# Presentations for wreath products involving symmetric inverse monoids and
categories

Chad Clark and James East111Supported by ARC Future Fellowship FT190100632.
Centre for Research in Mathematics and Data Science,
Western Sydney University, Locked Bag 1797, Penrith NSW 2751, Australia.
Chad.Clark @ WesternSydney.edu.au, J.East @ WesternSydney.edu.au

###### Abstract

Wreath products involving symmetric inverse monoids/semigroups/categories
arise in many areas of algebra and science, and presentations by generators
and relations are crucial tools in such studies. The current paper finds such
presentations for $M\wr\mathcal{I}_{n}$,
$M\wr\operatorname{Sing}(\mathcal{I}_{n})$ and $M\wr\mathcal{I}$. Here $M$ is
an arbitrary monoid, $\mathcal{I}_{n}$ is the symmetric inverse monoid,
$\operatorname{Sing}(\mathcal{I}_{n})$ its singular ideal, and $\mathcal{I}$
is the symmetric inverse category.

_Keywords_ : Presentations, wreath products, symmetric inverse
monoids/semigroups/categories.

MSC(2020): 20M05, 20M20, 20M50, 20M18, 18M05.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Categories
    2. 2.2 Symmetric inverse monoids and categories
    3. 2.3 Wreath products
    4. 2.4 Presentations
  3. 3 The monoid $M\wr\mathcal{I}_{n}$
    1. 3.1 Presentations for $\mathcal{I}_{n}$
    2. 3.2 First presentation for $M\wr\mathcal{I}_{n}$
    3. 3.3 Second presentation for $M\wr\mathcal{I}_{n}$
  4. 4 The category $M\wr\mathcal{I}$
    1. 4.1 Category presentation for $M\wr\mathcal{I}$
    2. 4.2 Tensor presentation for $M\wr\mathcal{I}$
  5. 5 The semigroup $M\wr\operatorname{Sing}(\mathcal{I}_{n})$
    1. 5.1 Presentation for $\operatorname{Sing}(\mathcal{I}_{n})$
    2. 5.2 Presentation for $\operatorname{Sing}(M_{0}^{n})$
    3. 5.3 Presentation for $M\wr\operatorname{Sing}(\mathcal{I}_{n})$

##  1 Introduction

Semigroups whose elements are fundamental mathematical entities such as
partitions, transformations and relations have a natural representation using
diagrams. Since the work of Brauer in the 1930s [5], so-called diagram
semigroups have appeared in a number diverse settings, including topology,
invariant theory, quantum mechanics, representation theory, statistical
mechanics and many other branches of mathematics and science. See for example
[24, 31, 22, 34, 52, 1, 30, 37, 38, 8], and especially [38] for an
overview/survey; more examples and references can be found in the
introductions to [17, 19].

Finding presentations of diagram/transformation semigroups and their singular
parts, which are ideals formed by complementing the group of units, has been
of considerable interest since at least the late 1950s. This began with the
work of Aĭzenštat [3, 2] and Popova [44, 45], and can be traced back to
Moore’s presentation for the symmetric group as a Coxeter group [41]; cf.
[29]. With the notable exceptions of the semigroups of (partitioned) binary
relations [43, 39], presentations are known for a great many important diagram
semigroups. These include the full/partial transformation monoids, (dual)
symmetric inverse monoids, Temperley-Lieb/Kauffman monoids, (partial) Brauer
monoids, Motzkin monoids, (rook) partition monoids, and the singular ideals of
many of the above; see for example [4, 3, 2, 44, 45, 14, 32, 36, 18, 16, 15,
25, 17, 21].

The symmetric inverse monoid $\mathcal{I}_{n}$ is the set of all partial
permutations of the set ${{\bf n}=\\{1,\ldots,n\\}}$, under the operation of
relational composition. This monoid plays the same role in the theory of
inverse semigroups as the symmetric group $\mathcal{S}_{n}$ plays in group
theory; for more details on this, see for example [33, 35]. The monoid
$\mathcal{I}_{n}$ is also known in the literature as the rook monoid because
of the alternate characterisation of partial permutations by
$\\{0,1\\}$-matrices with at most one non-zero entry in each row and each
column; such matrices are in one-to-one correspondence with placements of non-
attacking rooks on an $n\times n$ chess board. While the representation theory
of the rook monoid is well studied (see for example [49, 42]), there has also
been substantial recent interest in the representation theory of wreath
products $G\wr\mathcal{I}_{n}$, where $G$ is a group (definitions are given
below). See especially the work of Steinberg [50, 51] and Mazorchuk and
Srivastava [40]. The paper [40] concerns the important special case that
$G=C_{r}$ is a finite cyclic group, which stems from analogous wreath products
$C_{r}\wr\mathcal{S}_{n}$ and their deformations, the cyclotomic Hecke
algebras [10]. See also [6], which computes the congruence lattice of
$G\wr\mathcal{I}_{n}$ for an arbitrary group $G$. As the title of that paper
suggests, $G\wr\mathcal{I}_{n}$ is in fact isomorphic to the monoid of partial
automorphisms of a free $G$-act of rank $n$, providing another source of
motivation for studying such products.

The current article studies more general structures still, and concerns
_transformational_ wreath products $M\wr\mathcal{I}_{n}$ for an arbitrary
_monoid_ $M$, as studied for example in [6, 7] (full definitions are given in
Subsection 2.3). We also consider:

  * •

$M\wr\operatorname{Sing}(\mathcal{I}_{n})$, where
$\operatorname{Sing}(\mathcal{I}_{n})$ is the singular ideal of
$\mathcal{I}_{n}$, consisting of all _strictly_ partial permutations, and

  * •

$M\wr\mathcal{I}$, where $\mathcal{I}$ is the category of all partial
bijections ${\bf m}\to{\bf n}$ where $m$ and $n$ range over
$\mathbb{N}=\\{0,1,\dots\\}$.

Our main results give presentations by generators and relations for each of
these structures. Although several general results exist on presentations of
wreath products of monoids (see [11, 46, 28, 47]), these are not applicable to
$M\wr\mathcal{I}_{n}$ as they concern different kinds of wreath products to
the transformational types considered here. As for $M\wr\mathcal{I}$, we are
not aware of any previous results on presentations for wreath products
involving categories, but we will be aided by tools developed in [17], which
allow presentations for certain classes of categories to be constructed from
presentations of their endomorphism monoids. The situation for
$M\wr\operatorname{Sing}(\mathcal{I}_{n})$ is rather more complex, as
$\operatorname{Sing}(\mathcal{I}_{n})$ is not a monoid, and it is well known
that semigroups tend to behave quite badly even for much simpler constructions
such as Cartesian/direct products [48]. As a case in point, an entire 41-page
paper [20] has been devoted to the related wreath product
$M\wr\operatorname{Sing}(\mathcal{T}_{n})$, where $\mathcal{T}_{n}$ is the
full transformation monoid (consisting of all self-maps of $\mathbf{n}$).
While the recent paper [7] deals with a vast class of semigroup products, and
proves many general results, examples such as
$M\wr\operatorname{Sing}(\mathcal{I}_{n})$ were singled out as a special kind
of difficult case, and none of the results proved in [7] apply here. Thus, we
hope that these methods of working with non-monoid wreath products will be of
use in the study of other such complex structures.

The article is organised as follows. We begin with some preliminary material
in Section 2, which includes the definition of the wreath products that are
our focus. Section 3 gives presentations for the monoid $M\wr\mathcal{I}_{n}$;
see Theorems 3.22 and 3.29, each of which extends a known presentation for
$\mathcal{I}_{n}$ [44, 23]. In Section 4, we apply the results of Section 3
and the general machinery developed in [17] to obtain a category presentation
(Theorem 4.11) of $M\wr\mathcal{I}$, which is then used to obtain a tensor
presentation (Theorem 4.17) for $M\wr\mathcal{I}$. This tensor presentation
involves a particularly compact set of generators and relations, which is
afforded by the tensor (a.k.a monoidal) structure of $M\wr\mathcal{I}$. This
structure allows morphisms between large objects to be built from morphisms
between smaller objects, as made precise below. Finally, we treat the singular
wreath product $M\wr\operatorname{Sing}(\mathcal{I}_{n})$ in Section 5. This
involves the most technically-demanding arguments, and an intermediate result
of independent interest gives a presentation for the (singular) semigroup
$M_{0}^{n}\setminus M^{n}$, where here $M_{0}$ is the semigroup obtained by
adjoining a new zero element to the monoid $M$; see Theorem 5.20. This is then
combined with the presentation for $\operatorname{Sing}(\mathcal{I}_{n})$ from
[14] to prove our final main result, Theorem 5.37, which gives a presentation
for $M\wr\operatorname{Sing}(\mathcal{I}_{n})$.

### Acknowledgement

The authors would like to thank the anonymous referee for their careful
reading of the original manuscript and their valuable suggestions.

##  2 Preliminaries

We require only the most basic notions from semigroup/monoid theory, as may be
found in monographs such as [27, 26, 9]. We now gather the preliminary
material and background results we need concerning categories (Subsection
2.1), symmetric inverse monoids and categories (Subsection 2.2), wreath
products (Subsection 2.3) and presentations (Subsection 2.4).

###  2.1 Categories

As in [17], we are interested in very special kinds of categories. Throughout,
the term _category_ will always mean a small, strict tensor (a.k.a. monoidal)
category with object set $\mathbb{N}$; these terms will all be defined below.
Any such category $\mathcal{C}$ will be identified with its morphism set. The
domain and range of $a\in\mathcal{C}$ are denoted by $\mathbf{d}(a)$ and
$\mathbf{r}(a)$, respectively. Morphisms are composed left-to-right, so that
$ab=a\circ b$ is defined if and only if $\mathbf{r}(a)=\mathbf{d}(b)$, in
which case $\mathbf{d}(ab)=\mathbf{d}(a)$ and $\mathbf{r}(ab)=\mathbf{r}(b)$.
For $m,n\in\mathbb{N}$ we write
$\mathcal{C}_{m,n}=\\{a\in\mathcal{C}:\mathbf{d}(a)=m,\ \mathbf{r}(a)=n\\}$
for the set of all morphisms $m\to n$. For $n\in\mathbb{N}$ we write
$\mathcal{C}_{n}=\mathcal{C}_{n,n}$ for the endomorphism monoid at the object
$n$, and we denote the identity of this monoid by $\iota_{n}$ (or a similar
symbol). To say that $\mathcal{C}$ is a _strict tensor category_ means that it
has an additional (totally defined) operation
$\oplus:\mathcal{C}\times\mathcal{C}\to\mathcal{C}$ for which

  * •

$(\mathcal{C},\oplus)$ is a monoid with identity $\iota_{0}$,

  * •

$\mathbf{d}(a\oplus b)=\mathbf{d}(a)+\mathbf{d}(b)$ and $\mathbf{r}(a\oplus
b)=\mathbf{r}(a)+\mathbf{r}(b)$ for all $a,b\in\mathcal{C}$,

  * •

$\iota_{m}\oplus\iota_{n}=\iota_{m+n}$ for all $m,n\in\mathbb{N}$,

  * •

$(a\circ b)\oplus(c\circ d)=(a\oplus c)\circ(b\oplus d)$ for all
$a,b,c,d\in\mathcal{C}$ with $\mathbf{r}(a)=\mathbf{d}(b)$ and
$\mathbf{r}(c)=\mathbf{d}(d)$.

A _congruence_ on a category $\mathcal{C}$ (as above) is an equivalence
relation $\sigma$ on $\mathcal{C}$ for which

  * •

$(a,b)\in\sigma\ \Rightarrow\ \mathbf{d}(a)=\mathbf{d}(b)$ and
$\mathbf{r}(a)=\mathbf{r}(b)$ for all $a,b\in\mathcal{C}$,

  * •

$(a,b)\in\sigma\ \Rightarrow\ (xa,xb),(ay,by)\in\sigma$ for all
$a,b,x,y\in\mathcal{C}$ whenever the stated products are defined.

The _quotient category_ $\mathcal{C}/\sigma$ is the set of all
$\sigma$-classes under the induced operation. We say $\sigma$ is a _tensor
congruence_ if it also preserves $\oplus$, meaning that

  * •

$(a,b)\in\sigma\ \Rightarrow\ (x\oplus a,x\oplus b),(a\oplus y,b\oplus
y)\in\sigma$ for all $a,b,x,y\in\mathcal{C}$.

For a set $\Omega\subseteq\mathcal{C}\times\mathcal{C}$ satisfying
$(a,b)\in\Omega\ \Rightarrow\ \mathbf{d}(a)=\mathbf{d}(b)$ and
$\mathbf{r}(a)=\mathbf{r}(b)$, we denote by $\Omega^{\sharp}$ the congruence
on $\mathcal{C}$ generated by $\Omega$. So $\Omega^{\sharp}$ is the smallest
congruence on $\mathcal{C}$ containing $\Omega$. We also denote by
$\Omega_{\scriptscriptstyle{\oplus}}^{\sharp}$ the tensor congruence generated
by $\Omega$; note that
$\Omega^{\sharp}\subseteq\Omega_{\scriptscriptstyle{\oplus}}^{\sharp}$, but
the inclusion is sometimes strict.

Given categories $\mathcal{C}$ and $\mathcal{D}$ (as above), a _category
morphism_ $\phi:\mathcal{C}\to\mathcal{D}$ is a functor that acts as the
identity on objects, meaning that $\mathbf{d}(a\phi)=\mathbf{d}(a)$ and
$\mathbf{r}(a\phi)=\mathbf{r}(a)$ for all $a\in\mathcal{C}$. The _kernel_ of
$\phi$ is the congruence

$\ker(\phi)=\big{\\{}(a,b)\in\mathcal{C}\times\mathcal{C}:a\phi=b\phi\big{\\}},$

and we have $\mathcal{C}/{\ker(\phi)}\cong\operatorname{im}(\phi)$.

###  2.2 Symmetric inverse monoids and categories

A particularly important category for us is the _symmetric inverse category_
$\mathcal{I}$, defined as follows. For $n\in\mathbb{N}$ let ${\bf
n}=\\{1,\ldots,n\\}$, interpreting this to be empty when $n=0$. For
$m,n\in\mathbb{N}$ we denote by $\mathcal{I}_{m,n}$ the set of all injective
partial maps ${\bf m}\to{\bf n}$: i.e., all injective functions $A\to{\bf n}$
with $A\subseteq{\bf m}$. We then define

$\mathcal{I}=\big{\\{}(m,\alpha,n):m,n\in\mathbb{N},\
\alpha\in\mathcal{I}_{m,n}\big{\\}}.$

For $m,n,k\in\mathbb{N}$, and for $\alpha\in\mathcal{I}_{m,n}$ and
$\beta\in\mathcal{I}_{n,k}$, we define

$\mathbf{d}(m,\alpha,n)=m,\qquad\mathbf{r}(m,\alpha,n)=n\qquad\text{and}\qquad(m,\alpha,n)\circ(n,\beta,k)=(m,\alpha\beta,k),$

where $\alpha\beta=\alpha\circ\beta\in\mathcal{I}_{m,k}$ is the ordinary
relational composition. To avoid clutter in our notation, we will typically
identify an element $(m,\alpha,n)$ of $\mathcal{I}$ with the partial map
$\alpha\in\mathcal{I}_{m,n}$ itself, but regard $m$ and $n$ as ‘encoded’ in
$\alpha$, writing $\mathbf{d}(\alpha)=m$ and $\mathbf{r}(\alpha)=n$. In this
way, the morphism sets of $\mathcal{I}$ are the $\mathcal{I}_{m,n}$
($m,n\in\mathbb{N}$), and the endomorphism monoids are the _symmetric inverse
monoids_ $\mathcal{I}_{n}=\mathcal{I}_{n,n}$.

A partial bijection $\alpha\in\mathcal{I}_{m,n}$ is typically represented by a
graph consisting of a row of upper vertices labelled $1,\ldots,m$, a row of
lower vertices labelled $1,\ldots,n$, and an edge from upper vertex $i$ to
lower vertex $j$ if $i\in\operatorname{dom}(\alpha)$ and $i\alpha=j$. Here as
usual $\operatorname{dom}(\alpha)$ denotes the domain of $\alpha$; we also
write $\operatorname{im}(\alpha)$ for the image of $\alpha$. Figure 1 gives
representations of the partial bijections

$\alpha=\left(\begin{smallmatrix}1&2&3&4&5&6\\\
-&-&4&1&-&7\end{smallmatrix}\right)\in\mathcal{I}_{6,8}\qquad\text{and}\qquad\beta=\left(\begin{smallmatrix}1&2&3&4&5&6&7&8\\\
2&-&1&-&5&-&4&-\end{smallmatrix}\right)\in\mathcal{I}_{8,7},$

as well as their composition
$\alpha\beta=\left(\begin{smallmatrix}1&2&3&4&5&6\\\
-&-&-&2&-&4\end{smallmatrix}\right)\in\mathcal{I}_{6,7}$, noting that an entry
of ‘$-$’ in position $i$ indicates that $i$ does not belong to the domain of
the mapping. This diagrammatic interpretation of composition in $\mathcal{I}$
will help with many of the calculations to follow. We assume, unless otherwise
specified, that the vertices in each row are labelled in left-to-right
ascending order and generally omit these labels for convenience.

$\alpha=$$\beta=$$=\alpha\beta$ Figure 1: Partial bijections
$\alpha\in\mathcal{I}_{6,8}$ and $\beta\in\mathcal{I}_{8,7}$, and their
composition $\alpha\beta\in\mathcal{I}_{6,7}$.

The category $\mathcal{I}$ also has a tensor operation. For
$\alpha\in\mathcal{I}_{m,n}$ and $\beta\in\mathcal{I}_{k,l}$, we define the
mapping ${\alpha\oplus\beta\in\mathcal{I}_{m+k,n+l}}$ by

$x(\alpha\oplus\beta)=\begin{cases}x\alpha&\text{if $1\leq x\leq m$ and
$x\in\operatorname{dom}(\alpha)$,}\\\ (x-m)\beta+n&\text{if $m+1\leq x\leq
m+k$ and $x-m\in\operatorname{dom}(\beta)$,}\\\
-&\text{otherwise.}\end{cases}$

Diagrammatically, $\alpha\oplus\beta$ is obtained by stacking $\alpha$ and
$\beta$ horizontally, as in Figure 2.

${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\alpha}=$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\beta}=$$={\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\alpha}\oplus{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\beta}$
Figure 2: Partial bijections $\alpha\in\mathcal{I}_{6,8}$ and
$\beta\in\mathcal{I}_{7,6}$, and their sum
$\alpha\oplus\beta\in\mathcal{I}_{13,14}$.

We have already noted that the endomorphism monoids of the category
$\mathcal{I}$ are the symmetric inverse monoids
$\mathcal{I}_{n}=\mathcal{I}_{n,n}$. The group of units of $\mathcal{I}_{n}$
is the _symmetric group_ $\mathcal{S}_{n}$. Because the monoids here are
finite, the singular part
$\operatorname{Sing}(\mathcal{I}_{n})=\mathcal{I}_{n}\setminus\mathcal{S}_{n}$
is itself a semigroup; indeed, it is an ideal of $\mathcal{I}_{n}$. The set
$\mathcal{S}=\bigcup_{n\in\mathbb{N}}\mathcal{S}_{n}$ of all units of
$\mathcal{I}$ is of course a subcategory, but
$\operatorname{Sing}(\mathcal{I})=\mathcal{I}\setminus\mathcal{S}$ is not a
category; for example, if $\alpha$ and $\beta$ denote the unique elements of
$\mathcal{I}_{0,1}$ and $\mathcal{I}_{1,0}$, respectively, then
$\alpha,\beta\in\operatorname{Sing}(\mathcal{I})$, while
$\alpha\beta=\varnothing$ is the unique element of
$\mathcal{I}_{0}=\mathcal{S}_{0}$.

###  2.3 Wreath products

In this subsection we define the transformational wreath products that will be
the objects of our study.

