}$), in which case we write ${{\bf F}\cong{\bf G}}$.

######  Definition 5.4.

Consider two categories ${\bf C}$ and ${\bf D}$. An _adjunction_ ${\bf
C}\to{\bf D}$ is a triple $({\bf F},{\bf U},\eta)$, where ${\bf F}:{\bf
C}\to{\bf D}$ and ${\bf U}:{\bf D}\to{\bf C}$ are functors, and $\eta$ is a
natural transformation $\operatorname{id}_{\bf C}\to{\bf U}{\bf F}$, such that
the following condition holds:

  * •

For every pair of objects $C\in v{\bf C}$ and $D\in v{\bf D}$, and for every
morphism $\phi:C\to{\bf U}(D)$ in ${\bf C}$, there exists a unique morphism
$\overline{\phi}:{\bf F}(C)\to D$ in ${\bf D}$ such that the following diagram
commutes:

${C}$ ${{\bf U}({\bf F}(C))}$ ${{\bf
U}(D).}$$\scriptstyle{\phi}$$\scriptstyle{\eta_{C}}$$\scriptstyle{{\bf
U}(\overline{\phi})}$

In this set-up, ${\bf F}$ and ${\bf U}$ are called the _left_ and _right
adjoints_ , respectively, and $\eta$ is the _unit_ of the adjunction. The
_${\bf U}$ -free objects in ${\bf D}$_ are the objects in the image of ${\bf
F}$, i.e. those of the form ${\bf F}(C)$ for $C\in v{\bf C}$.

We are particularly interested in special adjunctions where we actually have
the _equality_ ${{\bf U}{\bf F}=\operatorname{id}_{{\bf C}}}$. Note that
$\operatorname{id}=(\operatorname{id}_{C})_{C\in v{\bf C}}$ is clearly a
natural transformation $\operatorname{id}_{\bf C}\to\operatorname{id}_{\bf C}$
for any category ${\bf C}$. Lemma 5.6 below concerns this situation, and
speaks of so-called _coreflective_ (sub)categories.

######  Definition 5.5.

A _coreflective subcategory_ of a category ${\bf D}$ is a full subcategory
${\bf B}$ whose inclusion functor ${\bf B}\to{\bf D}$ has a right adjoint. We
say a category is _coreflective in ${\bf D}$_ if it is isomorphic to a
coreflective subcategory of ${\bf D}$. That is, ${\bf C}$ is coreflective in
${\bf D}$ if there is a full embedding ${\bf C}\to{\bf D}$ that has a right
adjoint.

In the above, a subcategory ${\bf B}$ of ${\bf D}$ is _full_ if it contains
every morphism of ${\bf D}$ between objects of ${\bf B}$. A _full embedding_
is a functor ${\bf C}\to{\bf D}$ that is injective on objects and morphisms,
and whose image is full in ${\bf D}$.

######  Lemma 5.6.

Suppose ${\bf C}$ and ${\bf D}$ are categories, and ${\bf F}:{\bf C}\to{\bf
D}$ and ${\bf U}:{\bf D}\to{\bf C}$ are functors with ${\bf U}{\bf
F}=\operatorname{id}_{{\bf C}}$, for which the following condition holds:

  * •

For every pair of objects $C\in v{\bf C}$ and $D\in v{\bf D}$, and for every
morphism $\phi:C\to{\bf U}(D)$ in ${\bf C}$, there exists a unique morphism
$\overline{\phi}:{\bf F}(C)\to D$ in ${\bf D}$ such that $\phi={\bf
U}(\overline{\phi})$.

Then

  1. (i)

$({\bf F},{\bf U},\operatorname{id})$ is an adjunction,

  2. (ii)

${\bf C}\cong{\bf F}({\bf C})$ is coreflective in ${\bf D}$.

###### Proof.

(i). This is a direct translation of Definition 5.4 in the special case that
${\bf U}{\bf F}=\operatorname{id}_{\bf C}$ and $\eta=\operatorname{id}$.

(ii). This follows from the fact that all components of the unit
$\operatorname{id}=(\operatorname{id}_{C})$ are isomorphisms; see for example
[48, Theorem IV.3.1]. ∎

###### Proof of Theorem 5.2..

We denote the functors in question by

${\bf F}:{\bf PA}\to{\bf
CPG}:P\mapsto(P,\overline{\mathscr{C}},\nu)\qquad\text{and}\qquad{\bf U}:{\bf
CPG}\to{\bf PA}:(P,\mathcal{G},\varepsilon)\mapsto P.$

We prove the theorem by applying Lemma 5.6. It is clear that ${\bf U}{\bf
F}=\operatorname{id}_{\bf PA}$, so we are left to verify the following
condition:

  * •

For every projection algebra $P$, every chained projection groupoid
$(P^{\prime},\mathcal{G},\varepsilon)$, and every projection algebra morphism
${\phi:P\to{\bf U}(P^{\prime},\mathcal{G},\varepsilon)=P^{\prime}}$, there
exists a unique chained projection functor $\overline{\phi}:{\bf
F}(P)=(P,\overline{\mathscr{C}},\nu)\to(P^{\prime},\mathcal{G},\varepsilon)$
such that $\phi={\bf U}(\overline{\phi})$.

To establish the existence of $\overline{\phi}$, we first define

$\varphi:\mathscr{P}\to\mathcal{G}\qquad\text{by}\qquad\mathfrak{p}\varphi=\varepsilon([\mathfrak{p}]\mathscr{C}(\phi)),\qquad\text{i.e.
}(p_{1},\ldots,p_{k})\varphi=\varepsilon[p_{1}\phi,\ldots,p_{k}\phi]\text{ for
}\mathfrak{p}\in\mathscr{P}.$

where here ${\mathscr{P}=\mathscr{P}(P)}$ is the path category of $P$. As with
(4.18), $\varphi$ is the composite of three ordered $*$-morphisms, and is
hence itself an ordered $*$-morphism. We next show that
$\Xi\subseteq\ker(\varphi)$, i.e. that

$\mathfrak{s}\varphi=\mathfrak{t}\varphi\qquad\text{for all
$(\mathfrak{s},\mathfrak{t})\in\Xi$.}$ (5.7)

This is clear when $(\mathfrak{s},\mathfrak{t})$ has the form
($\mathsf{\Omega}$1) or ($\mathsf{\Omega}$2). Now suppose
$(\mathfrak{s},\mathfrak{t})$ has the form ($\mathsf{\Omega}$3), so that

$\mathfrak{s}=\lambda(e,p,f)=(e,e\theta_{p},f)\quad\text{and}\quad\mathfrak{t}=\rho(e,p,f)=(e,f\theta_{p},f)\qquad\text{for
some $p$-linked pair $(e,f)$.}$

We show that:

  1. (i)

$(e\phi,f\phi)$ is $b$-linked in $\mathcal{G}$ for the morphism
$b=p\phi\in\mathcal{G}$, and

  2. (ii)

$\lambda(e\phi,b,f\phi)=\mathfrak{s}\varphi$ and
$\rho(e\phi,b,f\phi)=\mathfrak{t}\varphi$.

Since $\lambda(e\phi,b,f\phi)=\rho(e\phi,b,f\phi)$ in $\mathcal{G}$, this will
complete the proof that $\mathfrak{s}\varphi=\mathfrak{t}\varphi$. For (i) we
need to show that

$f\phi=(e\phi)\Theta_{b}\theta_{f\phi}\qquad\text{and}\qquad
e\phi=(f\phi)\Theta_{b^{-1}}\theta_{e\phi}.$

Since $p\phi\in P^{\prime}$ is a projection, we have
$\Theta_{b}=\Theta_{p\phi}=\theta_{p\phi}$ by [28, equation (6.3)]. Combined
with the fact that $\phi$ is a projection algebra morphism, it follows that

$(e\phi)\Theta_{b}\theta_{f\phi}=(e\phi)\theta_{p\phi}\theta_{f\phi}=(e\theta_{p}\theta_{f})\phi=f\phi.$

An analogous calculation (keeping in mind $b^{-1}=(p\phi)^{-1}=p\phi$) gives
$e\phi=(f\phi)\Theta_{b^{-1}}\theta_{e\phi}$, and completes the proof of (i).
For (ii) we first note that