Let $M$ be a monoid with identity $1$ and let $0$ be a symbol not belonging to
$M$. We denote by $M_{0}=M\cup\\{0\\}$ the monoid obtained from $M$ by
adjoining $0$ as a multiplicative zero element, even if $M$ already had a
multiplicative zero element. For $n\in\mathbb{N}$ we write $M_{0}^{n}$ for the
set of all $n$-tuples over $M_{0}$, which is a monoid with identity ${\bf
1}_{n}=(1,\ldots,1)$. Note that $M_{0}^{0}=\\{\varnothing\\}$ consists only of
the empty tuple. We also write

$\mathcal{M}=M_{0}^{0}\cup M_{0}^{1}\cup M_{0}^{2}\cup\cdots$

for the set of all finite tuples over $M_{0}$, which is trivially a category
(in our sense). For ${\bf a}\in\mathcal{M}$ we write $\mathbf{d}({\bf
a})=\mathbf{r}({\bf a})$ for the unique $n\in\mathbb{N}$ such that ${\bf a}\in
M_{0}^{n}$. For ${\bf a}=(a_{1},\ldots,a_{m})\in M_{0}^{m}$ and ${\bf
b}=(b_{1},\ldots,b_{n})\in M_{0}^{n}$ we define ${\bf a}\oplus{\bf
b}=(a_{1},\ldots,a_{m},b_{1},\ldots,b_{n})\in M_{0}^{m+n}$.

The _support_ of ${\bf a}=(a_{1},\ldots,a_{n})\in M_{0}^{n}$ is defined by

$\operatorname{supp}({\bf a})=\\{i\in{\bf n}:a_{i}\not=0\\}.$

Since $\operatorname{supp}({\bf a}\cdot{\bf b})=\operatorname{supp}({\bf
a})\cap\operatorname{supp}({\bf b})$ for all ${\bf a},{\bf b}\in M_{0}^{n}$,
it follows that $\operatorname{supp}:M_{0}^{n}\to 2^{{\bf n}}$ is in fact a
monoid (sur)morphism, where $2^{\bf n}$ denotes the power set
$\\{A:A\subseteq{\bf n}\\}$, considered as a semilattice (semigroup of
commuting idempotents) under $\cap$.

Let $\alpha\in\mathcal{I}_{m,n}$ and let ${\bf a}=(a_{1},\ldots,a_{n})\in
M_{0}^{n}$. We define

${}^{\alpha}{{\bf a}}=(b_{1},\ldots,b_{m})\in
M_{0}^{m}\qquad\text{where}\qquad b_{i}=\begin{cases}a_{i\alpha}&\text{if
$i\in\operatorname{dom}(\alpha)$}\\\ 0&\text{otherwise.}\end{cases}$

Note that ${}^{\alpha}{\bf a}$ is defined if and only if $\mathbf{d}({\bf
a})=\mathbf{r}(\alpha)$, in which case $\mathbf{d}({}^{\alpha}{\bf
a})=\mathbf{d}(\alpha)$. Diagrammatically, one can represent a tuple ${\bf
a}=(a_{1},\ldots,a_{n})\in M_{0}^{n}$ as a row of $n$ vertices, with the $i$th
vertex labelled $a_{i}$. The action ${\bf a}\mapsto{}^{\alpha}{\bf a}$ can
then be calculated by ‘sliding’ the relevant entries of ${\bf a}$ up the edges
of $\alpha$; Figure 3 gives an example with
$\alpha=\left(\begin{smallmatrix}1&2&3&4&5&6\\\
-&-&4&1&-&7\end{smallmatrix}\right)\in\mathcal{I}_{6,8}$.

$a_{1}$$a_{2}$$a_{3}$$a_{4}$$a_{5}$$a_{6}$$a_{7}$$a_{8}$${\bf
a}$$a_{4}$$a_{1}$$a_{7}$$0$$0$$0$${}^{\alpha}{\bf a}$

Figure 3: The action of $\alpha\in\mathcal{I}_{6,8}$ on ${\bf a}\in
M_{0}^{8}$.

One can easily check that

${}^{\alpha}({\bf a}\cdot{\bf b})={}^{\alpha}{\bf a}\cdot{}^{\alpha}{\bf
b}\qquad\text{and}\qquad{}^{\alpha\beta}{\bf a}={}^{\alpha}({}^{\beta}{\bf
a})$

for all $\alpha,\beta\in\mathcal{I}$ and ${\bf a},{\bf b}\in\mathcal{M}$ for
which the above expressions are defined. We also have
${}^{\operatorname{id}_{n}}{\bf a}={\bf a}$ for all ${\bf a}\in M_{0}^{n}$.
However, we do not necessarily have ${}^{\alpha}{\bf 1}_{n}={\bf 1}_{m}$ for
$\alpha\in\mathcal{I}_{m,n}$; rather, ${}^{\alpha}{\bf 1}_{n}$ is the
$m$-tuple over $\\{0,1\\}$ with support equal to $\operatorname{dom}(\alpha)$.

The _wreath product_ $M\wr\mathcal{I}$ is defined as follows. As a set we have

$M\wr\mathcal{I}=\big{\\{}({\bf
a},\alpha)\in\mathcal{M}\times\mathcal{I}:\mathbf{d}({\bf
a})=\mathbf{d}(\alpha),\ \operatorname{supp}({\bf
a})=\operatorname{dom}(\alpha)\big{\\}}.$

For $({\bf a},\alpha)\in M\wr\mathcal{I}$ we define $\mathbf{d}({\bf
a},\alpha)=\mathbf{d}(\alpha)$ and $\mathbf{r}({\bf
a},\alpha)=\mathbf{r}(\alpha)$. For $({\bf a},\alpha),({\bf b},\beta)\in
M\wr\mathcal{I}$ with $\mathbf{r}(\alpha)=\mathbf{d}(\beta)$ we define

$({\bf a},\alpha)\circ({\bf b},\beta)=({\bf a}\cdot{}^{\alpha}{\bf
b},\alpha\beta).$

This composition is well defined (i.e., $\mathbf{d}({\bf
a}\cdot{}^{\alpha}{\bf b})=\mathbf{d}(\alpha\beta)$ and
${\operatorname{supp}({\bf a}\cdot{}^{\alpha}{\bf
b})=\operatorname{dom}(\alpha\beta)}$ for such ${\bf a},{\bf b},\alpha,\beta$)
and associative, and $M\wr\mathcal{I}$ forms a category. The tensor operation
is given by

$({\bf a},\alpha)\oplus({\bf b},\beta)=({\bf a}\oplus{\bf
b},\alpha\oplus\beta).$

Again there is a handy diagrammatic representation of the elements and
operations of $M\wr\mathcal{I}$. We represent $({\bf a},\alpha)$ by picturing
$\alpha$ in the way described above, and we label the $i$th upper vertex with
$a_{i}$. We always omit the label $0$ from an upper vertex, and sometimes also
do this with the label $1$; one can always tell if an unspecified upper vertex
label is $0$ or $1$ according to whether the corresponding element of ${\bf
m}$ belongs to $\operatorname{dom}(\alpha)$ or not. Figure 4 gives an example
calculation of a composition. The tensor operation again corresponds to a
horizontal stacking, as in Figure 2. Of course when $M=\\{1\\}$ is trivial,
$M\wr\mathcal{I}\cong\mathcal{I}$.

$({\bf a},\alpha)=$$a_{3}$$a_{4}$$a_{6}$$({\bf
b},\beta)=$$b_{1}$$b_{3}$$b_{5}$$b_{7}$$=({\bf a},\alpha)\circ({\bf
b},\beta)$$a_{4}b_{1}$$a_{6}b_{7}$

Figure 4: Elements of $M\wr\mathcal{I}$ (left) and their composition (right).

The endomorphism monoids of $M\wr\mathcal{I}$ are the ordinary monoid wreath
products

$M\wr\mathcal{I}_{n}=\big{\\{}({\bf a},\alpha)\in
M_{0}^{n}\times\mathcal{I}_{n}:\operatorname{supp}({\bf
a})=\operatorname{dom}(\alpha)\big{\\}},$

as studied for example in [6, 7].

Note that $M\wr\mathcal{I}$ contains natural isomorphic copies of
$\mathcal{M}$ and $\mathcal{I}$, and we will identify these with subcategories
of $M\wr\mathcal{I}$ as follows. For ${\bf a}\in M_{0}^{n}$ and
$\alpha\in\mathcal{I}_{m,n}$ we identify

${\bf a}\equiv({\bf a},\operatorname{id}_{\operatorname{supp}({\bf
a})})\qquad\text{and}\qquad\alpha\equiv({\bf
1}_{\operatorname{dom}(\alpha)},\alpha),$

where $\operatorname{id}_{\operatorname{supp}({\bf
a})}\in\mathcal{I}_{\mathbf{d}({\bf a})}$ is the identity map on
$\operatorname{supp}({\bf a})$ and where ${\bf
1}_{\operatorname{dom}(\alpha)}\in M_{0}^{\mathbf{d}(\alpha)}$ denotes the
$\mathbf{d}(\alpha)$-tuple over $\\{0,1\\}$ with support equal to
$\operatorname{dom}(\alpha)$.

######  Remark 2.1.

With the above identifications, we note that

$({\bf a},\alpha)={\bf a}\cdot\alpha\qquad\text{for any $({\bf a},\alpha)\in
M\wr\mathcal{I}$.}$

More generally, if ${\bf a}\in M_{0}^{m}$ and $\alpha\in\mathcal{I}_{m,n}$ are
arbitrary, then ${\bf a}\cdot\alpha=({\bf b},\beta)$, where:

  * •

${\bf b}$ is obtained from ${\bf a}$ by replacing $a_{i}$ by $0$ for any
$i\not\in\operatorname{dom}(\alpha)$, and

  * •

$\beta$ is the restriction of $\alpha$ to $\operatorname{supp}({\bf a})$.

For any $n\in\mathbb{N}$ and any subsemigroup $S$ of $\mathcal{I}_{n}$ we also
have the wreath product

$M\wr S=\big{\\{}({\bf a},\alpha)\in M_{0}^{n}\times
S:\operatorname{supp}({\bf a})=\operatorname{dom}(\alpha)\big{\\}},$

which is a subsemigroup of $M\wr\mathcal{I}_{n}$. Later we will be especially
concerned with the case that $S$ is the singular subsemigroup
$\operatorname{Sing}(\mathcal{I}_{n})=\mathcal{I}_{n}\setminus\mathcal{S}_{n}$.

###  2.4 Presentations

We now fix the notation we will be using for (monoid, semigroup, category and
tensor category) presentations. We also state two results from [17] that will
be required.

Fix an alphabet $X$ and let $X^{*}$ be the _free monoid_ over $X$. So $X^{*}$
consists of all words over $X$, including the empty word, which we will
typically denote by $\iota$. Let $R\subseteq X^{*}\times X^{*}$, and let
$R^{\sharp}$ be the congruence on $X^{*}$ generated by $R$. We say a monoid
$M$ has _presentation_ $\textup{{Mon}}\langle{X}:{R}\rangle$ if $M\cong
X^{*}/R^{\sharp}$: i.e., if there exists a monoid surmorphism $X^{*}\to M$
with kernel $R^{\sharp}$. If $\phi$ is such a surmorphism, we say $M$ has
_presentation $\textup{{Mon}}\langle{X}:{R}\rangle$ via $\phi$_. Elements of
$X$ and $R$ are called _generators_ and _relations_ , respectively, and a
relation $(u,v)\in R$ is typically displayed as an equation: $u=v$. (The same
conventions hold for the other kinds of presentations discussed below.) We
sometimes use the notation $\textup{{Mon}}\langle{X}:{R}\rangle$ to denote the
monoid $X^{*}/R^{\sharp}$ itself.

_Semigroup presentations_ are defined analogously in terms of the _free
semigroup_ ${X^{+}=X^{*}\setminus\\{\iota\\}}$. Relations in a semigroup
presentation $\textup{{Sgp}}\langle{X}:{R}\rangle$ always involve non-empty
words, meaning that ${R\subseteq X^{+}\times X^{+}}$.

For any alphabet $X$, we denote by $\ell=\ell_{X}:X^{*}\to\mathbb{N}$ the
length function. The following basic result will be useful on a number of
occasions.

######  Lemma 2.2.

Let $X$ and $Y$ be disjoint alphabets, let $\sim$ be a congruence on $(X\cup
Y)^{*}$ or $(X\cup Y)^{+}$, and suppose one of the following two conditions
holds:

  1. (i)

for all $x\in X$ and $y\in Y$, $yx\sim uv$ for some $u\in X^{*}$ and $v\in
Y^{*}$ with $\ell(u)\leq 1$,

  2. (ii)

for all $x\in X$ and $y\in Y$, $yx\sim uv$ for some $u\in X^{*}$ and $v\in
Y^{*}$ with $\ell(v)\leq 1$.

Then for all $w\in(X\cup Y)^{+}$ we have $w\sim uv$ for some $u\in X^{*}$ and
$v\in Y^{*}$.

###### Proof.

By symmetry, we assume that (i) holds, and we proceed by induction on
$k=\ell(w)$. The result being clear for $k=1$, we assume that $k\geq 2$, so
that $w=w^{\prime}z$ for some $w^{\prime}\in(X\cup Y)^{+}$ and $z\in X\cup Y$.
By induction, we have $w^{\prime}\sim u^{\prime}v^{\prime}$ for some
$u^{\prime}\in X^{*}$ and $v^{\prime}\in Y^{*}$, and so $w\sim
u^{\prime}v^{\prime}z$. If $z\in Y$ then we take $u=u^{\prime}$ and
$v=v^{\prime}z$, so we now assume that $z\in X$. Repeated application of
condition (i) gives $v^{\prime}z\sim u^{\prime\prime}v$ for some
$u^{\prime\prime}\in X^{*}$ and $v\in Y^{*}$ (with $\ell(u^{\prime\prime})\leq
1$), so $w\sim u^{\prime}v^{\prime}z\sim u^{\prime}u^{\prime\prime}v$, and we
take $u=u^{\prime}u^{\prime\prime}$. ∎

The assumption that $\ell(u)\leq 1$ in condition (i) of Lemma 2.2 cannot be
dropped (and similarly for condition (ii)). For example, suppose $X=\\{x\\}$,
$Y=\\{y\\}$, and that $\sim$ is the congruence on $\\{x,y\\}^{*}$ generated by
the single relation $yx=xxyy$. Then one can show by induction that any word
$\sim$-equivalent to $yxx$ must have the form $x^{2k}\cdot yxx\cdot y^{2k}$ or
$x^{2k}\cdot xxyyx\cdot y^{2k}$ for some $k\in\mathbb{N}$. In particular, the
conclusion of Lemma 2.2 does not hold in this case.

For category presentations, we must work with digraphs and paths in place of
alphabets and words. Keeping in mind the conventions about categories from
Subsection 2.1, let $\Gamma$ be a digraph with vertex set $\mathbb{N}$,
possibly with multiple/parallel edges, and possibly with loops. We identify
$\Gamma$ with its edge set, and denote the source and target of an edge
$x\in\Gamma$ by $\mathbf{d}(x)$ and $\mathbf{r}(x)$ respectively. The _free
category_ over $\Gamma$ is the set $\Gamma^{*}$ of all paths in $\Gamma$ under
concatenation (where defined). The empty path at $n\in\mathbb{N}$ will be
denoted by $\iota_{n}$. Every other path can be thought of as a word of the
form $w=x_{1}\cdots x_{k}$, where $k\geq 1$ and $x_{1},\ldots,x_{k}\in\Gamma$,
and where $\mathbf{r}(x_{i})=\mathbf{d}(x_{i+1})$ for all $1\leq i<k$. For
such a word/path, we have $\mathbf{d}(w)=\mathbf{d}(x_{1})$ and
$\mathbf{r}(w)=\mathbf{r}(x_{k})$. Now let
$\Omega\subseteq\Gamma^{*}\times\Gamma^{*}$ be a set of pairs of paths, such
that $\mathbf{d}(u)=\mathbf{d}(v)$ and $\mathbf{r}(u)=\mathbf{r}(v)$ for all
$(u,v)\in\Omega$. We say a category $\mathcal{C}$ (over $\mathbb{N}$) has
_presentation_ $\textup{{Cat}}\langle{\Gamma}:{\Omega}\rangle$ if
$\mathcal{C}\cong\Gamma^{*}/\Omega^{\sharp}$: i.e., if there exists a
surmorphism $\Gamma^{*}\to\mathcal{C}$ with kernel $\Omega^{\sharp}$. If
$\phi$ is such a surmorphism, we say $\mathcal{C}$ has _presentation
$\textup{{Cat}}\langle{\Gamma}:{\Omega}\rangle$ via $\phi$_.

Finally, we recall the formulation of (strict) tensor category presentations
from [17]. Let $\Delta$ be a digraph with vertex set $\mathbb{N}$, again
identified with its edge set. We will denote the _free tensor category_ over
$\Delta$ by $\Delta^{\oast}$. It consists of all terms constructed in the
following way:

  1. (T1)

All empty paths $\iota_{n}$ ($n\in\mathbb{N}$) are terms, with
$\mathbf{d}(\iota_{n})=\mathbf{r}(\iota_{n})=n$, acting as identities of
$\Delta^{\oast}$.

  2. (T2)

All edges $x\in\Delta$ are terms, with $\mathbf{d}(x)$ and $\mathbf{r}(x)$ the
source and target of $x$, respectively.

  3. (T3)

If $s$ and $t$ are terms, and if $\mathbf{r}(s)=\mathbf{d}(t)$, then the
formal expression $s\circ t$ is a term, with $\mathbf{d}(s\circ
t)=\mathbf{d}(s)$ and $\mathbf{r}(s\circ t)=\mathbf{r}(t)$.

  4. (T4)

If $s$ and $t$ are terms, then the formal expression $s\oplus t$ is a term,
with $\mathbf{d}(s\oplus t)=\mathbf{d}(s)+\mathbf{d}(t)$ and
$\mathbf{r}(s\oplus t)=\mathbf{r}(s)+\mathbf{r}(t)$.

Note that (T1)–(T4) describe the elements of $\Delta^{\oast}$, while (T3) and
(T4) also give the definition of the $\circ$ and $\oplus$ operations. Now let
$\Xi\subseteq\Delta^{\oast}\times\Delta^{\oast}$ be a set of pairs of terms,
such that $\mathbf{d}(u)=\mathbf{d}(v)$ and $\mathbf{r}(u)=\mathbf{r}(v)$ for
all $(u,v)\in\Xi$. We say a tensor category $\mathcal{C}$ (over $\mathbb{N}$)
has presentation $\textup{{TCat}}\langle{\Delta}:{\Xi}\rangle$ if
$\mathcal{C}\cong\Delta^{\oast}/\Xi_{\scriptscriptstyle{\oplus}}^{\sharp}$:
i.e., if there is a surmorphism $\Delta^{\oast}\to\mathcal{C}$ with kernel
$\Xi_{\scriptscriptstyle{\oplus}}^{\sharp}$. If $\phi$ is such a surmorphism,
we say $\mathcal{C}$ has presentation
$\textup{{TCat}}\langle{\Delta}:{\Xi}\rangle$ via $\phi$.