$\displaystyle\lambda(e\phi,p\phi,f\phi)$
$\displaystyle=\varepsilon[e\phi,e_{1}]\circ{}_{e_{1}}{\downharpoonleft}(p\phi)\circ\varepsilon[f_{1},f\phi]$
$\displaystyle=\varepsilon[e\phi,e_{1}]\circ
e_{1}\circ\varepsilon[f_{1},f\phi]$
$\displaystyle=\varepsilon[e\phi,e_{1}]\circ\varepsilon[f_{1},f\phi],$

where $e_{1}=(e\phi)\theta_{p\phi}=(e\theta_{p})\phi$ and
$f_{1}=(e\phi)\Theta_{p\phi}=(e\phi)\theta_{p\phi}=(e\theta_{p})\phi\
(=e_{1})$. Thus, continuing from above we have

$\lambda(e\phi,p\phi,f\phi)=\varepsilon[e\phi,e_{1}]\circ\varepsilon[f_{1},f\phi]=\varepsilon[e\phi,(e\theta_{p})\phi]\circ\varepsilon[(e\theta_{p})\phi,f\phi]=\varepsilon[e\phi,(e\theta_{p})\phi,f\phi]=\mathfrak{s}\varphi.$

Analogously, $\rho(e\phi,p\phi,f\phi)=\mathfrak{t}\varphi$, completing the
proof of (ii).

Now that we have proved (5.7), it follows that there is a well-defined ordered
groupoid functor

$\overline{\phi}:\overline{\mathscr{C}}(=\mathscr{P}/{\mathrel{\raise
1.07639pt\hbox{\oalign{$\approx$\crcr\kern-3.61664pt\cr$\approx$}}}})\to\mathcal{G}\qquad\text{given
by}\qquad[[\mathfrak{p}]]\overline{\phi}=\mathfrak{p}\varphi{}=\varepsilon([\mathfrak{p}]\mathscr{C}(\phi))\qquad\text{for
$\mathfrak{p}\in\mathscr{P}$.}$

We next check that $\overline{\phi}$ is in fact a chained projection functor
$(P,\overline{\mathscr{C}},\nu)\to(P^{\prime},\mathcal{G},\varepsilon)$, for
which we need to show that:

  1. (iii)

the object map $v\overline{\phi}=\overline{\phi}|_{P}$ is a projection algebra
morphism $P\to P^{\prime}$, and

  2. (iv)

$\overline{\phi}$ respects the evaluation maps, in the sense that the
following diagram commutes:

${\mathscr{C}(P)}$ ${\mathscr{C}(P^{\prime})}$ ${\overline{\mathscr{C}}}$
${\mathcal{G}.}$$\scriptstyle{\mathscr{C}(v\overline{\phi})}$$\scriptstyle{\nu}$$\scriptstyle{\varepsilon}$$\scriptstyle{\overline{\phi}}$

For (iii), we note that $p\overline{\phi}=p\varphi=\varepsilon[p\phi]=p\phi$
for all $p\in P$, so that $v\overline{\phi}=\phi$ is a projection algebra
morphism by assumption. For (iv), if $\mathfrak{p}\in\mathscr{P}$ then

$\varepsilon([\mathfrak{p}]\mathscr{C}(\phi))=\mathfrak{p}\varphi=[[\mathfrak{p}]]\overline{\phi}=(\nu[\mathfrak{p}])\overline{\phi}.$

So $\overline{\phi}$ is indeed a chained projection functor, and it follows
from the proof of (iii) above that ${\bf
U}(\overline{\phi})=\overline{\phi}|_{P}=\phi$.

We have now established the existence of $\overline{\phi}$. For uniqueness,
suppose the chained projection functor
${\psi:(P,\overline{\mathscr{C}},\nu)\to(P^{\prime},\mathcal{G},\varepsilon)}$
also satisfies ${\bf U}(\psi)=\phi$. Then for any
${\mathfrak{p}\in\mathscr{P}}$ we have

$\displaystyle[[\mathfrak{p}]]\psi=(\nu[\mathfrak{p}])\psi$
$\displaystyle=\varepsilon([\mathfrak{p}]\mathscr{C}(v\psi))$ since $\psi$
respects evaluation maps
$\displaystyle=\varepsilon([\mathfrak{p}]\mathscr{C}(\phi))$ since $v\psi={\bf
U}(\psi)=\phi$ $\displaystyle=[[\mathfrak{p}]]\overline{\phi},$