As an example, we state the tensor presentation for $\mathcal{I}$ from [17,
Section 4.5]. Let $\Delta_{\mathcal{I}}$ be the digraph with three edges
denoted $X$, $U$ and  $U$ , with

$\mathbf{d}(X)=\mathbf{r}(X)=2,\qquad\mathbf{d}(U)=\mathbf{r}(\rotatebox[origin={c}]{180.0}{$U$})=1\qquad\text{and}\qquad\mathbf{r}(U)=\mathbf{d}(\rotatebox[origin={c}]{180.0}{$U$})=0,$

and let $\Xi_{\mathcal{I}}$ be the following set of relations, where for
convenience we write $I=\iota_{1}\in\Delta^{\oast}$:

$\displaystyle X\circ X=\iota_{2},\qquad(X\oplus I)\circ(I\oplus
X)\circ(X\oplus I)=(I\oplus X)\circ(X\oplus I)\circ(I\oplus X),$ (2.3)
$\displaystyle\rotatebox[origin={c}]{180.0}{$U$}\circ U=\iota_{0},\qquad
X\circ(U\oplus I)=I\oplus U,\qquad(\rotatebox[origin={c}]{180.0}{$U$}\oplus
I)\circ X=I\oplus\rotatebox[origin={c}]{180.0}{$U$}.$ (2.4)

######  Theorem 2.5.

The symmetric inverse category $\mathcal{I}$ has presentation
$\textup{{TCat}}\langle{\Delta_{\mathcal{I}}}:{\Xi_{\mathcal{I}}}\rangle$ via

$X\mapsto{\lower 6.02773pt\hbox{ \leavevmode\hbox to11.44pt{\vbox
to19.97pt{\pgfpicture\makeatletter\hbox{\hskip-7.08482pt\lower-1.45108pt\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\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{8.5359pt}{17.07182pt}\pgfsys@moveto{9.98698pt}{17.07182pt}\pgfsys@curveto{9.98698pt}{17.87323pt}{9.33731pt}{18.5229pt}{8.5359pt}{18.5229pt}\pgfsys@curveto{7.7345pt}{18.5229pt}{7.08482pt}{17.87323pt}{7.08482pt}{17.07182pt}\pgfsys@curveto{7.08482pt}{16.27042pt}{7.7345pt}{15.62074pt}{8.5359pt}{15.62074pt}\pgfsys@curveto{9.33731pt}{15.62074pt}{9.98698pt}{16.27042pt}{9.98698pt}{17.07182pt}\pgfsys@closepath\pgfsys@moveto{8.5359pt}{17.07182pt}\pgfsys@fill\pgfsys@invoke{
}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{17.07182pt}{17.07182pt}\pgfsys@moveto{18.5229pt}{17.07182pt}\pgfsys@curveto{18.5229pt}{17.87323pt}{17.87323pt}{18.5229pt}{17.07182pt}{18.5229pt}\pgfsys@curveto{16.27042pt}{18.5229pt}{15.62074pt}{17.87323pt}{15.62074pt}{17.07182pt}\pgfsys@curveto{15.62074pt}{16.27042pt}{16.27042pt}{15.62074pt}{17.07182pt}{15.62074pt}\pgfsys@curveto{17.87323pt}{15.62074pt}{18.5229pt}{16.27042pt}{18.5229pt}{17.07182pt}\pgfsys@closepath\pgfsys@moveto{17.07182pt}{17.07182pt}\pgfsys@fill\pgfsys@invoke{
} } {
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{8.5359pt}{0.0pt}\pgfsys@moveto{9.98698pt}{0.0pt}\pgfsys@curveto{9.98698pt}{0.8014pt}{9.33731pt}{1.45108pt}{8.5359pt}{1.45108pt}\pgfsys@curveto{7.7345pt}{1.45108pt}{7.08482pt}{0.8014pt}{7.08482pt}{0.0pt}\pgfsys@curveto{7.08482pt}{-0.8014pt}{7.7345pt}{-1.45108pt}{8.5359pt}{-1.45108pt}\pgfsys@curveto{9.33731pt}{-1.45108pt}{9.98698pt}{-0.8014pt}{9.98698pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{8.5359pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{17.07182pt}{0.0pt}\pgfsys@moveto{18.5229pt}{0.0pt}\pgfsys@curveto{18.5229pt}{0.8014pt}{17.87323pt}{1.45108pt}{17.07182pt}{1.45108pt}\pgfsys@curveto{16.27042pt}{1.45108pt}{15.62074pt}{0.8014pt}{15.62074pt}{0.0pt}\pgfsys@curveto{15.62074pt}{-0.8014pt}{16.27042pt}{-1.45108pt}{17.07182pt}{-1.45108pt}\pgfsys@curveto{17.87323pt}{-1.45108pt}{18.5229pt}{-0.8014pt}{18.5229pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{17.07182pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} } {}{{}}{}
{}{}{}\pgfsys@moveto{8.5359pt}{17.07182pt}\pgfsys@lineto{17.07182pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{17.07182pt}{17.07182pt}\pgfsys@lineto{8.5359pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} }},\qquad
U\mapsto{\lower 6.02773pt\hbox{ \leavevmode\hbox to2.9pt{\vbox
to19.97pt{\pgfpicture\makeatletter\hbox{\hskip-7.08482pt\lower-1.45108pt\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\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{8.5359pt}{17.07182pt}\pgfsys@moveto{9.98698pt}{17.07182pt}\pgfsys@curveto{9.98698pt}{17.87323pt}{9.33731pt}{18.5229pt}{8.5359pt}{18.5229pt}\pgfsys@curveto{7.7345pt}{18.5229pt}{7.08482pt}{17.87323pt}{7.08482pt}{17.07182pt}\pgfsys@curveto{7.08482pt}{16.27042pt}{7.7345pt}{15.62074pt}{8.5359pt}{15.62074pt}\pgfsys@curveto{9.33731pt}{15.62074pt}{9.98698pt}{16.27042pt}{9.98698pt}{17.07182pt}\pgfsys@closepath\pgfsys@moveto{8.5359pt}{17.07182pt}\pgfsys@fill\pgfsys@invoke{
} } {} {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@moveto{8.5359pt}{0.0pt}\pgfsys@moveto{9.98698pt}{0.0pt}\pgfsys@curveto{9.98698pt}{0.8014pt}{9.33731pt}{1.45108pt}{8.5359pt}{1.45108pt}\pgfsys@curveto{7.7345pt}{1.45108pt}{7.08482pt}{0.8014pt}{7.08482pt}{0.0pt}\pgfsys@curveto{7.08482pt}{-0.8014pt}{7.7345pt}{-1.45108pt}{8.5359pt}{-1.45108pt}\pgfsys@curveto{9.33731pt}{-1.45108pt}{9.98698pt}{-0.8014pt}{9.98698pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{8.5359pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \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\rotatebox[origin={c}]{180.0}{$U$}\mapsto{\lower 6.02773pt\hbox{
\leavevmode\hbox to2.9pt{\vbox
to2.9pt{\pgfpicture\makeatletter\hbox{\hskip-7.08482pt\lower-1.45108pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {} {
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{8.5359pt}{0.0pt}\pgfsys@moveto{9.98698pt}{0.0pt}\pgfsys@curveto{9.98698pt}{0.8014pt}{9.33731pt}{1.45108pt}{8.5359pt}{1.45108pt}\pgfsys@curveto{7.7345pt}{1.45108pt}{7.08482pt}{0.8014pt}{7.08482pt}{0.0pt}\pgfsys@curveto{7.08482pt}{-0.8014pt}{7.7345pt}{-1.45108pt}{8.5359pt}{-1.45108pt}\pgfsys@curveto{9.33731pt}{-1.45108pt}{9.98698pt}{-0.8014pt}{9.98698pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{8.5359pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} } \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} }}.$

Note for example that $X\oplus I\mapsto{\lower 6.02773pt\hbox{
\leavevmode\hbox to19.97pt{\vbox
to19.97pt{\pgfpicture\makeatletter\hbox{\hskip-7.08482pt\lower-1.45108pt\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\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{8.5359pt}{17.07182pt}\pgfsys@moveto{9.98698pt}{17.07182pt}\pgfsys@curveto{9.98698pt}{17.87323pt}{9.33731pt}{18.5229pt}{8.5359pt}{18.5229pt}\pgfsys@curveto{7.7345pt}{18.5229pt}{7.08482pt}{17.87323pt}{7.08482pt}{17.07182pt}\pgfsys@curveto{7.08482pt}{16.27042pt}{7.7345pt}{15.62074pt}{8.5359pt}{15.62074pt}\pgfsys@curveto{9.33731pt}{15.62074pt}{9.98698pt}{16.27042pt}{9.98698pt}{17.07182pt}\pgfsys@closepath\pgfsys@moveto{8.5359pt}{17.07182pt}\pgfsys@fill\pgfsys@invoke{
}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{17.07182pt}{17.07182pt}\pgfsys@moveto{18.5229pt}{17.07182pt}\pgfsys@curveto{18.5229pt}{17.87323pt}{17.87323pt}{18.5229pt}{17.07182pt}{18.5229pt}\pgfsys@curveto{16.27042pt}{18.5229pt}{15.62074pt}{17.87323pt}{15.62074pt}{17.07182pt}\pgfsys@curveto{15.62074pt}{16.27042pt}{16.27042pt}{15.62074pt}{17.07182pt}{15.62074pt}\pgfsys@curveto{17.87323pt}{15.62074pt}{18.5229pt}{16.27042pt}{18.5229pt}{17.07182pt}\pgfsys@closepath\pgfsys@moveto{17.07182pt}{17.07182pt}\pgfsys@fill\pgfsys@invoke{
}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{25.60774pt}{17.07182pt}\pgfsys@moveto{27.05882pt}{17.07182pt}\pgfsys@curveto{27.05882pt}{17.87323pt}{26.40915pt}{18.5229pt}{25.60774pt}{18.5229pt}\pgfsys@curveto{24.80634pt}{18.5229pt}{24.15666pt}{17.87323pt}{24.15666pt}{17.07182pt}\pgfsys@curveto{24.15666pt}{16.27042pt}{24.80634pt}{15.62074pt}{25.60774pt}{15.62074pt}\pgfsys@curveto{26.40915pt}{15.62074pt}{27.05882pt}{16.27042pt}{27.05882pt}{17.07182pt}\pgfsys@closepath\pgfsys@moveto{25.60774pt}{17.07182pt}\pgfsys@fill\pgfsys@invoke{
} } {
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{8.5359pt}{0.0pt}\pgfsys@moveto{9.98698pt}{0.0pt}\pgfsys@curveto{9.98698pt}{0.8014pt}{9.33731pt}{1.45108pt}{8.5359pt}{1.45108pt}\pgfsys@curveto{7.7345pt}{1.45108pt}{7.08482pt}{0.8014pt}{7.08482pt}{0.0pt}\pgfsys@curveto{7.08482pt}{-0.8014pt}{7.7345pt}{-1.45108pt}{8.5359pt}{-1.45108pt}\pgfsys@curveto{9.33731pt}{-1.45108pt}{9.98698pt}{-0.8014pt}{9.98698pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{8.5359pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{17.07182pt}{0.0pt}\pgfsys@moveto{18.5229pt}{0.0pt}\pgfsys@curveto{18.5229pt}{0.8014pt}{17.87323pt}{1.45108pt}{17.07182pt}{1.45108pt}\pgfsys@curveto{16.27042pt}{1.45108pt}{15.62074pt}{0.8014pt}{15.62074pt}{0.0pt}\pgfsys@curveto{15.62074pt}{-0.8014pt}{16.27042pt}{-1.45108pt}{17.07182pt}{-1.45108pt}\pgfsys@curveto{17.87323pt}{-1.45108pt}{18.5229pt}{-0.8014pt}{18.5229pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{17.07182pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{25.60774pt}{0.0pt}\pgfsys@moveto{27.05882pt}{0.0pt}\pgfsys@curveto{27.05882pt}{0.8014pt}{26.40915pt}{1.45108pt}{25.60774pt}{1.45108pt}\pgfsys@curveto{24.80634pt}{1.45108pt}{24.15666pt}{0.8014pt}{24.15666pt}{0.0pt}\pgfsys@curveto{24.15666pt}{-0.8014pt}{24.80634pt}{-1.45108pt}{25.60774pt}{-1.45108pt}\pgfsys@curveto{26.40915pt}{-1.45108pt}{27.05882pt}{-0.8014pt}{27.05882pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{25.60774pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} } {}{{}}{}
{}{}{}\pgfsys@moveto{8.5359pt}{17.07182pt}\pgfsys@lineto{17.07182pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{17.07182pt}{17.07182pt}\pgfsys@lineto{8.5359pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{25.60774pt}{17.07182pt}\pgfsys@lineto{25.60774pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} }}$ and $I\oplus
X\mapsto{\lower 6.02773pt\hbox{ \leavevmode\hbox to19.97pt{\vbox
to19.97pt{\pgfpicture\makeatletter\hbox{\hskip-7.08482pt\lower-1.45108pt\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{
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second relation in (2.3) is an instance of a so-called braid relation:

$=$

The reader is invited to draw diagrams to check that the other relations
listed in (2.3) and (2.4) hold.

The main general results from [17] provide mechanisms for building
presentations for a category out of presentations for its endomorphism
monoids. Here we state special cases of these results that are sufficient for
our purposes. With this in mind, we fix a tensor category $\mathcal{C}$ (as
above), and we denote the identities of $\mathcal{C}$ by
$\overline{\iota}_{n}\in\mathcal{C}_{n}$ ($n\in\mathbb{N}$). In addition, we
assume that all of the following conditions hold:

  1. (A1)

Every hom-set $\mathcal{C}_{m,n}$ ($m,n\in\mathbb{N}$) is non-empty. (This
actually follows from the next assumption, but we list it anyway, for ease of
comparison with [17].)

  2. (A2)

For every $n\in\mathbb{N}$, there exist
$\overline{\lambda}_{n}\in\mathcal{C}_{n,n+1}$ and
$\overline{\rho}_{n}\in\mathcal{C}_{n+1,n}$ such that
$\overline{\lambda}_{n}\overline{\rho}_{n}=\overline{\iota}_{n}$.

  3. (A3)

For every $n\in\mathbb{N}$, the endomorphism monoid $\mathcal{C}_{n}$ has
presentation $\textup{{Mon}}\langle{X_{n}}:{R_{n}}\rangle$ via
${\phi_{n}:X_{n}^{*}\to\mathcal{C}_{n}}$. We assume the alphabets $X_{n}$
($n\in\mathbb{N}$) are pairwise disjoint, and we denote by $\iota_{n}$ the
empty word over $X_{n}$.

We now define a digraph $\Gamma$ with vertex set $\mathbb{N}$ and edge set

$\\{\lambda_{n},\rho_{n}:n\in\mathbb{N}\\}\cup\bigcup_{n\in\mathbb{N}}X_{n},$

with sources and targets given by

$\mathbf{d}(x)=\mathbf{r}(x)=\mathbf{d}(\lambda_{n})=\mathbf{r}(\rho_{n})=n\quad\text{and}\quad\mathbf{r}(\lambda_{n})=\mathbf{d}(\rho_{n})=n+1\qquad\text{for
all $n\in\mathbb{N}$ and $x\in X_{n}$.}$

We define a morphism $\phi:\Gamma^{*}\to\mathcal{C}$ by

$\lambda_{n}\phi=\overline{\lambda}_{n},\qquad\rho_{n}\phi=\overline{\rho}_{n}\qquad\text{and}\qquad
x\phi=x\phi_{n}\qquad\text{for all $n\in\mathbb{N}$ and $x\in X_{n}$.}$

For $w\in\Gamma^{*}$ we write $\overline{w}=w\phi\in\mathcal{C}$.

We now assume that we have a set of relations
$\Omega\subseteq\Gamma^{*}\times\Gamma^{*}$ such that, writing ${\sim}$ for
the congruence $\Omega^{\sharp}$ on $\Gamma^{*}$, the following all hold:

  1. (A4)

For every relation $(u,v)\in\Omega$, we have $\overline{u}=\overline{v}$.

  2. (A5)

For all $n\in\mathbb{N}$, $\Omega$ contains the relation
$\lambda_{n}\rho_{n}=\iota_{n}$, and a relation of the form
$\rho_{n}\lambda_{n}=w_{n}$ for some word $w_{n}\in X_{n+1}^{*}$.

  3. (A6)

For all $n\in\mathbb{N}$, we have $R_{n}\subseteq\Omega$.

  4. (A7)

For all $n\in\mathbb{N}$, there is a mapping $X_{n}\to X_{n+1}^{*}:x\mapsto
x^{+}$, and $\Omega$ contains the relations

$x\lambda_{n}=\lambda_{n}x^{+}\qquad\text{and}\qquad\rho_{n}x=x^{+}\rho_{n}\qquad\text{for
all $x\in X_{n}$.}$

  5. (A8)

For all $n\in\mathbb{N}$, and for all $u\in X_{n+1}^{*}$, we have
$w_{n}uw_{n}\sim w_{n}v^{+}w_{n}$ for some $v\in X_{n}^{*}$, where $w_{n}$ is
the word from (A5), and $X_{n}^{*}\to X_{n+1}^{*}:u\mapsto u^{+}$ is (the
extension of) the map from (A7).

The following is [17, Theorem 2.16], but see also [17, Remark 2.12]:

######  Theorem 2.6.

With the above notation, and subject to assumptions _(A1)–(A8)_, the category
$\mathcal{C}$ has presentation $\textup{{Cat}}\langle{\Gamma}:{\Omega}\rangle$
via $\phi$. ∎

The next result, also from [17], shows how to convert the presentation from
Theorem 2.6 into a tensor presentation. In what follows, $\mathcal{C}$ still
denotes a tensor category satisfying the assumptions (A1)–(A8), and we retain
the meaning of $\Gamma$, $\Omega$, $\phi$ and so on. We now also fix a digraph
$\Delta$ on vertex set $\mathbb{N}$, a set of relations
$\Xi\subseteq\Delta^{\oast}\times\Delta^{\oast}$, and a morphism
$\Phi:\Delta^{\oast}\to\mathcal{C}$. For $w\in\Delta^{\oast}$ we write
$\underline{w}=w\Phi\in\mathcal{C}$, and we denote by $\approx$ the congruence
$\Xi_{\scriptscriptstyle{\oplus}}^{\sharp}$ on $\Delta^{\oast}$. We also
assume that the following all hold:

  1. (A9)

For every relation $(u,v)\in\Xi$, we have $\underline{u}=\underline{v}$.

  2. (A10)

There is a mapping $\Gamma\to\Delta^{\oast}:x\mapsto\widehat{x}$ such that
$\underline{\hskip 1.99168pt\widehat{x}\hskip 1.99168pt}=\overline{x}$ (i.e.,
$\widehat{x}\Phi=x\phi$) for all $x\in\Gamma$. We extend this to a morphism
$\Gamma^{*}\to\Delta^{\oast}:w\mapsto\widehat{w}$. It quickly follows that
$\underline{\hskip 1.99168pt\widehat{w}\hskip 1.99168pt}=\overline{w}$ for all
$w\in\Gamma^{*}$.

  3. (A11)

For a generator $x\in\Delta$, and for natural numbers $m,n\in\mathbb{N}$, we
define the term

$x_{m,n}=\iota_{m}\oplus x\oplus\iota_{n}\in\Delta^{\oast}.$

We assume that for every such $x,m,n$, we have $x_{m,n}\approx\widehat{w}$ for
some $w\in\Gamma^{*}$.

  4. (A12)

For every relation $(u,v)\in\Omega$, we have $\widehat{u}\approx\widehat{v}$.

The following is [17, Theorem 2.17]:

######  Theorem 2.7.

With the above notation, and subject to assumptions _(A1)–(A12)_, the category
$\mathcal{C}$ has tensor presentation
$\textup{{TCat}}\langle{\Delta}:{\Xi}\rangle$ via $\Phi$. ∎

##  3 The monoid $M\wr\mathcal{I}_{n}$

Our goal in this section is to give monoid presentations for the wreath
product

$M\wr\mathcal{I}_{n}=\big{\\{}({\bf a},\alpha):{\bf a}\in M_{0}^{n},\
\alpha\in\mathcal{I}_{n},\ \operatorname{supp}({\bf
a})=\operatorname{dom}(\alpha)\big{\\}}.$

We give two such presentations (Theorems 3.22 and 3.29), each extending a
known presentation for $\mathcal{I}_{n}$ (stated in Theorems 3.10 and 3.15).
The first presentation for $M\wr\mathcal{I}_{n}$ is more symmetrical, and will
be used in Section 4 when we study the category $M\wr\mathcal{I}$. The second
presentation utilises a smaller generating set, and will be obtained by re-
writing the first. Examples 3.24 and 3.32 show how the presentations given in
Theorem 3.22 and Theorem 3.29 respectively look for small, explicit choices of
$M$ and $n$.

###  3.1 Presentations for $\mathcal{I}_{n}$

As just noted, our presentations for $M\wr\mathcal{I}_{n}$ will utilise
presentations for the symmetric inverse monoid $\mathcal{I}_{n}$, which we now
state. We begin by defining the alphabet

$X_{\mathcal{I}_{n}}=\\{s_{1},\ldots,s_{n-1}\\}\cup\\{e_{1},\ldots,e_{n}\\},$
(3.1)

and the morphism

$\phi_{\mathcal{I}_{n}}:X_{\mathcal{I}_{n}}^{*}\to\mathcal{I}_{n}\qquad\text{given
by}\qquad s_{i}\mapsto\overline{s}_{i}=(i,i+1)\qquad\text{and}\qquad
e_{i}\mapsto\overline{e}_{i}=\left(\begin{smallmatrix}1&\cdots&i-1&i&i+1&\cdots&n\\\
1&\cdots&i-1&-&i+1&\cdots&n\end{smallmatrix}\right).$

(Here $(i,i+1)$ is the transposition swapping $i$ and $i+1$.) Diagrammatic
representations of the $\overline{s}_{i}$ and $\overline{e}_{i}$ are given in
Figure 5. We denote by $R_{\mathcal{I}_{n}}$ the following set of relations
over $X_{\mathcal{I}_{n}}$, where in each relation the subscripts range over
all meaningful values, subject to any stated constraints:

$\displaystyle s_{i}^{2}$ $\displaystyle=\iota,$ (3.2) $\displaystyle
s_{i}s_{j}$ $\displaystyle=s_{j}s_{i}$ if $|i-j|>1$, (3.3) $\displaystyle
s_{i}s_{j}s_{i}$ $\displaystyle=s_{j}s_{i}s_{j}$ if $|i-j|=1$, (3.4)
$\displaystyle e_{i}^{2}$ $\displaystyle=e_{i},$ (3.5) $\displaystyle
e_{i}e_{j}$ $\displaystyle=e_{j}e_{i},$ (3.6) $\displaystyle s_{i}e_{j}$
$\displaystyle=e_{j}s_{i}$ if $j\not=i,i+1$, (3.7) $\displaystyle s_{i}e_{i}$
$\displaystyle=e_{i+1}s_{i},$ (3.8) $\displaystyle e_{i}e_{i+1}s_{i}$
$\displaystyle=e_{i}e_{i+1}.$ (3.9)

The next result follows quickly from [23, Theorem 4.8] and [13, Proposition
31]:

######  Theorem 3.10.

For any $n\geq 0$, the symmetric inverse monoid $\mathcal{I}_{n}$ has
presentation
${\textup{{Mon}}\langle{X_{\mathcal{I}_{n}}}:{R_{\mathcal{I}_{n}}}\rangle}$
via $\phi_{\mathcal{I}_{n}}$. ∎

It will also be convenient to state Popova’s original presentation for
$\mathcal{I}_{n}$ from [44], which utilises a smaller generating set. For
this, we define the alphabet

$X_{\mathcal{I}_{n}}^{\prime}=\\{s_{1},\ldots,s_{n-1},e\\},$ (3.11)

and morphism

$\phi_{\mathcal{I}_{n}}^{\prime}:(X_{\mathcal{I}_{n}}^{\prime})^{*}\to\mathcal{I}_{n}\qquad\text{given
by}\qquad s_{i}\mapsto\overline{s}_{i}=(i,i+1)\qquad\text{and}\qquad
e\mapsto\overline{e}=\left(\begin{smallmatrix}1&2&\cdots&n\\\
-&2&\cdots&n\end{smallmatrix}\right).$

We also denote by $R_{\mathcal{I}_{n}}^{\prime}$ the set of relations
consisting of (3.2)–(3.4) together with

$\displaystyle e^{2}$ $\displaystyle=e,$ (3.12) $\displaystyle
es_{1}es_{1}=es_{1}e$ $\displaystyle=s_{1}es_{1}e,$ (3.13) $\displaystyle
es_{i}$ $\displaystyle=s_{i}e$ $\displaystyle\text{if $i\geq 2$}.$ (3.14)

The following is attributed in [12] to Popova [44]:

######  Theorem 3.15.

For any $n\geq 1$, the symmetric inverse monoid $\mathcal{I}_{n}$ has
presentation
${\textup{{Mon}}\langle{X_{\mathcal{I}_{n}}^{\prime}}:{R_{\mathcal{I}_{n}}^{\prime}}\rangle}$
via $\phi_{\mathcal{I}_{n}}^{\prime}$. ∎

It should be noted that a symmetrical version of Popova’s presentation is
often stated, where the generator $e$ maps instead to
$\left(\begin{smallmatrix}1&\cdots&n-1&n\\\
1&\cdots&n-1&-\end{smallmatrix}\right)$. One then has to modify the relations
accordingly, so for example (3.13) becomes
$es_{n-1}es_{n-1}=es_{n-1}e=s_{n-1}es_{n-1}e$.

###  3.2 First presentation for $M\wr\mathcal{I}_{n}$

Our first presentation for $M\wr\mathcal{I}_{n}$ will extend the presentation
for $\mathcal{I}_{n}$ from Theorem 3.10, and will incorporate a presentation
for $M$. Thus, for the rest of this section we fix a presentation
$\textup{{Mon}}\langle{X_{M}}:{R_{M}}\rangle$ for $M$, via
$\phi_{M}:X_{M}^{*}\to M$. For $w\in X_{M}^{*}$ we write
$\overline{w}=w\phi_{M}$.

We begin by defining $n$ disjoint copies of $X_{M}$:

$X_{M}^{(i)}=\\{x^{(i)}:x\in X_{M}\\}\qquad\text{for each $i\in{\bf n}$,}$

and we set

$X_{M\wr\mathcal{I}_{n}}=X_{\mathcal{I}_{n}}\cup X_{M}^{(1)}\cup\cdots\cup
X_{M}^{(n)},$

where $X_{\mathcal{I}_{n}}$ is as in (3.1).

In order to define a morphism $X_{M\wr\mathcal{I}_{n}}^{*}\to
M\wr\mathcal{I}_{n}$, we first introduce a piece of notation. For $a\in M$ and
$i\in{\bf n}$, we define the tuple $a^{(i)}=(1,\ldots,1,a,1,\ldots,1)\in
M^{n}$, where $a$ is in position $i$. We then define the morphism

$\phi_{M\wr\mathcal{I}_{n}}:X_{M\wr\mathcal{I}_{n}}^{*}\to
M\wr\mathcal{I}_{n}:\begin{cases}x\mapsto\overline{x}&\text{for $x\in
X_{\mathcal{I}_{n}}$,}\\\ x^{(i)}\mapsto\overline{x}^{(i)}&\text{for $x\in
X_{M}$ and $i\in{\bf n}$,}\end{cases}$

where again we identify both $M^{n}$ and $\mathcal{I}_{n}$ as submonoids of
$M\wr\mathcal{I}_{n}$, as explained at the end of Subsection 2.3. Diagrammatic
representations of the images of the letters from $X_{M\wr\mathcal{I}_{n}}$
are shown in Figure 5. The reader is reminded of the convention regarding
suppressed upper labels. As with words in $X_{M}^{*}$, we extend the over-line
notation for words $w\in X_{M\wr\mathcal{I}_{n}}^{*}$ by writing
$\overline{w}=w\phi_{M\wr\mathcal{I}_{n}}$.

$1$$i$$n$$1$$i$$n$$1$$i$$n$$\overline{x}$ Figure 5: Generators of
$M\wr\mathcal{I}_{n}$ (left to right): $\overline{s}_{i}$, $\overline{e}_{i}$
and $\overline{x}^{(i)}$.

To state the relations for our presentation of $M\wr\mathcal{I}_{n}$ we need
another piece of notation. For any word ${w=x_{1}\cdots x_{k}\in X_{M}^{*}}$
and for any $i\in{\bf n}$, we define the word

$w^{(i)}=x_{1}^{(i)}\cdots x_{k}^{(i)}\in(X_{M}^{(i)})^{*}.$

We now define $R_{M\wr\mathcal{I}_{n}}$ to be the set of relations consisting
of $R_{\mathcal{I}_{n}}$ (i.e., (3.2)–(3.9)) together with the following,
again with sub/superscripts $i$ and $j$ as well as elements $x$ and $y$ of
$X_{M}$ ranging over all meaningful values, subject to any stated constraints:

$\displaystyle u^{(i)}$ $\displaystyle=v^{(i)}$ for all $(u,v)\in R_{M}$ and
$i\in{\bf n}$, (3.16) $\displaystyle x^{(i)}y^{(j)}$
$\displaystyle=y^{(j)}x^{(i)}$ if $i\not=j$, (3.17) $\displaystyle
s_{i}x^{(j)}$ $\displaystyle=x^{(j)}s_{i}$ if $j\not=i,i+1$, (3.18)
$\displaystyle s_{i}x^{(i)}$ $\displaystyle=x^{(i+1)}s_{i},$ (3.19)
$\displaystyle e_{i}x^{(j)}$ $\displaystyle=x^{(j)}e_{i}$ if $i\not=j$, (3.20)
$\displaystyle e_{i}x^{(i)}=e_{i}$ $\displaystyle=x^{(i)}e_{i}.$ (3.21)

######  Theorem 3.22.

For $n\geq 0$, the wreath product $M\wr\mathcal{I}_{n}$ has presentation
$\textup{{Mon}}\langle{X_{M\wr\mathcal{I}_{n}}}:{R_{M\wr\mathcal{I}_{n}}}\rangle$
via $\phi_{M\wr\mathcal{I}_{n}}$.

###### Proof.

To show that $\phi_{M\wr\mathcal{I}_{n}}$ is surjective, let $({\bf
a},\alpha)\in M\wr\mathcal{I}_{n}$. Let ${\bf b}=(b_{1},\ldots,b_{n})\in
M^{n}$ be the tuple obtained from ${\bf a}$ by replacing any $0$ entries by
$1$, so that $({\bf a},\alpha)={\bf b}\cdot\alpha$ (cf. Remark 2.1), and note
that ${\bf b}=b_{1}^{(1)}\cdots b_{n}^{(n)}$. For each $i\in{\bf n}$, we have
$b_{i}=\overline{w}_{i}$ for some $w_{i}\in X_{M}^{*}$, and we also have
$\alpha=\overline{w}$ for some $w\in X_{\mathcal{I}_{n}}^{*}$. It then follows
that

$({\bf a},\alpha)=b_{1}^{(1)}\cdots
b_{n}^{(n)}\cdot\alpha=\overline{w_{1}^{(1)}\cdots w_{n}^{(n)}\cdot w}.$

Next, it is easy to check that $\phi_{M\wr\mathcal{I}_{n}}$ preserves all the
relations from $R_{M\wr\mathcal{I}_{n}}$; a couple of sample calculations are
given in Figure 6 for (3.19) and (3.21). It follows from this that
${R_{M\wr\mathcal{I}_{n}}^{\sharp}\subseteq\ker(\phi_{M\wr\mathcal{I}_{n}})}$.
The rest of the proof is devoted to establishing the reverse inclusion. For
this, we write $\sim$ for the congruence $R_{M\wr\mathcal{I}_{n}}^{\sharp}$.
We begin with a technical lemma:

######  Lemma 3.23.

For any $w\in X_{M\wr\mathcal{I}_{n}}^{*}$, we have $w\sim w_{1}^{(1)}\cdots
w_{n}^{(n)}\cdot w^{\prime}$ for some $w^{\prime}\in X_{\mathcal{I}_{n}}^{*}$,
and some $w_{1},\ldots,w_{n}\in X_{M}^{*}$ with $w_{i}=\iota$ for any
$i\in{\bf n}\setminus\operatorname{dom}(\overline{w}^{\prime})$.

###### Proof.

We first claim that

$w\sim w^{\prime\prime}w^{\prime}\qquad\text{for some
$w^{\prime\prime}\in(X_{M}^{(1)}\cup\cdots\cup X_{M}^{(n)})^{*}$ and
$w^{\prime}\in X_{\mathcal{I}_{n}}^{*}$.}$

Indeed, this follows from Lemma 2.2(i). Note that to apply this lemma, we need
to check that for any $x\in X_{M}^{(1)}\cup\cdots\cup X_{M}^{(n)}$ and $y\in
X_{\mathcal{I}_{n}}$, we have $yx\sim uv$ for some
$u\in(X_{M}^{(1)}\cup\cdots\cup X_{M}^{(n)})^{*}$ and $v\in
X_{\mathcal{I}_{n}}^{*}$ with $\ell(u)\leq 1$. Now, $R_{M\wr\mathcal{I}_{n}}$
itself contains such a relation $(yx,uv)$ in every case except $x=x^{(i+1)}$
and $y=s_{i}$. But for this we use (3.2) and (3.19) to calculate

$yx=s_{i}x^{(i+1)}\sim s_{i}x^{(i+1)}s_{i}s_{i}\sim s_{i}s_{i}x^{(i)}s_{i}\sim
x^{(i)}s_{i},$

and we can take $u=x^{(i)}$ and $v=s_{i}$. This completes the proof of the
claim, and we now fix $w^{\prime}$ and $w^{\prime\prime}$ as above.

Next we use (3.17) to ‘unshuffle’ $w^{\prime\prime}$: i.e., to move all the
letters in $w^{\prime\prime}$ from $X_{M}^{(1)}$ to the left, followed by all
those from $X_{M}^{(2)}$, and so on. Thus, we have $w^{\prime\prime}\sim
w_{1}^{(1)}\cdots w_{n}^{(n)}$ for some ${w_{1},\ldots,w_{n}\in X_{M}^{*}}$,
from which it follows by our first claim that $w\sim w_{1}^{(1)}\cdots
w_{n}^{(n)}\cdot w^{\prime}$.

It remains to show that any $w_{i}^{(i)}$ with
$i\not\in\operatorname{dom}(\overline{w}^{\prime})$ can be deleted (i.e.,
replaced by $\iota$). To do so, fix some such $i$. We then have
$\overline{w}^{\prime}=\overline{e}_{i}\overline{w}^{\prime}$ (in
$\mathcal{I}_{n}$), and so $w^{\prime}\sim e_{i}w^{\prime}$ by Theorem 3.10.
Combining this with (3.20) and (3.21), and writing $u=w_{1}^{(1)}\cdots
w_{i-1}^{(i-1)}$ and $v=w_{i+1}^{(i+1)}\cdots w_{n}^{(n)}$ for convenience, we
then have

$\displaystyle w\sim u\cdot w_{i}^{(i)}\cdot v\cdot w^{\prime}\sim u\cdot
w_{i}^{(i)}\cdot v\cdot e_{i}w^{\prime}\sim u\cdot w_{i}^{(i)}e_{i}\cdot
v\cdot w^{\prime}\sim u\cdot e_{i}\cdot v\cdot w^{\prime}\sim u\cdot v\cdot
e_{i}w^{\prime}\sim u\cdot v\cdot w^{\prime},$

completing the proof. ∎

Returning now to the main proof, let
$(u,v)\in\ker(\phi_{M\wr\mathcal{I}_{n}})$, so that $u,v\in
X_{M\wr\mathcal{I}_{n}}^{*}$ and $\overline{u}=\overline{v}$; we must show
that $u\sim v$. By Lemma 3.23, we have

$u\sim u_{1}^{(1)}\cdots u_{n}^{(n)}\cdot u^{\prime}\qquad\text{and}\qquad
v\sim v_{1}^{(1)}\cdots v_{n}^{(n)}\cdot v^{\prime},$

for appropriate $u_{i},v_{i}\in X_{M}^{*}$ and $u^{\prime},v^{\prime}\in
X_{\mathcal{I}_{n}}^{*}$. Define ${\bf a},{\bf b}\in M_{0}^{n}$ by

$a_{i}=\begin{cases}\overline{u}_{i}&\text{if
$i\in\operatorname{dom}(\overline{u}^{\prime})$}\\\
0&\text{otherwise}\end{cases}\qquad\text{and}\qquad
b_{i}=\begin{cases}\overline{v}_{i}&\text{if
$i\in\operatorname{dom}(\overline{v}^{\prime})$}\\\
0&\text{otherwise.}\end{cases}$

Then, keeping Remark 2.1 in mind, we have

$((a_{1},\ldots,a_{n}),\overline{u}^{\prime})=(\overline{u}_{1},\ldots,\overline{u}_{n})\cdot\overline{u}^{\prime}=\overline{u}=\overline{v}=(\overline{v}_{1},\ldots,\overline{v}_{n})\cdot\overline{v}^{\prime}=((b_{1},\ldots,b_{n}),\overline{v}^{\prime}).$

In particular we have $\overline{u}^{\prime}=\overline{v}^{\prime}$, and it
follows from Theorem 3.10 that $u^{\prime}\sim v^{\prime}$. We also have
$a_{i}=b_{i}$ for all $i$, so in particular when
$i\in\operatorname{dom}(\overline{u}^{\prime})=\operatorname{dom}(\overline{v}^{\prime})$
we have $\overline{u}_{i}=\overline{v}_{i}$. Since $M$ has presentation
$\textup{{Mon}}\langle{X_{M}}:{R_{M}}\rangle$, it follows that
$(u_{i},v_{i})\in R_{M}^{\sharp}$ for any such $i$, and hence $u_{i}^{(i)}\sim
v_{i}^{(i)}$ by (3.16). When
$i\not\in\operatorname{dom}(\overline{u}_{i}^{\prime})=\operatorname{dom}(\overline{v}_{i}^{\prime})$
we have $u_{i}=v_{i}=\iota$, so also $u_{i}^{(i)}\sim v_{i}^{(i)}$ for such
$i$. It follows from all of this that

$u\sim u_{1}^{(1)}\cdots u_{n}^{(n)}\cdot u^{\prime}\sim v_{1}^{(1)}\cdots
v_{n}^{(n)}\cdot v^{\prime}\sim v,$

and we have completed the proof of the theorem. ∎

$=$$\overline{x}$$\overline{x}$$=$$\overline{x}$$=$$\overline{x}$ Figure 6:
Left: relation (3.19), picturing only strings $i$ and $i+1$. Right: relation
(3.21), picturing only string $i$.

######  Example 3.24.

Before moving on it is worth considering an explicit, small example of the
presentation
$\textup{{Mon}}\langle{X_{M\wr\mathcal{I}_{n}}}:{R_{M\wr\mathcal{I}_{n}}}\rangle$
from Theorem 3.22. Specifically, we consider the case in which $n=2$ and
$M=\langle{a,b}:{ab=\iota}\rangle$ is the bicyclic monoid. Here we have

$X_{M\wr\mathcal{I}_{2}}=\\{s_{1},e_{1},e_{2},a^{(1)},b^{(1)},a^{(2)},b^{(2)}\\}.$

Relations (3.3), (3.4), (3.7) and (3.18) are empty for $n=2$. The remaining
relations from (3.2)–(3.9) have the form:

  *   * (3.2)

$s_{1}^{2}=\iota$,

  * (3.5)

$e_{1}^{2}=e_{1}$, $e_{2}^{2}=e_{2}$,

  * (3.6)

$e_{1}e_{2}=e_{2}e_{1}$,

  * (3.8)

$s_{1}e_{1}=e_{2}s_{1}$,

  * (3.9)

$e_{1}e_{2}s_{1}=e_{1}e_{2}$,

  * a

while the remaining relations in (3.16)–(3.21) have the form:

  * (3.16)

$a^{(1)}b^{(1)}=a^{(2)}b^{(2)}=\iota$,

  * (3.17)

$a^{(1)}a^{(2)}=a^{(2)}a^{(1)}$, $a^{(1)}b^{(2)}=b^{(2)}a^{(1)}$,
$b^{(1)}a^{(2)}=a^{(2)}b^{(1)}$, $b^{(1)}b^{(2)}=b^{(2)}b^{(1)}$,

  * (3.19)

$s_{1}a^{(1)}=a^{(2)}s_{1}$, $s_{1}b^{(1)}=b^{(2)}s_{1}$,

  * (3.20)

$e_{1}a^{(2)}=a^{(2)}e_{1}$, $e_{1}b^{(2)}=b^{(2)}e_{1}$,
$e_{2}a^{(1)}=a^{(1)}e_{2}$, $e_{2}b^{(1)}=b^{(1)}e_{2}$,

  * (3.21)

$e_{1}a^{(1)}=a^{(1)}e_{1}=e_{1}=e_{1}b^{(1)}=b^{(1)}e_{1}$,
$e_{2}a^{(2)}=a^{(2)}e_{2}=e_{2}=e_{2}b^{(2)}=b^{(2)}e_{2}$.

###  3.3 Second presentation for $M\wr\mathcal{I}_{n}$

The presentation for $M\wr\mathcal{I}_{n}$ from Theorem 3.22 extended the
presentation for $\mathcal{I}_{n}$ from Theorem 3.10. We now give a second
presentation, in terms of a smaller generating set, which will extended
Popova’s presentation for $\mathcal{I}_{n}$ from Theorem 3.15.