and so $\psi=\overline{\phi}$. ∎

Now that we have proved Theorem 5.1 (via Theorem 5.2 and the isomorphism ${\bf
RSS}\cong{\bf CPG}$), it follows that the chain semigroups are the ${\bf
P}$-free objects in ${\bf RSS}$. Henceforth, we call $\mathcal{C}_{P}$ the
_free (projection-generated) regular $*$-semigroup over the projection algebra
$P$_. By analogy with the free (idempotent-generated) semigroup over a
biordered set $E$, which is denoted $\operatorname{\textup{{IG}}}(E)$, from
this point forward we will denote $\mathcal{C}_{P}$ by
$\operatorname{\textup{{PG}}}(P)$. The relationship between
$\operatorname{\textup{{PG}}}(P)$ and $\operatorname{\textup{{IG}}}(E)$ will
be a key topic for the remainder of the paper, starting in Section 6.

We conclude the current section by drawing out two purely semigroup-
theoretical results.

######  Theorem 5.8.

If $P$ is a projection algebra, then

  1. (i)

$\operatorname{\textup{{PG}}}(P)$ is a regular $*$-semigroup with projection
algebra $P$,

  2. (ii)

for any regular $*$-semigroup $S$, and any projection algebra morphism
$\phi:P\to{\bf P}(S)$, there is a unique $*$-semigroup homomorphism
${\overline{\phi}:\operatorname{\textup{{PG}}}(P)\to S}$ such that the
following diagram commutes (where both vertical maps are inclusions):

${P}$ ${{\bf P}(S)}$ ${\operatorname{\textup{{PG}}}(P)}$ ${S,}$$\scriptstyle{\
\ \phi}$$\scriptstyle{\overline{\phi}\ \ }$

  1. and moreover any $*$-semigroup homomorphism $\operatorname{\textup{{PG}}}(P)\to S$ has the form $\overline{\phi}$ for some projection algebra morphism $\phi:P\to{\bf P}(S)$.

###### Proof.

(i). This is contained in Theorem 4.22.

(ii). The existence and uniqueness of $\overline{\phi}$ follows from the proof
of Theorem 5.2 and the isomorphism ${\bf RSS}\cong{\bf CPG}$. Given any
$*$-semigroup morphism $\psi:\operatorname{\textup{{PG}}}(P)\to S$, we have
$\psi=\overline{v\psi}$. ∎

The previous theorem has the following consequence:

######  Theorem 5.9.

Let $P$ and $P^{\prime}$ be projection algebras. Then for any projection
algebra morphism $\phi:P\to P^{\prime}$, there is a unique $*$-semigroup
homomorphism
${\overline{\phi}:\operatorname{\textup{{PG}}}(P)\to\operatorname{\textup{{PG}}}(P^{\prime})}$
such that the following diagram commutes (where both vertical maps are
inclusions):

${P}$ ${P^{\prime}}$ ${\operatorname{\textup{{PG}}}(P)}$
${\operatorname{\textup{{PG}}}(P^{\prime}),}$$\scriptstyle{\phi}$$\scriptstyle{\overline{\phi}}$

and moreover any $*$-semigroup homomorphism
$\operatorname{\textup{{PG}}}(P)\to\operatorname{\textup{{PG}}}(P^{\prime})$
has the form $\overline{\phi}$ for some projection algebra morphism $\phi:P\to
P^{\prime}$. ∎

######  Remark 5.10.

Let us pause briefly to illustrate the significance of coreflectivity in
Theorems 5.1 and 5.2, by a comparison with the category ${\bf Sgp}$ of
semigroups and the set-based free objects within it. The latter are the
semigroups $X^{+}$, consisting of all words over a set $X$ under the operation
of concatenation. The assignment $X\mapsto X^{+}$ can be viewed as a functor
${{\bf F}^{\prime}:{\bf Set}\rightarrow{\bf Sgp}}$. It is a left adjoint to
the forgetful functor ${\bf U}^{\prime}:{\bf Sgp}\rightarrow{\bf Set}$, which
maps any semigroup to its underlying set. However, ${\bf U}^{\prime}{\bf
F}^{\prime}$ is not naturally isomorphic, let alone equal, to the identity on
${\bf Sgp}$: indeed, ${\bf U}^{\prime}{\bf F}^{\prime}(X)$ is the underlying
set of the free semigroup $X^{+}$. On the level of morphisms this is
manifested by the fact that not every morphism $X^{+}\rightarrow Y^{+}$ arises
from a mapping $X\rightarrow Y$.