We continue to fix the presentation
$\textup{{Mon}}\langle{X_{M}}:{R_{M}}\rangle$ for $M$, via
$\phi_{M}:X_{M}^{*}\to M$. We begin by defining the alphabet

$X_{M\wr\mathcal{I}_{n}}^{\prime}=X_{\mathcal{I}_{n}}^{\prime}\cup X_{M},$

where $X_{\mathcal{I}_{n}}^{\prime}$ is as in (3.11), and we define the
morphism

$\phi_{M\wr\mathcal{I}_{n}}^{\prime}:(X_{M\wr\mathcal{I}_{n}}^{\prime})^{*}\to
M\wr\mathcal{I}_{n}:\begin{cases}x\mapsto\overline{x}&\text{for $x\in
X_{\mathcal{I}_{n}}^{\prime}$,}\\\ x\mapsto(\overline{x},1,\dots,1)&\text{for
$x\in X_{M}$.}\end{cases}$

Here we continue to identify $M^{n}$ and $\mathcal{I}_{n}$ with submonoids of
$M\wr\mathcal{I}_{n}$. Again we write
$\overline{w}=w\phi_{M\wr\mathcal{I}_{n}}^{\prime}$ for
$w\in(X_{M\wr\mathcal{I}_{n}}^{\prime})^{*}$. Denote by
$R_{M\wr\mathcal{I}_{n}}^{\prime}$ the relations consisting of
$R_{\mathcal{I}_{n}}^{\prime}$ (i.e., (3.2)–(3.4) and (3.12)–(3.14)), the
relations in $R_{M}$, as well as the following relations where $x,y\in X_{M}$:

$\displaystyle s_{i}x$ $\displaystyle=xs_{i}$ if $i\geq 2$, (3.25)
$\displaystyle xs_{1}ys_{1}$ $\displaystyle=s_{1}ys_{1}x,$ (3.26)
$\displaystyle es_{1}xs_{1}$ $\displaystyle=s_{1}xs_{1}e,$ (3.27)
$\displaystyle ex=xe$ $\displaystyle=e.$ (3.28)

######  Theorem 3.29.

For $n\geq 1$, the wreath product $M\wr\mathcal{I}_{n}$ has presentation
$\textup{{Mon}}\langle{X_{M\wr\mathcal{I}_{n}}^{\prime}}:{R_{M\wr\mathcal{I}_{n}}^{\prime}}\rangle$
via $\phi_{M\wr\mathcal{I}_{n}}^{\prime}$.

###### Proof.

To simplify notation, throughout the proof we will write

$Y=X_{M\wr\mathcal{I}_{n}},\qquad Z=X_{M\wr\mathcal{I}_{n}}^{\prime},\qquad
S=R_{M\wr\mathcal{I}_{n}}\qquad\text{and}\qquad
T=R_{M\wr\mathcal{I}_{n}}^{\prime}.$

By Theorem 3.22 we know that
$M\wr\mathcal{I}_{n}\cong\textup{{Mon}}\langle{Y}:{S}\rangle$, and we will
prove the current theorem by showing that
$\textup{{Mon}}\langle{Y}:{S}\rangle\cong\textup{{Mon}}\langle{Z}:{T}\rangle$.
(This can be thought of as a variation of the Tietze transformation
technique.) For the rest of the proof we write ${\sim}$ for $S^{\sharp}$ and
${\approx}$ for $T^{\sharp}$. For $u\in Y^{*}$ and $v\in Z^{*}$ we write $[u]$
and $[[v]]$ for the $\sim$-class of $u$ and the $\approx$-class of $v$,
respectively.

We begin by defining two morphisms:

$\psi_{1}:Y^{*}\to Z^{*}:\begin{cases}s_{i}\mapsto s_{i},\\\ e_{i}\mapsto
s_{i-1}\cdots s_{1}es_{1}\cdots s_{i-1},\\\ x^{(i)}\mapsto s_{i-1}\cdots
s_{1}xs_{1}\cdots s_{i-1},\end{cases}\qquad\text{and}\qquad\psi_{2}:Z^{*}\to
Y^{*}:\begin{cases}s_{i}\mapsto s_{i},\\\ e\mapsto e_{1},\\\ x\mapsto
x^{(1)}.\end{cases}$

The main substance of the proof involves demonstrating that the following all
hold:

  1. (i)

For every relation $(u,v)\in S$, we have $u\psi_{1}\approx v\psi_{1}$.

  2. (ii)

For every relation $(u,v)\in T$, we have $u\psi_{2}\sim v\psi_{2}$.

  3. (iii)

For every generator $y\in Y$, we have $y\psi_{1}\psi_{2}\sim y$.

  4. (iv)

For every generator $z\in Z$, we have $z\psi_{2}\psi_{1}\approx z$.

Before we prove these, we first note that they imply the theorem. Indeed, it
follows from (i) and (ii) that we have well-defined morphisms

$\Psi_{1}:\textup{{Mon}}\langle{Y}:{S}\rangle\to\textup{{Mon}}\langle{Z}:{T}\rangle\qquad\text{and}\qquad\Psi_{2}:\textup{{Mon}}\langle{Z}:{T}\rangle\to\textup{{Mon}}\langle{Y}:{S}\rangle,$

given by $[u]\Psi_{1}=[[u\psi_{1}]]$ and $[[v]]\Psi_{2}=[v\psi_{2}]$, for
$u\in Y^{*}$ and $v\in Z^{*}$. By (iii) and (iv) we have
$[y]\Psi_{1}\Psi_{2}=[y]$ and $[[z]]\Psi_{2}\Psi_{1}=[[z]]$ for all $y\in Y$
and $z\in Z$. It follows from these that $\Psi_{1}\Psi_{2}$ and
$\Psi_{2}\Psi_{1}$ are identity maps, so that $\Psi_{1}$ is an isomorphism
(with $\Psi_{2}=\Psi_{1}^{-1}$).

It now remains to prove items (i)–(iv), and we cover these roughly in order of
difficulty.

(iv). This is trivial, as we actually have equality: $z\psi_{2}\psi_{1}=z$ for
all $z\in Z$.

(ii). Fix some $(u,v)\in T$. Because of Theorem 3.22, we can show that
$u\psi_{2}\sim v\psi_{2}$ by showing that $\phi_{M\wr\mathcal{I}_{n}}$ maps
$u\psi_{2}$ and $v\psi_{2}$ to the same element of $M\wr\mathcal{I}_{n}$. This
can be verified diagrammatically for each such relation $(u,v)$, similarly to
Figure 6.

(iii). By Theorem 3.22, this again just amounts to a diagrammatic check that
$\phi_{M\wr\mathcal{I}_{n}}$ maps $y\psi_{1}\psi_{2}$ and $y$ to the same
element of $M\wr\mathcal{I}_{n}$.

(i). We treat the relations from $S=R_{M\wr\mathcal{I}_{n}}$ (i.e.,
(3.2)–(3.9) and (3.16)–(3.21)) one at a time. In what follows, we will write
$\widehat{u}=u\psi_{1}$ for all $u\in Y^{*}$. We must show that
$\widehat{u}\approx\widehat{v}$ for every relation $(u,v)\in S$.

(3.2)–(3.9). This follows immediately from the fact that $\mathcal{I}_{n}$ has
presentations
$\textup{{Mon}}\langle{X_{\mathcal{I}_{n}}}:{R_{\mathcal{I}_{n}}}\rangle$ and
$\textup{{Mon}}\langle{X_{\mathcal{I}_{n}}^{\prime}}:{R_{\mathcal{I}_{n}}^{\prime}}\rangle$,
by Theorems 3.10 and 3.15, noting that $R_{\mathcal{I}_{n}}^{\prime}\subseteq
T$.

(3.16). Fix some $(u,v)\in R_{M}$ and $i\in{\bf n}$. Writing $u=x_{1}\cdots
x_{k}$, and using (3.2), we have

$\displaystyle\widehat{u^{(i)}}=\widehat{x_{1}^{(i)}}\cdots\widehat{x_{k}^{(i)}}$
$\displaystyle=(s_{i-1}\cdots s_{1}x_{1}s_{1}\cdots
s_{i-1})\cdots(s_{i-1}\cdots s_{1}x_{k}s_{1}\cdots s_{i-1})$
$\displaystyle\approx s_{i-1}\cdots s_{1}(x_{1}\cdots x_{k})s_{1}\cdots
s_{i-1}$ $\displaystyle=s_{i-1}\cdots s_{1}\cdot u\cdot s_{1}\cdots s_{i-1}.$

(If $u=\iota$, then the previous conclusion still holds because of (3.2).) A
similar calculation gives $\widehat{v^{(i)}}\approx s_{i-1}\cdots s_{1}\cdot
v\cdot s_{1}\cdots s_{i-1}$. Since $R_{M}\subseteq T$, it follows that

$\widehat{u^{(i)}}\approx s_{i-1}\cdots s_{1}\cdot u\cdot s_{1}\cdots
s_{i-1}\approx s_{i-1}\cdots s_{1}\cdot v\cdot s_{1}\cdots
s_{i-1}\approx\widehat{v^{(i)}}.$

(3.17). First consider some word $w\in\\{s_{1},\ldots,s_{n-1}\\}^{*}$, and let
$x\in X_{M}$ and $i\in{\bf n}$. We denote by $w^{-1}$ the reverse of the word
$w$: i.e., if $w=s_{i_{1}}\cdots s_{i_{k}}$ then $w^{-1}=s_{i_{k}}\cdots
s_{i_{1}}$. Noting that $\overline{w}=w\phi_{\mathcal{I}_{n}}^{\prime}$ is a
permutation of ${\bf n}$, we claim that

$\widehat{x^{(i)}}\cdot w\approx
w\cdot\widehat{x^{(i\overline{w})}}\qquad\text{and}\qquad
w^{-1}\cdot\widehat{x^{(i)}}\cdot w\approx\widehat{x^{(i\overline{w})}}.$
(3.30)

By induction, it suffices to prove (3.30) in the case that $w=s_{k}$ is a
single letter. For this, and given (3.2), it suffices to show that

$s_{k}\cdot\widehat{x^{(i)}}\cdot
s_{k}\approx\begin{cases}\widehat{x^{(i)}}&\text{if $k\not=i,i-1$,}\\\
\widehat{x^{(i+1)}}&\text{if $k=i$,}\\\ \widehat{x^{(i-1)}}&\text{if
$k=i-1$.}\end{cases}$

When $k>i$ the result follows quickly from relations (3.3) and (3.25), which
say that $s_{k}$ commutes with each letter appearing in $\widehat{x^{(i)}}$.
The $k=i$ case holds by equality, and the $k=i-1$ case by (3.2). For the final
case, fix $k<i-1$. We first note that

$s_{1}\cdots s_{i-1}\cdot s_{k}\approx s_{k+1}\cdot s_{1}\cdots
s_{i-1}\qquad\text{and}\qquad s_{k}\cdot s_{i-1}\cdots s_{1}\approx
s_{i-1}\cdots s_{1}\cdot s_{k+1}.$

Indeed, these follow quickly from (3.3) and (3.4), or by a diagrammatic check
in $\mathcal{S}_{n}$. Using the relations (3.25) and (3.2) we then calculate

$\displaystyle s_{k}\cdot\widehat{x^{(i)}}\cdot s_{k}=s_{k}\cdot s_{i-1}\cdots
s_{1}xs_{1}\cdots s_{i-1}\cdot s_{k}$ $\displaystyle\approx s_{i-1}\cdots
s_{1}\cdot s_{k+1}xs_{k+1}\cdot s_{1}\cdots s_{i-1}$ $\displaystyle\approx
s_{i-1}\cdots s_{1}\cdot s_{k+1}s_{k+1}x\cdot s_{1}\cdots s_{i-1}$
$\displaystyle\approx s_{i-1}\cdots s_{1}\cdot x\cdot s_{1}\cdots
s_{i-1}=\widehat{x^{(i)}}.$

This completes the proof of (3.30).

Returning now to relation (3.17), fix some $x,y\in X_{M}$ and some $i,j\in{\bf
n}$ with $i\not=j$. We must show that
$\widehat{x^{(i)}}\cdot\widehat{y^{(j)}}\approx\widehat{y^{(j)}}\cdot\widehat{x^{(i)}}$.
Also fix some $w\in\\{s_{1},\ldots,s_{n-1}\\}^{*}$ with $i=1\overline{w}$ and
$j=2\overline{w}$. Then by (3.30) and (3.2) we have

$\widehat{x^{(i)}}\cdot\widehat{y^{(j)}}=\widehat{x^{(1\overline{w})}}\cdot\widehat{y^{(2\overline{w})}}\approx
w^{-1}\cdot\widehat{x^{(1)}}\cdot w\cdot w^{-1}\cdot\widehat{y^{(2)}}\cdot
w\approx w^{-1}\cdot\widehat{x^{(1)}}\cdot\widehat{y^{(2)}}\cdot w,$

and similarly $\widehat{y^{(j)}}\cdot\widehat{x^{(i)}}\approx
w^{-1}\cdot\widehat{y^{(2)}}\cdot\widehat{x^{(1)}}\cdot w$. We then obtain
$\widehat{x^{(i)}}\cdot\widehat{y^{(j)}}\approx\widehat{y^{(j)}}\cdot\widehat{x^{(i)}}$
from (3.26), which says precisely
$\widehat{x^{(1)}}\cdot\widehat{y^{(2)}}\approx\widehat{y^{(2)}}\cdot\widehat{x^{(1)}}$.

(3.18) and (3.19). These are special cases of (3.30).

(3.20). Similarly to (3.30), we can show that

$\widehat{e}_{i}\cdot w\approx
w\cdot\widehat{e}_{i\overline{w}}\qquad\text{and}\qquad\widehat{w}^{-1}\cdot
e_{i}\cdot w\approx\widehat{e}_{i\overline{w}}\qquad\text{for all $i\in{\bf
n}$ and $w\in\\{s_{1},\ldots,s_{n-1}\\}^{*}$.}$ (3.31)

(This also follows by a diagrammatic check, given that $\mathcal{I}_{n}$ has
presentation
$\textup{{Mon}}\langle{X_{\mathcal{I}_{n}}^{\prime}}:{R_{\mathcal{I}_{n}}^{\prime}}\rangle$.)
But then for any $x\in X_{M}$ and $i,j\in{\bf n}$ with $i\not=j$, we fix some
$w\in\\{s_{1},\ldots,s_{n-1}\\}^{*}$ such that $i=1\overline{w}$ and
$j=2\overline{w}$, and we have

$\widehat{e}_{i}\cdot\widehat{x^{(j)}}\approx w^{-1}\cdot\widehat{e}_{1}\cdot
w\cdot w^{-1}\cdot\widehat{x^{(2)}}\cdot w\approx
w^{-1}\cdot\widehat{e}_{1}\cdot\widehat{x^{(2)}}\cdot w\qquad\text{and
similarly}\qquad\widehat{x^{(j)}}\cdot\widehat{e}_{i}\approx
w^{-1}\cdot\widehat{x^{(2)}}\cdot\widehat{e}_{1}\cdot w.$

We then obtain
$\widehat{e}_{i}\cdot\widehat{x^{(j)}}\approx\widehat{x^{(j)}}\cdot\widehat{e}_{i}$
from (3.27), which says that
$\widehat{e}_{1}\cdot\widehat{x^{(2)}}\approx\widehat{x^{(2)}}\cdot\widehat{e}_{1}$.

(3.21). This is similar to the previous case, but we use relation (3.28) in
place of (3.27). ∎

######  Example 3.32.

As in Example 3.24, we now consider the presentation
$\textup{{Mon}}\langle{X_{M\wr\mathcal{I}_{n}}^{\prime}}:{R_{M\wr\mathcal{I}_{n}}^{\prime}}\rangle$
from Theorem 3.29 where $M=\langle{a,b}:{ab=\iota}\rangle$ and $n=2$. Here we
have

$X_{M\wr\mathcal{I}_{2}}^{\prime}=\\{s_{1},e,a,b\\}.$

This time, $R_{\mathcal{I}_{2}}^{\prime}$ consists of the relations
$s_{1}^{2}=\iota$, $e^{2}=e$ and $es_{1}es_{1}=es_{1}e=s_{1}es_{1}e$, $R_{M}$
consists of the relation $ab=\iota$, while relations (3.25)–(3.28) have the
form:

  * (3.25)

empty for $n=2$,

  * (3.26)

$as_{1}as_{1}=s_{1}as_{1}a$, $as_{1}bs_{1}=s_{1}bs_{1}a$,
$bs_{1}as_{1}=s_{1}as_{1}b$, $bs_{1}bs_{1}=s_{1}bs_{1}b$,

  * (3.27)

$es_{1}as_{1}=s_{1}as_{1}e$, $es_{1}bs_{1}=s_{1}bs_{1}e$,

  * (3.28)

$ea=ae=e=eb=be$.

##  4 The category $M\wr\mathcal{I}$

In this section we turn our attention to the category

$M\wr\mathcal{I}=\big{\\{}({\bf
a},\alpha)\in\mathcal{M}\times\mathcal{I}:\mathbf{d}({\bf
a})=\mathbf{d}(\alpha),\ \operatorname{supp}({\bf
a})=\operatorname{dom}(\alpha)\big{\\}}.$

Our main results here are Theorems 4.11 and 4.17, which give category and
tensor category presentations for $M\wr\mathcal{I}$. These theorems are proved
by combining Theorems 2.6 and 2.7 (from [17]) with Theorem 3.22, which gives
presentations for the endomorphism monoids of $M\wr\mathcal{I}$.

###  4.1 Category presentation for $M\wr\mathcal{I}$

We begin by giving a category presentation for $M\wr\mathcal{I}$, using
Theorem 2.6. To apply this theorem, we require presentations for the
endomorphism monoids $M\wr\mathcal{I}_{n}$ ($n\in\mathbb{N}$), and the most
convenient such presentations are those from Theorem 3.22. However, to ensure
that the generating sets are pairwise disjoint, the generators $s_{i},e_{i}$
and $x^{(i)}$ from $X_{M\wr\mathcal{I}_{n}}$ for any $n\in\mathbb{N}$ will be
renamed $s_{i;n}$, $e_{i;n}$ and $x^{(i;n)}$ respectively. We again fix the
presentation $\textup{{Mon}}\langle{X_{M}}:{R_{M}}\rangle$ for $M$ via
$\phi_{M}:X_{M}^{*}\to M$.

With the above notational comments in mind, we begin by defining the digraph
$\Gamma_{M\wr\mathcal{I}}$ over vertex set $\mathbb{N}$, and with edges

  * •

$s_{i;n}$ for each $n\in\mathbb{N}$ and $1\leq i<n$, with
$\mathbf{d}(s_{i;n})=\mathbf{r}(s_{i;n})=n$,

  * •

$e_{i;n}$ for each $n\in\mathbb{N}$ and $1\leq i\leq n$, with
$\mathbf{d}(e_{i;n})=\mathbf{r}(e_{i;n})=n$,

  * •

$x^{(i;n)}$ for each $x\in X_{M}$, $n\in\mathbb{N}$ and $1\leq i\leq n$, with
$\mathbf{d}(x^{(i;n)})=\mathbf{r}(x^{(i;n)})=n$,

  * •

$\lambda_{n},\rho_{n}$ for each $n\in\mathbb{N}$, with
$\mathbf{d}(\lambda_{n})=\mathbf{r}(\rho_{n})=n$ and
$\mathbf{r}(\lambda_{n})=\mathbf{d}(\rho_{n})=n+1$.

We define the morphism

$\phi_{M\wr\mathcal{I}}:\Gamma_{M\wr\mathcal{I}}^{*}\to
M\wr\mathcal{I}:x\mapsto\overline{x}$

diagrammatically in Figure 7, which also pictures the identities
$\overline{\iota}_{n}$ ($n\in\mathbb{N}$). (Here as usual
$\iota_{n}\in\Gamma_{M\wr\mathcal{I}}^{*}$ denotes the empty path $n\to n$.)

$\overline{\lambda}_{n}=$$1$$n$$\overline{\rho}_{n}=$$1$$n$$\overline{\iota}_{n}=$$1$$n$$\overline{s}_{i;n}=$$1$$i$$n$$\overline{e}_{i;n}=$$1$$i$$n$$\overline{x}^{(i;n)}=$$1$$i$$n$$\overline{x}$
Figure 7: Generators of $M\wr\mathcal{I}$, as well as the identity
$\overline{\iota}_{n}$.

For a word $u=x_{1}\cdots x_{k}\in X_{M}^{*}$, and for $n\in\mathbb{N}$ and
$1\leq i\leq n$, we define

$u^{(i;n)}=x_{1}^{(i;n)}\cdots x_{k}^{(i;n)}\in\Gamma_{M\wr\mathcal{I}}^{*}.$

######  Remark 4.1.

If $u$ is empty, we interpret $u^{(i;n)}=\iota_{n}$.