######  Remark 5.11.

Lawson in [47, Theorem 2.2.4] states that groups form a _reflective_
subcategory of inverse semigroups. The _left_ adjoint to the inclusion maps an
inverse semigroup to its maximum group image (the quotient by the least
congruence that identifies all idempotents). On the other hand one can check
from the definitions that semilattices (commutative idempotent semigroups)
form a coreflective subcategory of inverse semigroups. Similarly, the category
of (regular) biordered sets is coreflective in the category of (regular)
semigroups; see [55, Theorem 3.40] and [54, Theorem 6.10]. Coreflectivity of
${\bf PA}$ in ${\bf RSS}$ can be viewed as a ‘regular $*$-analogue’ of these
last two facts.

######  Remark 5.12.

There is another way to view the semigroups
$\operatorname{\textup{{PG}}}(P)=\mathcal{C}_{P}$ as free objects. Namely, for
any (fixed) projection algebra $P$, there is a category ${\bf RSS}(P)$ with:

  * •

objects all regular $*$-semigroups with projection algebra $P$, and

  * •

morphisms all $*$-morphisms that map projections identically.

We see then that $\operatorname{\textup{{PG}}}(P)$ is an initial object in
this category, meaning that for every object $S$ of ${\bf RSS}(P)$ there is
exactly one morphism $\operatorname{\textup{{PG}}}(P)\to S$ (in this
category). This follows by applying Theorem 5.8(ii) to the identity morphism
$\phi=\operatorname{id}_{P}:P\to P={\bf P}(S)$. Note that the image of the
morphism $\operatorname{\textup{{PG}}}(P)\to S$ is the projection-generated
subsemigroup of $S$, which of course is equal to the idempotent-generated
subsemigroup of $S$.

##  6 Projection algebras and biordered sets

We have just seen that the chain semigroups are the ${\bf P}$-free objects in
${\bf RSS}$, where ${{\bf P}:{\bf RSS}\to{\bf PA}}$ is the forgetful functor
that maps a regular $*$-semigroup $S$ to its underlying projection algebra
${\bf P}(S)$. Another forgetful functor ${\bf E}:{\bf RSS}\to{\bf RSBS}$ has
been considered (implicitly) in the literature [57], where ${\bf RSBS}$
denotes the category of _regular $*$-biordered sets_; this will be defined
formally below. It is then natural to ask whether this forgetful functor has
an adjoint, and if so what the ${\bf E}$-free objects are. Perhaps more
fundamentally, we would like to understand the relationship between the
categories ${\bf PA}$ and ${\bf RSBS}$. It turns out that these categories are
in fact _equivalent_ , as we show in Theorem 6.19 below. This then has the
consequence that ${\bf E}$-free objects do indeed exist, but that they are the
same as the ${\bf P}$-free objects.

To establish this equivalence, we need functors

${\bf E}:{\bf PA}\to{\bf RSBS}\qquad\text{and}\qquad{\bf P}:{\bf RSBS}\to{\bf
PA}$

with natural isomorphisms ${\bf P}\circ{\bf E}\cong\operatorname{id}_{\bf PA}$
and ${\bf E}\circ{\bf P}\cong\operatorname{id}_{\bf RSBS}$. These functors are
constructed in Subsections 6.2 and 6.3, and the category equivalence is
established in Subsection 6.4. A salient part of the argument is Proposition
6.10, which shows that regular $*$-semigroups with the same projection algebra
have isomorphic $*$-biordered sets.

###  6.1 Preliminaries on biordered sets

We begin with the necessary definitions; for more background, and proofs, see
[54, Chapter 1] and [57, Section 2].

The set $E={\bf E}(S)=\\{e\in S:e^{2}=e\\}$ of all idempotents of a semigroup
$S$ can be given the structure of a partial algebra called a _biordered set_.
This structure can be conveniently described using Easdown’s arrow notation
[26]. For $e,f\in E$ we write