Now let $\Omega_{M\wr\mathcal{I}}$ be the following set of relations over
$\Gamma_{M\wr\mathcal{I}}$, where as usual the sub/superscripts and elements
of $X_{M}$ range over all meaningful values, subject to any stated
constraints:

$\displaystyle s_{i;n}^{2}=\iota_{n},\qquad e_{i;n}^{2}=e_{i;n},\qquad
e_{i;n}e_{j;n}=e_{j;n}e_{i;n},\qquad s_{i;n}e_{j;n}=e_{j;n}s_{i;n}\text{ if
$j\not=i,i+1$,}$ (4.2) $\displaystyle s_{i;n}e_{i;n}=e_{i+1;n}s_{i;n},\qquad
e_{i;n}e_{i+1;n}s_{i;n}=e_{i;n}e_{i+1;n},$ (4.3) $\displaystyle
s_{i;n}s_{j;n}=s_{j;n}s_{i;n}\text{ if $|i-j|>1$},\qquad
s_{i;n}s_{j;n}s_{i;n}=s_{j;n}s_{i;n}s_{j;n}\text{ if $|i-j|=1$,}$ (4.4)
$\displaystyle u^{(i;n)}=v^{(i;n)}\text{ for $(u,v)\in R_{M}$},\qquad
x^{(i;n)}y^{(j;n)}=y^{(j;n)}x^{(i;n)}\text{ if $i\not=j$},$ (4.5)
$\displaystyle s_{i;n}x^{(j;n)}=x^{(j;n)}s_{i;n}\text{ if
$j\not=i,i+1$},\qquad s_{i;n}x^{(i;n)}=x^{(i+1;n)}s_{i;n},$ (4.6)
$\displaystyle e_{i;n}x^{(j;n)}=x^{(j;n)}e_{i;n}\text{ if $i\not=j$},\qquad
e_{i;n}x^{(i;n)}=e_{i;n}=x^{(i;n)}e_{i;n},$ (4.7)
$\displaystyle\lambda_{n}\rho_{n}=\iota_{n},\qquad\rho_{n}\lambda_{n}=e_{n+1;n+1},$
(4.8) $\displaystyle s_{i;n}\lambda_{n}=\lambda_{n}s_{i;n+1},\qquad
e_{i;n}\lambda_{n}=\lambda_{n}e_{i;n+1},\qquad
x^{(i;n)}\lambda_{n}=\lambda_{n}x^{(i;n+1)},$ (4.9)
$\displaystyle\rho_{n}s_{i;n}=s_{i;n+1}\rho_{n},\qquad\rho_{n}e_{i;n}=e_{i;n+1}\rho_{n},\qquad\rho_{n}x^{(i;n)}=x^{(i;n+1)}\rho_{n}.$
(4.10)

######  Theorem 4.11.

The category $M\wr\mathcal{I}$ has presentation
$\textup{{Cat}}\langle{\Gamma_{M\wr\mathcal{I}}}:{\Omega_{M\wr\mathcal{I}}}\rangle$
via $\phi_{M\wr\mathcal{I}}$.

###### Proof.

As we have already said, we will be applying Theorem 2.6. To do so, we must
check that each of assumptions (A1)–(A8) hold, as listed in Subsection 2.4. In
the following we write $\overline{w}=w\phi_{M\wr\mathcal{I}}$ for
$w\in\Gamma_{M\wr\mathcal{I}}^{*}$, and we also write ${\sim}$ for
$\Omega_{M\wr\mathcal{I}}^{\sharp}$ – the congruence on
$\Gamma_{M\wr\mathcal{I}}^{*}$ generated by the relations
$\Omega_{M\wr\mathcal{I}}$.

(A1), (A5) and (A6). These are clear.

(A2) and (A4). These are easily checked diagrammatically (see Figure 7 for
(A2)).

(A3). We require a presentation $\textup{{Mon}}\langle{X_{n}}:{R_{n}}\rangle$
for each endomorphism monoid $M\wr\mathcal{I}_{n}$ ($n\in\mathbb{N}$). For
this we take the presentation from Theorem 3.22, modified to ensure that the
alphabets $X_{n}$ are pairwise disjoint, as explained at the beginning of this
subsection. Explicitly, we have

$X_{n}=\\{s_{1;n},\ldots,s_{n-1;n}\\}\cup\\{e_{1;n},\ldots,e_{n;n}\\}\cup\\{x^{(i;n)}:x\in
X_{M},\ i\in{\bf n}\\},$

and we take $R_{n}$ to be the suitably modified versions of relations
$R_{M\wr\mathcal{I}_{n}}$. The union of these modified relations are precisely
(4.2)–(4.7).

(A7). Here we have $s_{i;n}^{+}=s_{i;n+1}$, and similarly for $e_{i;n}$ and
$x^{(i;n)}$; see (4.9) and (4.10).

(A8). Fix some $n\in\mathbb{N}$ and some $u\in X_{n+1}^{*}$. We must show that

$e_{n+1;n+1}ue_{n+1;n+1}\sim e_{n+1;n+1}v^{+}e_{n+1;n+1}\qquad\text{for some
$v\in X_{n}^{*}$.}$ (4.12)

Here $X_{n}^{*}\to X_{n+1}^{*}:v\mapsto v^{+}$ is the morphism extending the
map $X_{n}\to X_{n+1}^{*}$ from (A7). For simplicity, write $e=e_{n+1;n+1}$,
and consider the element $({\bf a},\alpha)=\overline{eue}\in
M\wr\mathcal{I}_{n+1}$. Because of the terms
$\overline{e}=\overline{e}_{n+1;n+1}$, we see that
$\operatorname{dom}(\alpha)$, $\operatorname{im}(\alpha)$ and
$\operatorname{supp}({\bf a})$ are all contained in ${\bf n}$. Now consider
$({\bf b},\beta)\in M\wr\mathcal{I}_{n}$, where $\beta$ is the restriction of
$\alpha$ to ${\bf n}$, and where ${\bf b}$ consists of the first $n$ entries
of ${\bf a}$. Since $({\bf b},\beta)\in M\wr\mathcal{I}_{n}$, we have $({\bf
b},\beta)=\overline{v}$ for some word $v\in X_{n}^{*}$. By construction, we
have $({\bf a},\alpha)=\overline{ev^{+}e}$. So
$\overline{eue}=\overline{ev^{+}e}$, and since $M\wr\mathcal{I}_{n+1}$ has
presentation $\textup{{Mon}}\langle{X_{n+1}}:{R_{n+1}}\rangle$, it follows
that $eue\sim ev^{+}e$. This completes the proof of (4.12), and hence of the
theorem. ∎

###  4.2 Tensor presentation for $M\wr\mathcal{I}$

We now use Theorem 2.7 to convert the category presentation for
$M\wr\mathcal{I}$ from Theorem 4.11 into a tensor presentation.

We continue to fix the presentation
$\textup{{Mon}}\langle{X_{M}}:{R_{M}}\rangle$ for $M$ via
$\phi_{M}:X_{M}^{*}\to M$. We begin by defining the digraph
$\Delta_{M\wr\mathcal{I}}$ on vertex set $\mathbb{N}$ with edges
$\\{X,U,\rotatebox[origin={c}]{180.0}{$U$}\\}\cup X_{M}$, with sources and
targets given by

$\mathbf{d}(X)=\mathbf{r}(X)=2,\quad\mathbf{d}(U)=\mathbf{r}(\rotatebox[origin={c}]{180.0}{$U$})=\mathbf{d}(x)=\mathbf{r}(x)=1\text{
for all $x\in
X_{M}$}\quad\text{and}\quad\mathbf{r}(U)=\mathbf{d}(\rotatebox[origin={c}]{180.0}{$U$})=0.$

(Note here that ‘$X$’ is the name of a single edge of
$\Delta_{M\wr\mathcal{I}}$.) We define the morphism

$\Phi_{M\wr\mathcal{I}}:\Delta_{M\wr\mathcal{I}}^{\oast}\to
M\wr\mathcal{I}:\begin{cases}X\mapsto\underline{X}=\overline{s}_{1;2},\\\
U\mapsto\underline{U}=\overline{\rho}_{0},\\\
\rotatebox[origin={c}]{180.0}{$U$}\mapsto\underline{\rotatebox[origin={c}]{180.0}{$U$}}=\overline{\lambda}_{0},\\\
x\mapsto\underline{x}=\overline{x}^{(1;1)},\end{cases}$

using the notation of the previous subsection. The images of the generators
under $\Phi_{M\wr\mathcal{I}}$ are shown in Figure 8. In what follows, it is
convenient to write $I=\iota_{1}$ for the empty path $1\to 1$; note that
$\underline{I}$ is also shown in Figure 8. For a term
$w\in\Delta_{M\wr\mathcal{I}}^{\oast}$ we write
$\underline{w}=w\Phi_{M\wr\mathcal{I}}\in M\wr\mathcal{I}$.

$\underline{X}=$$\underline{U}=$$\underline{\rotatebox[origin={c}]{180.0}{$U$}}=$$\underline{x}=$$\overline{x}$$\underline{I}=$
Figure 8: Generators of $M\wr\mathcal{I}$, as well as the identity
$\underline{I}=\overline{\iota}_{1}$.

We now let $\Xi_{M\wr\mathcal{I}}$ be the set of relations over
$\Delta_{M\wr\mathcal{I}}$ consisting of $R_{M}$ (see Remark 4.16) together
with the following, where $x\in X_{M}$ in the third line:

$\displaystyle X\circ X=\iota_{2},\qquad(X\oplus I)\circ(I\oplus
X)\circ(X\oplus I)=(I\oplus X)\circ(X\oplus I)\circ(I\oplus X),$ (4.13)
$\displaystyle\rotatebox[origin={c}]{180.0}{$U$}\circ U=\iota_{0},\qquad
X\circ(U\oplus I)=I\oplus U,\qquad(\rotatebox[origin={c}]{180.0}{$U$}\oplus
I)\circ X=I\oplus\rotatebox[origin={c}]{180.0}{$U$},$ (4.14) $\displaystyle
X\circ(x\oplus I)=(I\oplus x)\circ X,\qquad x\circ
U=U,\qquad\rotatebox[origin={c}]{180.0}{$U$}\circ
x=\rotatebox[origin={c}]{180.0}{$U$}.$ (4.15)

######  Remark 4.16.

We have stipulated that $R_{M}\subseteq\Xi_{M\wr\mathcal{I}}$, and a
clarification is in order here. Any word $u=x_{1}\cdots x_{k}\in X_{M}^{*}$
involved in a relation from $R_{M}$ is identified with the term $u\equiv
x_{1}\circ\cdots\circ x_{k}$ from $\Delta_{M\wr\mathcal{I}}^{\oast}$. If $u$
is empty, then this term is interpreted as the empty path $\iota_{1}=I$.

######  Theorem 4.17.

The category $M\wr\mathcal{I}$ has tensor presentation
$\textup{{TCat}}\langle{\Delta_{M\wr\mathcal{I}}}:{\Xi_{M\wr\mathcal{I}}}\rangle$
via $\Phi_{M\wr\mathcal{I}}$.

###### Proof.

The proof will be via an application of Theorem 2.7. We keep the notation of
Subsection 4.1, in particular $\Gamma_{M\wr\mathcal{I}}$,
$\Omega_{M\wr\mathcal{I}}$, $\phi_{M\wr\mathcal{I}}$ and so on. Throughout the
proof we write ${\approx}$ for the tensor congruence
$(\Xi_{M\wr\mathcal{I}})_{\scriptscriptstyle{\oplus}}^{\sharp}$ on
$\Delta_{M\wr\mathcal{I}}^{\oast}$ generated by the relations
$\Xi_{M\wr\mathcal{I}}$. In the proof of Theorem 4.11 we checked that
assumptions (A1)–(A8) hold, so it remains to check (A9)–(A12).

(A9). It is easy to check that $\underline{u}=\underline{v}$ in
$M\wr\mathcal{I}$ for each relation $(u,v)$ from $\Xi_{M\wr\mathcal{I}}$. For
example, the left-hand diagram of Figure 6 already does this for the first
relation in (4.15).

(A10). We define the mapping
$\Gamma_{M\wr\mathcal{I}}\to\Delta_{M\wr\mathcal{I}}^{\oast}$ as follows, for
all appropriate values of $i,n\in\mathbb{N}$:

  *   * •

$\widehat{s}_{i;n}=\iota_{i-1}\oplus X\oplus\iota_{n-i-1}$,

  * •

$\widehat{e}_{i;n}=\iota_{i-1}\oplus
U\oplus\rotatebox[origin={c}]{180.0}{$U$}\oplus\iota_{n-i}$,

  * •

$\widehat{x}^{(i;n)}=\iota_{i-1}\oplus x\oplus\iota_{n-i}$ for $x\in X_{M}$,

  * •

$\widehat{\lambda}_{n}=\iota_{n}\oplus\rotatebox[origin={c}]{180.0}{$U$}$,

  * •

$\widehat{\rho}_{n}=\iota_{n}\oplus U$.

  *

It is routine to check diagrammatically that $\underline{\hskip
1.99168pt\widehat{x}\hskip 1.99168pt}=\overline{x}$ (i.e.,
$\widehat{x}\Phi_{M\wr\mathcal{I}}=x\phi_{M\wr\mathcal{I}}$) for all
$x\in\Gamma_{M\wr\mathcal{I}}$. We extend this mapping to a morphism
$\Gamma_{M\wr\mathcal{I}}^{*}\to\Delta_{M\wr\mathcal{I}}^{\oast}:w\mapsto\widehat{w}$,
and note that $\widehat{\iota}_{n}=\iota_{n}$ for all $n$.

(A11). We must show that each term $x_{m,n}=\iota_{m}\oplus
x\oplus\iota_{n}\in\Delta_{M\wr\mathcal{I}}^{\oast}$
($x\in\Delta_{M\wr\mathcal{I}}$, $m,n\in\mathbb{N}$) is $\approx$-equivalent
to some term of the form $\widehat{w}$, with
$w\in\Gamma_{M\wr\mathcal{I}}^{*}$. This is clear for $x=X$ or for $x\in
X_{M}$, as

$X_{m,n}=\iota_{m}\oplus
X\oplus\iota_{n}=\widehat{s}_{m+1;m+n+2}\qquad\text{and}\qquad
x_{m,n}=\iota_{m}\oplus x\oplus\iota_{n}=\widehat{x}_{m+1;m+n+1}.$

For the case $x=U$ ($x=\rotatebox[origin={c}]{180.0}{$U$}$ is symmetrical) we
have

$U_{m,n}\approx\widehat{\sigma}_{m+1;m+n+1}\circ\cdots\circ\widehat{\sigma}_{m+n;m+n+1}\circ\widehat{\rho}_{m+n},$

which follows from a diagrammatic check and an application of Theorem 2.5.
(Alternatively it can be proved directly in exactly the same way as [17, Lemma
3.7]; the proof from [17] concerned a different category containing
$\mathcal{I}$, but used only relations from $\Xi_{M\wr\mathcal{I}}$.)

(A12). Consider a relation $(u,v)\in\Omega_{M\wr\mathcal{I}}$ (i.e., one of
(4.2)–(4.10)). We must show that $\widehat{u}\approx\widehat{v}$. In any case
that the terms $\widehat{u}$ and $\widehat{v}$ involve only the generators
$X$, $U$ and  $U$ , this follows immediately from Theorem 2.5 and a simple
diagrammatic check that
$\widehat{u}\Phi_{M\wr\mathcal{I}}=\widehat{v}\Phi_{M\wr\mathcal{I}}$. This
takes care of all relations from $\Omega_{M\wr\mathcal{I}}$ other than
(4.5)–(4.7), and the third parts of each of (4.9) and (4.10). We now consider
these remaining relations.

(4.5). For the first part of this relation, let $(u,v)\in R_{M}$ and $1\leq
i\leq n$; we must show that $\widehat{u}^{(i;n)}\approx\widehat{v}^{(i;n)}$.
Writing $u=x_{1}\cdots x_{k}$, we first use the tensor category axioms to
calculate

$\displaystyle\widehat{u}^{(i;n)}=\widehat{x}_{1}^{(i;n)}\circ\cdots\circ\widehat{x}_{k}^{(i;n)}$
$\displaystyle=(\iota_{i-1}\oplus
x_{1}\oplus\iota_{n-i})\circ\cdots\circ(\iota_{i-1}\oplus
x_{k}\oplus\iota_{n-i})$
$\displaystyle=(\iota_{i-1}\circ\cdots\circ\iota_{i-1})\oplus(x_{1}\circ\cdots\circ
x_{k})\oplus(\iota_{n-i}\circ\cdots\circ\iota_{n-i})$
$\displaystyle=\iota_{i-1}\oplus u\oplus\iota_{n-i}.$

(Keeping Remarks 4.1 and 4.16 in mind, we note that if $u=\iota_{1}$ is empty,
then we still have
$\widehat{u}^{(i;n)}=\widehat{\iota}_{n}=\iota_{n}=\iota_{i-1}\oplus\iota_{1}\oplus\iota_{n-i}=\iota_{i-1}\oplus
u\oplus\iota_{n-i}$.) A similar calculation gives
$\widehat{v}^{(i;n)}=\iota_{i-1}\oplus v\oplus\iota_{n-i}$, so
$\widehat{u}^{(i;n)}\approx\widehat{v}^{(i;n)}$ now follows from the fact that
$R_{M}\subseteq\Xi_{M\wr\mathcal{I}}$.

For the second part of (4.5), let $x,y\in X_{M}$ and $i,j\in{\bf n}$ with
$i\not=j$; by symmetry we can assume that $i<j$. We must show that
$\widehat{x}^{(i;n)}\circ\widehat{y}^{(j;n)}\approx\widehat{y}^{(j;n)}\circ\widehat{x}^{(i;n)}$,
and in fact we have equality using only the tensor category axioms. Indeed, we
have

$\displaystyle\widehat{x}^{(i;n)}\circ\widehat{y}^{(j;n)}$
$\displaystyle=(\iota_{i-1}\oplus x\oplus\iota_{j-1-i}\oplus
I\oplus\iota_{n-j})\circ(\iota_{i-1}\oplus I\oplus\iota_{j-1-i}\oplus
y\oplus\iota_{n-j})$ $\displaystyle=(\iota_{i-1}\circ\iota_{i-1})\oplus(x\circ
I)\oplus(\iota_{j-1-i}\circ\iota_{j-1-i})\oplus(I\circ
y)\oplus(\iota_{n-j}\circ\iota_{n-j})$ $\displaystyle=\iota_{i-1}\oplus
x\oplus\iota_{j-1-i}\oplus y\oplus\iota_{n-j},$

and similarly $\widehat{y}^{(j;n)}\circ\widehat{x}^{(i;n)}=\iota_{i-1}\oplus
x\oplus\iota_{j-1-i}\oplus y\oplus\iota_{n-j}$.

(4.6). The first part is treated in the same fashion as the second part of
(4.5), where we again need only the tensor category axioms. For the second
part of (4.6), it follows quickly from the axioms that

$\widehat{s}_{i;n}\circ\widehat{x}^{(i;n)}=\iota_{i-1}\oplus(X\circ(x\oplus
I))\oplus\iota_{n-i-1}\qquad\text{and}\qquad\widehat{x}^{(i+1;n)}\circ\widehat{s}_{i;n}=\iota_{i-1}\oplus((I\oplus
x)\circ X)\oplus\iota_{n-i-1},$

so
$\widehat{s}_{i;n}\circ\widehat{x}^{(i;n)}\approx\widehat{x}^{(i+1;n)}\circ\widehat{s}_{i;n}$
follows from the first relation in (4.15).

(4.7). The first part is again similar to the second part of (4.5). For the
second part of (4.7), we have

$\widehat{e}_{i;n}\circ\widehat{x}^{(i;n)}=\iota_{i-1}\oplus((U\oplus\rotatebox[origin={c}]{180.0}{$U$})\circ
x)\oplus\iota_{n-i}\qquad\text{and}\qquad\widehat{x}^{(i;n)}\circ\widehat{e}_{i;n}=\iota_{i-1}\oplus(x\circ(U\oplus\rotatebox[origin={c}]{180.0}{$U$}))\oplus\iota_{n-i},$

so here we need to show that $(U\oplus\rotatebox[origin={c}]{180.0}{$U$})\circ
x\approx U\oplus\rotatebox[origin={c}]{180.0}{$U$}\approx
x\circ(U\oplus\rotatebox[origin={c}]{180.0}{$U$})$. But this follows quickly
from the second and third parts of (4.15), and the fact that
$U\oplus\rotatebox[origin={c}]{180.0}{$U$}=U\circ\rotatebox[origin={c}]{180.0}{$U$}$
(more generally, once can easily check that $a\oplus b=a\circ b$ in any tensor
category whenever $\mathbf{r}(a)=\mathbf{d}(b)=0$). Indeed,

$(U\oplus\rotatebox[origin={c}]{180.0}{$U$})\circ
x=(U\circ\rotatebox[origin={c}]{180.0}{$U$})\circ
x=U\circ(\rotatebox[origin={c}]{180.0}{$U$}\circ x)\approx
U\circ\rotatebox[origin={c}]{180.0}{$U$}=U\oplus\rotatebox[origin={c}]{180.0}{$U$},$

and similarly $x\circ(U\oplus\rotatebox[origin={c}]{180.0}{$U$})\approx
U\oplus\rotatebox[origin={c}]{180.0}{$U$}$.