$e\mathrel{\leavevmode\hbox to14.63pt{\vbox
to5.2pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-2.59998pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{}{{ {\pgfsys@beginscope{{}}
\pgfsys@setdash{}{0.0pt}\pgfsys@roundcap\pgfsys@roundjoin{} {}{}{} {}{}{}
\pgfsys@moveto{2.07999pt}{2.39998pt}\pgfsys@curveto{1.69998pt}{0.95998pt}{0.85318pt}{0.28pt}{0.0pt}{0.0pt}\pgfsys@curveto{0.85318pt}{-0.28pt}{1.69998pt}{-0.95998pt}{2.07999pt}{-2.39998pt}\pgfsys@stroke\pgfsys@endscope}}
}{}{}{{}}\pgfsys@moveto{2.07999pt}{0.0pt}\pgfsys@lineto{14.22638pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{2.27998pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}f\ \Leftrightarrow\
e=ef\text{ ($e$ is a left zero for $f$)}\qquad\text{and}\qquad
e\mathrel{\leavevmode\hbox to14.63pt{\vbox
to5.2pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-2.59998pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{13.82639pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{14.02638pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}f\ \Leftrightarrow\
e=fe\text{ ($e$ is a right zero for $f$)}.$

These relations were originally denoted $\omega^{l}={\mathrel{\leavevmode\hbox
to14.63pt{\vbox to5.2pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-2.59998pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{2.07999pt}{0.0pt}\pgfsys@lineto{14.22638pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{2.27998pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}$ and
$\omega^{r}={\mathrel{\leavevmode\hbox to14.63pt{\vbox
to5.2pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-2.59998pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{13.82639pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{14.02638pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}$ in [54]. Both are
pre-orders (reflexive and transitive relations), and if
$e\mathrel{\leavevmode\hbox to14.63pt{\vbox
to5.2pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-2.59998pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{2.07999pt}{0.0pt}\pgfsys@lineto{14.22638pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{2.27998pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}f$,
$e\mathrel{\leavevmode\hbox to14.63pt{\vbox
to5.2pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-2.59998pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{13.82639pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{14.02638pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}f$,
$e\mathrel{\leavevmode\hbox to14.63pt{\vbox
to5.2pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-2.59998pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{12.1464pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{11.9464pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}f$ or
$e\mathrel{\leavevmode\hbox to14.63pt{\vbox
to5.2pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-2.59998pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{0.4pt}{0.0pt}\pgfsys@lineto{14.22638pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{0.2pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}f$, then $ef$ and
$fe$ are both idempotents (at least one of which is equal to $e$ or $f$). It
follows that we can define a partial operation $\cdot$ on $E$ with domain

$\operatorname{\textup{{BP}}}(E)={\mathrel{\leavevmode\hbox to14.63pt{\vbox
to5.2pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-2.59998pt\hbox
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}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{2.07999pt}{0.0pt}\pgfsys@lineto{14.22638pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{2.27998pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\cup{\mathrel{\leavevmode\hbox
to14.63pt{\vbox to5.2pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-2.59998pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{13.82639pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{14.02638pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\cup{\mathrel{\leavevmode\hbox
to14.63pt{\vbox to5.2pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-2.59998pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{12.1464pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{11.9464pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\cup{\mathrel{\leavevmode\hbox
to14.63pt{\vbox to5.2pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-2.59998pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{0.4pt}{0.0pt}\pgfsys@lineto{14.22638pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{0.2pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}=\\{(e,f)\in
E:\\{e,f\\}\cap\\{ef,fe\\}\not=\varnothing\\},$

and with $e\cdot f=ef$ for $(e,f)\in\operatorname{\textup{{BP}}}(E)$. The
pairs in $\operatorname{\textup{{BP}}}(E)$ are called _basic pairs_. The
partial algebra $(E,{\cdot})$ is the _biordered set_ of $S$, or _boset_ for
short. Since the (partial) product in $E$ is just a restriction of the (total)
product in $S$, we denote it simply by juxtaposition. In what follows, we will
use various combinations of arrows, specifically

${\mathrel{\leavevmode\hbox to14.63pt{\vbox
to5.2pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-2.59998pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{2.07999pt}{0.0pt}\pgfsys@lineto{12.1464pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{2.27998pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{11.9464pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}={\mathrel{\leavevmode\hbox
to14.63pt{\vbox to5.2pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-2.59998pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{2.07999pt}{0.0pt}\pgfsys@lineto{14.22638pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{2.27998pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\cap{\mathrel{\leavevmode\hbox
to14.63pt{\vbox to5.2pt{\pgfpicture\makeatletter\hbox{\hskip
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{12.1464pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{11.9464pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}},\qquad{\mathrel{\leavevmode\hbox
to14.63pt{\vbox to5.2pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-2.59998pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
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{}{}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{0.4pt}{0.0pt}\pgfsys@lineto{13.82639pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{0.2pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{14.02638pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}={\mathrel{\leavevmode\hbox
to14.63pt{\vbox to5.2pt{\pgfpicture\makeatletter\hbox{\hskip
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{13.82639pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{14.02638pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\cap{\mathrel{\leavevmode\hbox
to14.63pt{\vbox to5.2pt{\pgfpicture\makeatletter\hbox{\hskip
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
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{}{}{}{}{}{}{{}}\pgfsys@moveto{0.4pt}{0.0pt}\pgfsys@lineto{14.22638pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
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to14.63pt{\vbox to5.2pt{\pgfpicture\makeatletter\hbox{\hskip
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}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}={\mathrel{\leavevmode\hbox
to14.63pt{\vbox to5.2pt{\pgfpicture\makeatletter\hbox{\hskip
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}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{2.07999pt}{0.0pt}\pgfsys@lineto{14.22638pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{2.27998pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\cap{\mathrel{\leavevmode\hbox
to14.63pt{\vbox to5.2pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-2.59998pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{13.82639pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{14.02638pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}},$

so for example $e\mathrel{\leavevmode\hbox to14.63pt{\vbox
to5.2pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-2.59998pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{2.07999pt}{0.0pt}\pgfsys@lineto{12.1464pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{2.27998pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{11.9464pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}f\ \Leftrightarrow\
[e\mathrel{\leavevmode\hbox to14.63pt{\vbox
to5.2pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-2.59998pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{2.07999pt}{0.0pt}\pgfsys@lineto{14.22638pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{2.27998pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}f$ and
$e\mathrel{\leavevmode\hbox to14.63pt{\vbox
to5.2pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-2.59998pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{12.1464pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{11.9464pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}f]\ \Leftrightarrow\
[e=ef$ and $f=fe]$. Note that $\mathrel{\leavevmode\hbox to14.63pt{\vbox
to5.2pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-2.59998pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{2.07999pt}{0.0pt}\pgfsys@lineto{13.82639pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{2.27998pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{14.02638pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$ is a partial order.

There is an axiomatic definition of abstract bosets (with no reference to any
over-semigroup), but we will not need to give that here, as it is known that
any abstract boset is the boset of idempotents of some semigroup [26].

We denote by ${\bf BS}$ the category of bosets. Morphisms in ${\bf BS}$ are
called _bimorphisms_ ,333The term ‘bimorphism’ comes from Nambooripad’s
original work [54], as a contraction of ‘biordered set morphism’, and should
not be confused with other uses of the term, for example to mean a bijective
morphism of graphs. and are simply morphisms of partial algebras, i.e.
functions $\phi:E\to E^{\prime}$ such that for all $e,f\in E$,

$(e,f)\in\operatorname{\textup{{BP}}}(E)\quad\Rightarrow\quad(e\phi,f\phi)\in\operatorname{\textup{{BP}}}(E^{\prime})\quad\text{and}\quad(ef)\phi=(e\phi)(f\phi).$
(6.1)

An isomorphism in ${\bf BS}$ will be called a _bisomorphism_ ; these are the
bijections $\phi:E\to E^{\prime}$ for which $\phi$ and $\phi^{-1}$ are both
bimorphisms. It is easy to see that a bijection $\phi:E\to E^{\prime}$ is a
bisomorphism if and only if

$\text{$\phi$ is a
bimorphism}\qquad\text{and}\qquad\operatorname{\textup{{BP}}}(E^{\prime})=\operatorname{\textup{{BP}}}(E)\phi=\\{(e\phi,f\phi):(e,f)\in\operatorname{\textup{{BP}}}(E)\\}.$

A boset is called _regular_ if it is the boset of a regular semigroup. Such
regular bosets can be abstractly characterised as the bosets $E$ whose
_sandwich sets_ $\SS(e,f)$ are non-empty for all $e,f\in E$. These sandwich
sets are defined as follows. We begin by defining the set (of ‘mixed’ common
lower bounds):

$\operatorname{\textup{{M}}}(e,f)=\\{g\in E:e\mathrel{\leavevmode\hbox
to14.63pt{\vbox to5.2pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-2.59998pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{12.1464pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{11.9464pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}g\mathrel{\leavevmode\hbox
to14.63pt{\vbox to5.2pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-2.59998pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{13.82639pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{14.02638pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}f\\}=\\{g\in
E:ge=g=fg\\}=\\{g\in E:fge=g\\}.$