(4.9) and (4.10). As noted earlier, we need only consider the third relation
in each, which amounts to showing that

$\widehat{x}^{(i;n)}\circ\widehat{\lambda}_{n}\approx\widehat{\lambda}_{n}\circ\widehat{x}^{(i;n+1)}\qquad\text{and}\qquad\widehat{\rho}_{n}\circ\widehat{x}^{(i;n)}\approx\widehat{x}^{(i;n+1)}\circ\widehat{\rho}_{n}.$

By symmetry it suffices to just deal with the first, and here again we have
equality, as

$\displaystyle\widehat{x}^{(i;n)}\circ\lambda_{n}=(\widehat{x}^{(i;n)}\oplus\iota_{0})\circ(\iota_{n}\oplus\rotatebox[origin={c}]{180.0}{$U$})=(\widehat{x}^{(i;n)}\circ\iota_{n})\oplus(\iota_{0}\circ\rotatebox[origin={c}]{180.0}{$U$})=\widehat{x}^{(i;n)}\oplus\rotatebox[origin={c}]{180.0}{$U$},$
and
$\displaystyle\widehat{\lambda}_{n}\circ\widehat{x}^{(i;n+1)}=(\iota_{n}\oplus\rotatebox[origin={c}]{180.0}{$U$})\circ(\widehat{x}^{(i;n)}\oplus
I)=(\iota_{n}\circ\widehat{x}^{(i;n)})\oplus(\rotatebox[origin={c}]{180.0}{$U$}\circ
I)=\widehat{x}^{(i;n)}\oplus\rotatebox[origin={c}]{180.0}{$U$}.$

This completes the proof. ∎

##  5 The semigroup $M\wr\operatorname{Sing}(\mathcal{I}_{n})$

In this final section we give a semigroup presentation for the singular wreath
product

$M\wr\operatorname{Sing}(\mathcal{I}_{n})=\big{\\{}({\bf a},\alpha)\in
M_{0}^{n}\times\operatorname{Sing}(\mathcal{I}_{n}):\operatorname{supp}({\bf
a})=\operatorname{dom}(\alpha)\big{\\}}.$

(Recall that
$\operatorname{Sing}(\mathcal{I}_{n})=\mathcal{I}_{n}\setminus\mathcal{S}_{n}$
is the singular part of $\mathcal{I}_{n}$.) Our presentation (see Theorem
5.37) will be built out of the presentation for
$\operatorname{Sing}(\mathcal{I}_{n})$ from [14], stated below in Theorem
5.10, and a new presentation for a certain (singular) subsemigroup of
$M_{0}^{n}$, which we will give in Theorem 5.20. Since
$M\wr\operatorname{Sing}(\mathcal{I}_{0})=\varnothing$ and
$M\wr\operatorname{Sing}(\mathcal{I}_{1})=\\{\varnothing\\}$, we assume that
$n\geq 2$ from here on.

###  5.1 Presentation for $\operatorname{Sing}(\mathcal{I}_{n})$

To state the presentation for $\operatorname{Sing}(\mathcal{I}_{n})$ from
[14], we begin by defining the alphabet

$X_{\operatorname{Sing}(\mathcal{I}_{n})}=\\{f_{i,j}:i,j\in{\bf n},\
i\not=j\\},$ (5.1)

and the morphism

$\phi_{\operatorname{Sing}(\mathcal{I}_{n})}:X_{\operatorname{Sing}(\mathcal{I}_{n})}^{+}\to\operatorname{Sing}(\mathcal{I}_{n}):f_{i,j}\mapsto\overline{f}_{i,j},$
(5.2)

where for distinct $i,j\in{\bf n}$, the partial map
$\overline{f}_{i,j}\in\operatorname{Sing}(\mathcal{I}_{n})$ is defined by

$k\overline{f}_{i,j}=\begin{cases}k&\text{if $k\not=i,j$,}\\\ i&\text{if
$k=j$,}\\\ -&\text{if $k=i$.}\end{cases}$

The maps $\overline{f}_{i,j}$ are pictured in Figure 9. Note that
$\operatorname{dom}(\overline{f}_{i,j})=\\{i\\}^{\complement}$ and
$\operatorname{im}(\overline{f}_{i,j})=\\{j\\}^{\complement}$. Here and
throughout, for $A\subseteq{\bf n}$ we write $A^{\complement}={\bf n}\setminus
A$ for the complement of $A$ in ${\bf n}$.

$1$$i$$j$$n$$1$$j$$i$$n$ Figure 9: The generators
$\overline{f}_{i,j}\in\operatorname{Sing}(\mathcal{I}_{n})$ for $i<j$ (left)
and $i>j$ (right).

Let $R_{\operatorname{Sing}(\mathcal{I}_{n})}$ be the following set of
relations, where $i,j,k,l\in{\bf n}$ are distinct in each:

$\displaystyle f_{i,j}f_{j,i}f_{i,j}$ $\displaystyle=f_{i,j},$ (5.3)
$\displaystyle f_{i,j}^{3}=f_{i,j}^{2}$ $\displaystyle=f_{j,i}^{2},$ (5.4)
$\displaystyle f_{i,j}f_{k,l}$ $\displaystyle=f_{k,l}f_{i,j},$ (5.5)
$\displaystyle f_{i,j}f_{j,i}$ $\displaystyle=f_{i,k}f_{k,i},$ (5.6)
$\displaystyle f_{i,j}f_{i,k}=f_{j,k}f_{i,j}$ $\displaystyle=f_{i,k}f_{j,k},$
(5.7) $\displaystyle f_{k,i}f_{i,j}f_{j,k}$
$\displaystyle=f_{k,j}f_{j,i}f_{i,k},$ (5.8) $\displaystyle
f_{k,i}f_{i,j}f_{j,k}f_{k,l}$ $\displaystyle=f_{k,l}f_{l,i}f_{i,j}f_{j,l}.$
(5.9)

The next result is [14, Theorem 2.1], but we note that the $f_{i,j}$ were
called $e_{ij}$ in [14]:

######  Theorem 5.10.

For $n\geq 2$, the semigroup $\operatorname{Sing}(\mathcal{I}_{n})$ has
presentation
$\textup{{Sgp}}\langle{X_{\operatorname{Sing}(\mathcal{I}_{n})}}:{R_{\operatorname{Sing}(\mathcal{I}_{n})}}\rangle$
via $\phi_{\operatorname{Sing}(\mathcal{I}_{n})}$. ∎

###  5.2 Presentation for $\operatorname{Sing}(M_{0}^{n})$

In this subsection we aim to give a presentation for the semigroup

$\operatorname{Sing}(M_{0}^{n})=M_{0}^{n}\setminus M^{n}$

of all $n$-tuples over $M_{0}$ with at least one zero entry. The notation
$\operatorname{Sing}(M_{0}^{n})$ is for convenience only, and is not meant to
imply that $M^{n}$ is the group of units of $M_{0}^{n}$ (which is only the
case when $M$ is a group). We continue to fix the monoid presentation
$\textup{{Mon}}\langle{X_{M}}:{R_{M}}\rangle$ for $M$ via
$\phi_{M}:X_{M}^{*}\to M$, and we write $\overline{w}=w\phi_{M}$ for $w\in
X_{M}^{*}$. Since we are incorporating this _monoid_ presentation into a
_semigroup_ presentation for $\operatorname{Sing}(M_{0}^{n})$, we will have to
tread carefully when considering empty words.

We begin by defining the alphabet

$X_{\operatorname{Sing}(M_{0}^{n})}=\\{e_{1},\ldots,e_{n}\\}\cup\\{x^{(i;j)}:x\in
X_{M},\ i,j\in{\bf n},\ i\not=j\\},$ (5.11)

and the morphism

$\phi_{\operatorname{Sing}(M_{0}^{n})}:X_{\operatorname{Sing}(M_{0}^{n})}^{+}\to\operatorname{Sing}(M_{0}^{n}):e_{i}\mapsto\overline{e}_{i},\
x^{(i;j)}\mapsto\overline{x}^{(i;j)},$ (5.12)

which is defined diagrammatically in Figure 10. Here as usual we identify
$\operatorname{Sing}(M_{0}^{n})$ with a subsemigroup of
${M\wr\operatorname{Sing}(\mathcal{I}_{n})}$, and we recall the usual
convention about unspecified upper vertex labels. We note that
$\overline{x}^{(i;j)}=\overline{x}^{(i)}\overline{e}_{j}=\overline{e}_{j}\overline{x}^{(i)}$
in the notation of Subsection 3.2, but we also note that
$\overline{x}^{(i)}\not\in\operatorname{Sing}(M_{0}^{n})$. It is worth noting
that

$\operatorname{supp}(\overline{e}_{i})=\\{i\\}^{\complement}\qquad\text{and}\qquad\operatorname{supp}(\overline{x}^{(i;j)})=\\{j\\}^{\complement}$

for appropriate $i,j,x$, again using the $A^{\complement}={\bf n}\setminus A$
notation. As usual, we write
$\overline{w}=w\phi_{\operatorname{Sing}(M_{0}^{n})}$ for $w\in
X_{\operatorname{Sing}(M_{0}^{n})}^{+}$.

$1$$i$$j$$n$$\overline{x}$$1$$j$$i$$n$$\overline{x}$$1$$i$$n$ Figure 10:
Generators of $\operatorname{Sing}(M_{0}^{n})$: $\overline{e}_{i}$ (top), and
$\overline{x}^{(i;j)}$ for $i<j$ (left) and $i>j$ (right).

For a (possibly empty) word $w=x_{1}\cdots x_{k}\in X_{M}^{*}$, and for
distinct $i,j\in{\bf n}$, we define

$w^{(i;j)}=\begin{cases}x_{1}^{(i;j)}\cdots x_{k}^{(i;j)}&\text{if $k\geq
1$,}\\\ e_{j}&\text{if $k=0$.}\end{cases}$

Note that $w^{(i;j)}$ is always a non-empty word over
$X_{\operatorname{Sing}(M_{0}^{n})}$. We now define
$R_{\operatorname{Sing}(M_{0}^{n})}$ to be the following set of relations,
where $x,y\in X_{M}$ and $i,j,k,l\in{\bf n}$ range over all meaningful values,
subject to any stated constraints:

$\displaystyle e_{i}^{2}$ $\displaystyle=e_{i},$ (5.13) $\displaystyle
e_{i}e_{j}$ $\displaystyle=e_{j}e_{i},$ (5.14) $\displaystyle u^{(i;j)}$
$\displaystyle=v^{(i;j)}$ for $(u,v)\in R_{M}$, (5.15) $\displaystyle
x^{(i;j)}e_{k}$ $\displaystyle=e_{k}x^{(i;j)}=e_{j}x^{(i;k)}$ if $k\not=i,j$,
(5.16) $\displaystyle x^{(i;j)}e_{j}$
$\displaystyle=e_{j}x^{(i;j)}=x^{(i;j)},$ (5.17) $\displaystyle
x^{(i;j)}e_{i}$ $\displaystyle=e_{i}x^{(i;j)}=e_{i}e_{j},$ (5.18)
$\displaystyle x^{(i;k)}y^{(j;k)}$ $\displaystyle=y^{(j;k)}x^{(i;k)}$ if
$i\not=j$. (5.19)

For the following proof it is convenient to introduce a piece of notation. For
$a\in M$, and for distinct $i,j\in{\bf n}$, we define the tuple

$a^{(i;j)}=(b_{1},\ldots,b_{n})\qquad\text{where}\qquad
b_{k}=\begin{cases}a&\text{if $k=i$,}\\\ 0&\text{if $k=j$,}\\\
1&\text{otherwise.}\end{cases}$

So $a^{(i;j)}\in\operatorname{Sing}(M_{0}^{n})$ for all such $a,i,j$. When
$a=\overline{x}$ for some $x\in X_{M}$, this agrees with our earlier
definition of $\overline{x}^{(i;j)}$; see Figure 10.

######  Theorem 5.20.

For $n\geq 2$, the semigroup
$\operatorname{Sing}(M_{0}^{n})=M_{0}^{n}\setminus M^{n}$ has presentation

$\textup{{Sgp}}\langle{X_{\operatorname{Sing}(M_{0}^{n})}}:{R_{\operatorname{Sing}(M_{0}^{n})}}\rangle$

via $\phi_{\operatorname{Sing}(M_{0}^{n})}$.

###### Proof.

To show that $\phi_{\operatorname{Sing}(M_{0}^{n})}$ is surjective, let ${\bf
a}=(a_{1},\ldots,a_{n})\in\operatorname{Sing}(M_{0}^{n})$; we must show that
${\bf a}=\overline{w}$ for some $w\in X_{\operatorname{Sing}(M_{0}^{n})}^{+}$.
Re-naming the elements of ${\bf n}$ if necessary, we may assume without loss
of generality that $\operatorname{supp}({\bf a})=\\{1,\ldots,q\\}$ for some
$0\leq q<n$. So in fact, ${\bf a}=(a_{1},\ldots,a_{q},0,\ldots,0)$ with
$a_{1},\ldots,a_{q}\in M$. For $1\leq i\leq q$, we have $a_{i}=w_{i}\phi_{M}$
for some $w_{i}\in X_{M}^{*}$, and it follows that

${\bf a}=\overline{e}_{q+1}\cdots\overline{e}_{n}\cdot a_{1}^{(1;n)}\cdots
a_{q}^{(q;n)}=\overline{e_{q+1}\cdots e_{n}\cdot w_{1}^{(1;n)}\cdots
w_{q}^{(q;n)}},$

as required. (When $q=0$, the above says that ${\bf a}=\overline{e_{1}\cdots
e_{n}}$. If $w_{i}=\iota$ for some $1\leq i\leq q$, then
$a_{i}=\overline{w}_{i}=1$, and
$a_{i}^{(i;n)}=1^{(i;n)}=\overline{e}_{n}=\overline{\iota^{(i;n)}}=\overline{w_{i}^{(i;n)}}$.)

Next, one can easily check that the relations from
$R_{\operatorname{Sing}(M_{0}^{n})}$ are preserved by
$\phi_{\operatorname{Sing}(M_{0}^{n})}$; see Figure 11 for an example
calculation for relation (5.16). It follows that
${R_{\operatorname{Sing}(M_{0}^{n})}^{\sharp}\subseteq\ker(\phi_{\operatorname{Sing}(M_{0}^{n})})}$.
The rest of the proof involves establishing the reverse inclusion. The bulk of
the work goes into proving the following technical lemma. For the rest of the
proof we write ${\sim}$ for $R_{\operatorname{Sing}(M_{0}^{n})}^{\sharp}$.

######  Lemma 5.21.

If $w\in X_{\operatorname{Sing}(M_{0}^{n})}^{+}$ is such that
$\operatorname{supp}(\overline{w})=\\{1,\ldots,q\\}$ for some $0\leq q<n$,
then

$w\sim e_{q+1}\cdots e_{n}\cdot w_{1}^{(1;n)}\cdots
w_{q}^{(q;n)}\qquad\text{for some $w_{1},\ldots,w_{q}\in X_{M}^{*}$.}$

###### Proof.

Before we begin the proof of the lemma, we note that while some of the $w_{i}$
($1\leq i\leq q$) could be empty, the $w_{i}^{(i;n)}$ never are. Indeed, if
$w_{i}=\iota$ for some $i$, then $w_{i}^{(i;n)}=\iota^{(i;n)}=e_{n}$.

Beginning the proof now, we first observe that (either part of) Lemma 2.2 and
relations (5.16)–(5.18) give

$w\sim uv\qquad\text{for some $u\in\\{e_{1},\ldots,e_{n}\\}^{*}$ and
$v\in\\{x^{(i;j)}:x\in X_{M},\ i,j\in{\bf n},\ i\not=j\\}^{*}$.}$

Since $w$ is non-empty, certainly $u$ and $v$ are not both empty, and we write

$u=e_{i_{1}}\cdots e_{i_{s}}\qquad\text{and}\qquad
v=x_{1}^{(j_{1};k_{1})}\cdots x_{t}^{(j_{t};k_{t})}.$

Because of (5.13) and (5.14) we can assume that $i_{1}<\cdots<i_{s}$. Now,

$\displaystyle\\{1,\ldots,q\\}=\operatorname{supp}(\overline{w})$
$\displaystyle=\operatorname{supp}(\overline{e}_{i_{1}})\cap\cdots\cap\operatorname{supp}(\overline{e}_{i_{s}})\cap\operatorname{supp}(\overline{x}_{1}^{(j_{1};k_{1})})\cap\cdots\cap\operatorname{supp}(\overline{x}_{t}^{(j_{t};k_{t})})$
$\displaystyle=\\{i_{1}\\}^{\complement}\cap\cdots\cap\\{i_{s}\\}^{\complement}\cap\\{k_{1}\\}^{\complement}\cap\cdots\cap\\{k_{t}\\}^{\complement}=\\{i_{1},\ldots,i_{s},k_{1},\ldots,k_{t}\\}^{\complement},$

so it follows that

$\\{i_{1},\ldots,i_{s},k_{1},\ldots,k_{t}\\}=\\{q+1,\ldots,n\\}.$ (5.22)

Next we claim that

$w\sim u^{\prime}v\qquad\text{where}\qquad u^{\prime}=e_{q+1}\cdots e_{n}.$
(5.23)

Indeed, this is obvious if $\\{i_{1},\ldots,i_{s}\\}=\\{q+1,\ldots,n\\}$, so
suppose instead that there exists some
$l\in\\{q+1,\ldots,n\\}\setminus\\{i_{1},\ldots,i_{s}\\}$. By (5.22) we have
$l=k_{m}$ for some $1\leq m\leq t$. For convenience we write

$v_{1}=x_{1}^{(j_{1};k_{1})}\cdots
x_{m-1}^{(j_{m-1};k_{m-1})}\qquad\text{and}\qquad
v_{2}=x_{m+1}^{(j_{m+1};k_{m+1})}\cdots x_{t}^{(j_{t};k_{t})}.$ (5.24)

Then by (5.16)–(5.18) we have

$v=v_{1}\cdot x_{m}^{(j_{m};l)}\cdot v_{2}\sim v_{1}\cdot
e_{l}x_{m}^{(j_{m};l)}\cdot v_{2}\sim e_{l}\cdot v_{1}\cdot
x_{m}^{(j_{m};l)}\cdot v_{2}=e_{l}\cdot v.$

It follows that $w\sim uv\sim ue_{l}\cdot v$. Continuing in this way, we have

$w\sim ue_{l_{1}}\cdots e_{l_{r}}\cdot
v\qquad\text{where}\qquad\\{q+1,\ldots,n\\}\setminus\\{i_{1},\ldots,i_{s}\\}=\\{l_{1},\ldots,l_{r}\\}.$

Since

$ue_{l_{1}}\cdots e_{l_{r}}=e_{i_{1}}\cdots e_{i_{s}}\cdot e_{l_{1}}\cdots
e_{l_{r}}\sim e_{q+1}\cdots e_{n}=u^{\prime},$

by (5.14), this completes the proof of (5.23). In what follows, we continue to
write ${u^{\prime}=e_{q+1}\cdots e_{n}}$.