This set has a pre-order $\preceq$ defined by $h\preceq g\ \Leftrightarrow\
[eh\mathrel{\leavevmode\hbox to14.63pt{\vbox
to5.2pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-2.59998pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{13.82639pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{14.02638pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}eg$ and
$hf\mathrel{\leavevmode\hbox to14.63pt{\vbox
to5.2pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-2.59998pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{2.07999pt}{0.0pt}\pgfsys@lineto{14.22638pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{2.27998pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}gf]$. The sandwich
set is then

$\SS(e,f)=\\{g\in\operatorname{\textup{{M}}}(e,f):h\preceq g\text{ for all
}h\in\operatorname{\textup{{M}}}(e,f)\\},$

the set of all $\preceq$-maximum elements in
$\operatorname{\textup{{M}}}(e,f)$. It turns out (see [54, Theorem 1.1]) that
if $E$ is a regular boset, and if $S$ is _any_ regular semigroup with $E={\bf
E}(S)$, then

$\SS(e,f)=\\{g\in E:egf=ef\text{ and }fge=g\text{ in }S\\}\qquad\text{for any
$e,f\in E$}.$ (6.2)

Note here that the products $egf$ and $ef$ are taken in $S$, but need not be
defined in $E$ itself; these products need not even be idempotents.

We write ${\bf RBS}$ for the category of regular bosets. Morphisms in ${\bf
RBS}$ are called _regular bimorphisms_ , and are the bimorphisms $\phi:E\to
E^{\prime}$ (as above) that map sandwich sets into sandwich sets, meaning that

$\SS(e,f)\phi\subseteq\SS(e\phi,f\phi)\qquad\text{for all $e,f\in E$.}$

Regular bosets are isomorphic in ${\bf RBS}$ if and only if they are
isomorphic in ${\bf BS}$. This is because sandwich sets are defined directly
from the (partial) ‘multiplication tables’ of bosets, and are hence preserved
by bisomorphisms.

Following [57], a _$*$ -boset_ is a partial algebra $E=(E,{\cdot},{}^{*})$,
where $(E,{\cdot})$ is a boset, and ∗ is a unary operation $E\to E:e\mapsto
e^{*}$ satisfying the following, for all $e,f\in E$:

  1. (SB1)

$(e^{*})^{*}=e$,

  2. (SB2)

$(e,f)\in\operatorname{\textup{{BP}}}(E)\ \Rightarrow\
(e^{*},f^{*})\in\operatorname{\textup{{BP}}}(E)$ and $(ef)^{*}=f^{*}e^{*}$.

We say $E=(E,{\cdot},{}^{*})$ is a _regular $*$-boset_ if $(E,\cdot)$ is
regular, and we additionally have:

  1. (SB3)

For all $e\in E$, there exist elements $s=s(e)$ and $t=t(e)$ of $E$ such that

$e$$s$$t$$e^{*}$,

and such that $e^{*}(gs)=(tg)e^{*}$ for all $g\mathrel{\leavevmode\hbox
to14.63pt{\vbox to5.2pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-2.59998pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{2.07999pt}{0.0pt}\pgfsys@lineto{13.82639pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{2.27998pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{14.02638pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}e$.

(We note that these were called ‘special $*$-biordered sets’ in [57], where
regular $*$-semigroups were also called ‘special $*$-semigroups’. Condition
(SB3) is known as _$\tau$ -commutativity_, as it can be stated in terms of
commutative diagrams involving natural maps between down-sets of idempotents
in the poset $(E,{\mathrel{\leavevmode\hbox to14.63pt{\vbox
to5.2pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-2.59998pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{2.07999pt}{0.0pt}\pgfsys@lineto{13.82639pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{2.27998pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{14.02638pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}})$.)

If $S$ is a regular $*$-semigroup, then the boset $E={\bf E}(S)$ becomes a
regular $*$-boset whose unary operation is the restriction of the involution
of $S$. The elements $s,t\in E$ in (SB3) are $s=ee^{*}$ and $t=e^{*}e$; for
any $g\mathrel{\leavevmode\hbox to14.63pt{\vbox
to5.2pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-2.59998pt\hbox