Next we claim that

$w\sim u^{\prime}v^{\prime}\qquad\text{for some
$v^{\prime}\in\\{x^{(i;n)}:x\in X_{M},\ 1\leq i<n\\}^{*}$.}$ (5.25)

This follows immediately from (5.23) if $k_{m}=n$ for all $1\leq m\leq t$ (in
which case we take $v^{\prime}=v$), so suppose instead that $k_{m}\not=n$ for
some such $m$. We prove (5.25) by showing that we can use the relations to
either remove the letter $x_{m}^{(j_{m};k_{m})}$ from $v$ or else replace it
by $x_{m}^{(j_{m};n)}$. Again for convenience we let $v_{1}$ and $v_{2}$ be as
in (5.24). Continuing from (5.23), and using relations (5.13) and
(5.16)–(5.18), we first note that

$w\sim u^{\prime}v\sim u^{\prime}e_{n}\cdot v_{1}\cdot
x_{m}^{(j_{m};k_{m})}\cdot v_{2}\sim u^{\prime}\cdot v_{1}\cdot
e_{n}x_{m}^{(j_{m};k_{m})}\cdot v_{2}.$ (5.26)

From this point we must consider separate cases for $j_{m}=n$ and
$j_{m}\not=n$. For both cases, we use relations (5.13), (5.14) and
(5.16)–(5.18), and we keep $q+1\leq k_{m}<n$ in mind; cf. (5.22). If
$j_{m}=n$, then continuing from (5.26), we have

$w\sim u^{\prime}\cdot v_{1}\cdot e_{n}e_{k_{m}}\cdot v_{2}\sim
u^{\prime}e_{n}e_{k_{m}}\cdot v_{1}\cdot v_{2}\sim u^{\prime}\cdot v_{1}\cdot
v_{2},$

so we have deleted $x_{m}^{(j_{m};k_{m})}$ in this case. On the other hand, if
$j_{m}\not=n$, then continuing from (5.26), we have

$w\sim u^{\prime}\cdot v_{1}\cdot e_{n}x_{m}^{(j_{m};k_{m})}\cdot v_{2}\sim
u^{\prime}\cdot v_{1}\cdot e_{k_{m}}x_{m}^{(j_{m};n)}\cdot v_{2}\sim
u^{\prime}e_{k_{m}}\cdot v_{1}\cdot x_{m}^{(j_{m};n)}\cdot v_{2}\sim
u^{\prime}\cdot v_{1}\cdot x_{m}^{(j_{m};n)}\cdot v_{2},$

so we have replaced $x_{m}^{(j_{m};k_{m})}$ by $x_{m}^{(j_{m};n)}$ in this
case. As noted above, this completes the proof of (5.25).

Next we claim that

$w\sim u^{\prime}v^{\prime\prime}\qquad\text{for some
$v^{\prime\prime}\in\\{x^{(i;n)}:x\in X_{M},\ 1\leq i\leq q\\}^{*}$.}$ (5.27)

To prove this, we continue from (5.25), and we show that any letter
$x^{(i;n)}$ appearing in $v^{\prime}$ with $q<i<n$ can be deleted (using the
relations). To do so, suppose some such letter exists, and write
$v^{\prime}=v_{1}^{\prime}\cdot x^{(i;n)}\cdot v_{2}^{\prime}$. Then using
relations (5.13), (5.14) and (5.16)–(5.18), we have

$w\sim u^{\prime}v^{\prime}\sim u^{\prime}e_{i}\cdot v_{1}^{\prime}\cdot
x^{(i;n)}\cdot v_{2}^{\prime}\sim u^{\prime}\cdot v_{1}^{\prime}\cdot
e_{i}x^{(i;n)}\cdot v_{2}^{\prime}\sim u^{\prime}\cdot v_{1}^{\prime}\cdot
e_{i}e_{n}\cdot v_{2}^{\prime}\sim u^{\prime}e_{i}e_{n}\cdot
v_{1}^{\prime}\cdot v_{2}^{\prime}\sim u^{\prime}\cdot v_{1}^{\prime}\cdot
v_{2}^{\prime},$

as required.

Next, we use (5.19) to ‘unshuffle’ $v^{\prime\prime}$ to obtain

$w\sim u^{\prime}v^{\prime\prime}\sim u^{\prime}\cdot v_{1}\cdots
v_{q}\qquad\text{where $v_{i}\in\\{x^{(i;n)}:x\in X_{M}\\}^{*}$ for each
$1\leq i\leq q$.}$ (5.28)

If some $v_{i}$ is non-empty, then of course it has the form $w_{i}^{(i;n)}$
for some $w_{i}\in X_{M}^{+}$. In particular, if every $v_{i}$ is non-empty,
then we have proved the lemma. So now suppose some $v_{i}=\iota$ is empty.
Then by (5.13) and (5.17) we have

$w\sim u^{\prime}\cdot v_{1}\cdots v_{i-1}\cdot v_{i+1}\cdots v_{q}\sim
u^{\prime}e_{n}\cdot v_{1}\cdots v_{i-1}\cdot v_{i+1}\cdots v_{q}\sim
u^{\prime}\cdot v_{1}\cdots v_{i-1}\cdot e_{n}\cdot v_{i+1}\cdots v_{q},$

so we can replace $v_{i}=\iota$ by $e_{n}=\iota^{(i;n)}$ in (5.28). After
doing this for every $i$ for which $v_{i}=\iota$, the lemma has been proved. ∎

Before returning to the main proof, we also record the following simple lemma.
For the proof, we observe that for any $u,v\in X_{M}^{*}$ and distinct
$i,j\in{\bf n}$, we have $(uv)^{(i;j)}\sim u^{(i;j)}v^{(i;j)}$. Indeed, this
follows immediately from the definitions if $u$ and $v$ are both non-empty,
from (5.13) if both are empty, or from (5.17) if only one is empty.

######  Lemma 5.29.

If $(u,v)\in R_{M}^{\sharp}$, and if $i,j\in{\bf n}$ are distinct, then
$u^{(i;j)}\sim v^{(i;j)}$.

###### Proof.

It suffices to consider the case that $u$ and $v$ differ by a single
application of a relation from $R_{M}$. So we assume that

$u=su^{\prime}t\qquad\text{and}\qquad v=sv^{\prime}t\qquad\text{for some
$s,t\in X_{M}^{*}$ and $(u^{\prime},v^{\prime})\in R_{M}$.}$

But then combining the observation before the lemma with (5.15), we have

$u^{(i;j)}\sim s^{(i;j)}(u^{\prime})^{(i;j)}t^{(i;j)}\sim
s^{(i;j)}(v^{\prime})^{(i;j)}t^{(i;j)}\sim v^{(i;j)},$

as required. ∎

Returning now to the proof of the theorem, suppose
$(u,v)\in\ker(\phi_{\operatorname{Sing}(M_{0}^{n})})$. So ${u,v\in
X_{\operatorname{Sing}(M_{0}^{n})}^{+}}$ and $\overline{u}=\overline{v}$; we
must show that $u\sim v$. Re-naming the elements of ${\bf n}$ if necessary, we
may assume for convenience that
$\operatorname{supp}(\overline{u})=\operatorname{supp}(\overline{v})=\\{1,\ldots,q\\}$
for some $0\leq q<n$. By Lemma 5.21, we have

$u\sim e_{q+1}\cdots e_{n}\cdot u_{1}^{(1;n)}\cdots
u_{q}^{(q;n)}\qquad\text{and}\qquad v\sim e_{q+1}\cdots e_{n}\cdot
v_{1}^{(1;n)}\cdots v_{q}^{(q;n)}\qquad\text{for some $u_{i},v_{i}\in
X_{M}^{*}$.}$

But then

$(\overline{u}_{1},\ldots,\overline{u}_{q},0,\ldots,0)=\overline{u}=\overline{v}=(\overline{v}_{1},\ldots,\overline{v}_{q},0,\ldots,0),$

and so $\overline{u}_{i}=\overline{v}_{i}$ for all $1\leq i\leq q$. Since $M$
has presentation $\textup{{Mon}}\langle{X_{M}}:{R_{M}}\rangle$, it follows
that $(u_{i},v_{i})\in R_{M}^{\sharp}$ for all $i$. Lemma 5.29 then gives
$u_{i}^{(i;n)}\sim v_{i}^{(i;n)}$ for all $i$. Putting everything together, we
finally deduce that

$u\sim e_{q+1}\cdots e_{n}\cdot u_{1}^{(1;n)}\cdots u_{q}^{(q;n)}\sim
e_{q+1}\cdots e_{n}\cdot v_{1}^{(1;n)}\cdots v_{q}^{(q;n)}\sim v,$

and this completes the proof. ∎

$=$$\overline{x}$$i$$j$$k$$=$$i$$j$$k$$\overline{x}$$i$$j$$k$$\overline{x}$
Figure 11: Relation (5.16), picturing only strings $i,j,k$. (Note that no
ordering on $i,j,k$ is implied.)

###  5.3 Presentation for $M\wr\operatorname{Sing}(\mathcal{I}_{n})$

We are now ready to give our presentation for the singular wreath product
$M\wr\operatorname{Sing}(\mathcal{I}_{n})$. We begin by defining the alphabet

$X_{M\wr\operatorname{Sing}(\mathcal{I}_{n})}=X_{\operatorname{Sing}(\mathcal{I}_{n})}\cup
X_{\operatorname{Sing}(M_{0}^{n})},$

where $X_{\operatorname{Sing}(\mathcal{I}_{n})}$ and
$X_{\operatorname{Sing}(M_{0}^{n})}$ are as in (5.1) and (5.11). We define the
morphism

$\phi_{M\wr\operatorname{Sing}(\mathcal{I}_{n})}:X_{M\wr\operatorname{Sing}(\mathcal{I}_{n})}^{+}\to
M\wr\operatorname{Sing}(\mathcal{I}_{n})$

to be the extension of the morphisms

$\phi_{\operatorname{Sing}(\mathcal{I}_{n})}:X_{\operatorname{Sing}(\mathcal{I}_{n})}^{+}\to\operatorname{Sing}(\mathcal{I}_{n})\qquad\text{and}\qquad\phi_{\operatorname{Sing}(M_{0}^{n})}:X_{\operatorname{Sing}(M_{0}^{n})}^{+}\to\operatorname{Sing}(M_{0}^{n})$

from (5.2) and (5.12), where as usual we identify
$\operatorname{Sing}(\mathcal{I}_{n})$ and $\operatorname{Sing}(M_{0}^{n})$
with subsemigroups of $M\wr\operatorname{Sing}(\mathcal{I}_{n})$.

Finally, we let $R_{M\wr\operatorname{Sing}(\mathcal{I}_{n})}$ be the set of
relations over $X_{M\wr\operatorname{Sing}(\mathcal{I}_{n})}$ consisting of
${R_{\operatorname{Sing}(\mathcal{I}_{n})}\cup
R_{\operatorname{Sing}(M_{0}^{n})}}$ (i.e., (5.3)–(5.9) and (5.13)–(5.19)),
together with the following, where $x\in X_{M}$ and $i,j,k,l\in{\bf n}$ range
over all meaningful values in each relation:

$\displaystyle f_{i,j}f_{j,i}$ $\displaystyle=e_{i},$ (5.30) $\displaystyle
f_{i,j}x^{(i;j)}$ $\displaystyle=x^{(j;i)}f_{i,j},$ (5.31) $\displaystyle
f_{i,j}x^{(i;k)}$ $\displaystyle=x^{(j;k)}f_{i,j},$ (5.32) $\displaystyle
f_{i,j}x^{(k;i)}$ $\displaystyle=x^{(k;i)}e_{j},$ (5.33) $\displaystyle
f_{i,j}x^{(j;k)}$ $\displaystyle=f_{i,j}f_{k,j},$ (5.34) $\displaystyle
f_{i,j}x^{(k;j)}$ $\displaystyle=x^{(k;i)}f_{i,j},$ (5.35) $\displaystyle
f_{i,j}x^{(k;l)}$ $\displaystyle=x^{(k;l)}f_{i,j}$ if
$\\{i,j\\}\cap\\{k,l\\}=\varnothing$. (5.36)

Note for example that the ‘all meaningful values’ assumption implies that
$i,j,k$ are distinct in (5.32). On the other hand, (5.34) includes the case in
which $i,j,k$ are distinct, as well as the case that $k=i\not=j$.

######  Theorem 5.37.

For $n\geq 2$, the singular wreath product
$M\wr\operatorname{Sing}(\mathcal{I}_{n})$ has presentation

$\textup{{Sgp}}\langle{X_{M\wr\operatorname{Sing}(\mathcal{I}_{n})}}:{R_{M\wr\operatorname{Sing}(\mathcal{I}_{n})}}\rangle$

via $\phi_{M\wr\operatorname{Sing}(\mathcal{I}_{n})}$.

###### Proof.

Surjectivity of $\phi_{M\wr\operatorname{Sing}(\mathcal{I}_{n})}$ follows
quickly from the surjectivity of $\phi_{\operatorname{Sing}(\mathcal{I}_{n})}$
and $\phi_{\operatorname{Sing}(M_{0}^{n})}$, given that $({\bf a},\alpha)={\bf
a}\cdot\alpha$ for any $({\bf a},\alpha)\in
M\wr\operatorname{Sing}(\mathcal{I}_{n})$; cf. Remark 2.1. One can also easily
check that $\phi_{M\wr\operatorname{Sing}(\mathcal{I}_{n})}$ preserves the
relations; for example, Figure 12 does this for (5.34). This shows that
${R_{M\wr\operatorname{Sing}(\mathcal{I}_{n})}^{\sharp}\subseteq\ker(\phi_{M\wr\operatorname{Sing}(\mathcal{I}_{n})})}$.
Thus, as ever, the bulk of the proof goes into showing the reverse inclusion.
For the rest of the proof, we write ${\sim}$ for
$R_{M\wr\operatorname{Sing}(\mathcal{I}_{n})}^{\sharp}$.

######  Lemma 5.38.

For any $w\in X_{M\wr\operatorname{Sing}(\mathcal{I}_{n})}^{+}$, we have

$w\sim w^{\prime}w^{\prime\prime}\qquad\text{for some $w^{\prime}\in
X_{\operatorname{Sing}(M_{0}^{n})}^{+}$ and $w^{\prime\prime}\in
X_{\operatorname{Sing}(\mathcal{I}_{n})}^{+}$ with
$\operatorname{supp}(\overline{w}^{\prime})=\operatorname{dom}(\overline{w}^{\prime\prime})$.}$

###### Proof.

We first use Lemma 2.2(i) with $X=X_{\operatorname{Sing}(M_{0}^{n})}$ and
$Y=X_{\operatorname{Sing}(\mathcal{I}_{n})}$ to deduce that

$w\sim uv\qquad\text{for some $u\in X_{\operatorname{Sing}(M_{0}^{n})}^{*}$
and $v\in X_{\operatorname{Sing}(\mathcal{I}_{n})}^{*}$.}$ (5.39)

To show that the lemma applies, we need to check that for each $x\in
X_{\operatorname{Sing}(M_{0}^{n})}$ and $y\in
X_{\operatorname{Sing}(\mathcal{I}_{n})}$, we have $yx\sim uv$ for some $u\in
X_{\operatorname{Sing}(M_{0}^{n})}^{*}$ and $v\in
X_{\operatorname{Sing}(\mathcal{I}_{n})}^{*}$ with $\ell(u)\leq 1$. Now, $y$
of course has the form $y=f_{i,j}$. The case that $x=e_{k}$ follows from
(5.30), as then

$yx=f_{i,j}e_{k}\sim f_{i,j}f_{k,j}f_{j,k},$

and we take $u=\iota$ and $v=f_{i,j}f_{k,j}f_{j,k}$. For the case that
$x=x^{(k;l)}$, we just need to observe that (5.31)–(5.36) contains a relation
$(yx,uv)$ of the desired form in each case. We now fix $u$ and $v$ as in
(5.39). Next we claim that

$w\sim w^{\prime}v\qquad\text{for some $w^{\prime}\in
X_{\operatorname{Sing}(M_{0}^{n})}^{+}$ with
$\operatorname{supp}(\overline{w}^{\prime})\subseteq\operatorname{dom}(\overline{v})$.}$
(5.40)

Indeed, there is nothing to show if $\operatorname{dom}(\overline{v})={\bf n}$
(i.e., if $v=\iota$), so suppose instead that
${\operatorname{dom}(\overline{v})\not={\bf n}}$, and fix some
$i\in\operatorname{dom}(\overline{v})^{\complement}$. Since $n\geq 2$, we may
also fix some $j\in\\{i\\}^{\complement}$. Since
$i\not\in\operatorname{dom}(\overline{v})$, and since
$\overline{f}_{i,j}\overline{f}_{j,i}=\operatorname{id}_{\\{i\\}^{\complement}}$,
we have $\overline{v}=\overline{f}_{i,j}\overline{f}_{j,i}\overline{v}$. It
follows from Theorem 5.10 that $v\sim f_{i,j}f_{j,i}v$. We then combine this
with (5.30) to calculate

$w\sim uv\sim uf_{i,j}f_{j,i}\cdot v\sim ue_{i}\cdot v.$

Continuing in this way, and writing
$\operatorname{dom}(\overline{v})^{\complement}=\\{i_{1},\ldots,i_{k}\\}$, we
have

$w\sim ue_{i_{1}}\cdots e_{i_{k}}\cdot v.$

We see then that (5.40) holds with $w^{\prime}=ue_{i_{1}}\cdots e_{i_{k}}$, as

$\displaystyle\operatorname{supp}(\overline{w}^{\prime})$
$\displaystyle=\operatorname{supp}(\overline{u})\cap\operatorname{supp}(\overline{e}_{i_{1}})\cap\cdots\cap\operatorname{supp}(\overline{e}_{i_{k}})$
$\displaystyle=\operatorname{supp}(\overline{u})\cap\\{i_{1}\\}^{\complement}\cap\cdots\cap\\{i_{k}\\}^{\complement}$
$\displaystyle=\operatorname{supp}(\overline{u})\cap\\{i_{1},\ldots,i_{k}\\}^{\complement}\subseteq\\{i_{1},\ldots,i_{k}\\}^{\complement}=\operatorname{dom}(\overline{v}).$

We now fix $w^{\prime}$ and $v$ as in (5.40). Also fix some
$i\in\operatorname{supp}(\overline{w}^{\prime})^{\complement}$, and some
$j\in\\{i\\}^{\complement}$. Since
$i\not\in\operatorname{supp}(\overline{w}^{\prime})$, we have
$\overline{w}^{\prime}=\overline{w}^{\prime}\overline{e}_{i}$, and so
$w^{\prime}\sim w^{\prime}e_{i}$ by Theorem 5.20. Combining this with (5.30),
it follows that

$w\sim w^{\prime}v\sim w^{\prime}\cdot e_{i}v\sim w^{\prime}\cdot
f_{i,j}f_{j,i}v.$

Continuing in this way, writing
$\operatorname{supp}(\overline{w}^{\prime})^{\complement}=\\{i_{1},\ldots,i_{k}\\}$,
and fixing some $j_{s}\in\\{i_{s}\\}^{\complement}$ for each $1\leq s\leq k$,
we have

$w\sim
w^{\prime}\cdot(f_{i_{1},j_{1}}f_{j_{1},i_{1}})\cdots(f_{i_{k},j_{k}}f_{j_{k},i_{k}})\cdot
v.$

We see then that the lemma holds with
$w^{\prime\prime}=(f_{i_{1},j_{1}}f_{j_{1},i_{1}})\cdots(f_{i_{k},j_{k}}f_{j_{k},i_{k}})\cdot
v$. Indeed, we first note that

$\overline{(f_{i_{1},j_{1}}f_{j_{1},i_{1}})\cdots(f_{i_{k},j_{k}}f_{j_{k},i_{k}})}=\operatorname{id}_{\\{i_{1}\\}^{\complement}}\cdots\operatorname{id}_{\\{i_{k}\\}^{\complement}}=\operatorname{id}_{\\{i_{1},\ldots,i_{k}\\}^{\complement}}=\operatorname{id}_{\operatorname{supp}(\overline{w}^{\prime})}.$

But then

$\operatorname{dom}(\overline{w}^{\prime\prime})=\operatorname{dom}(\operatorname{id}_{\operatorname{supp}(\overline{w}^{\prime})}\cdot\overline{v})=\operatorname{dom}(\operatorname{id}_{\operatorname{supp}(\overline{w}^{\prime})})\cap\operatorname{dom}(\overline{v})=\operatorname{supp}(\overline{w}^{\prime})\cap\operatorname{dom}(\overline{v})=\operatorname{supp}(\overline{w}^{\prime}),$

where the last equality follows from the fact that
$\operatorname{supp}(\overline{w}^{\prime})\subseteq\operatorname{dom}(\overline{v})$;
cf. (5.40). ∎

Returning to the proof of the theorem, let
$(u,v)\in\ker(\phi_{M\wr\operatorname{Sing}(\mathcal{I}_{n})})$, so that
$u,v\in X_{M\wr\operatorname{Sing}(\mathcal{I}_{n})}^{+}$ and
$\overline{u}=\overline{v}$; we must show that $u\sim v$. By Lemma 5.38, we
have

$u\sim u^{\prime}u^{\prime\prime}\qquad\text{and}\qquad v\sim
v^{\prime}v^{\prime\prime}$

for some $u^{\prime},v^{\prime}\in X_{\operatorname{Sing}(M_{0}^{n})}^{+}$ and
$u^{\prime\prime},v^{\prime\prime}\in
X_{\operatorname{Sing}(\mathcal{I}_{n})}^{+}$ with
$\operatorname{supp}(\overline{u}^{\prime})=\operatorname{dom}(\overline{u}^{\prime\prime})$
and
${\operatorname{supp}(\overline{v}^{\prime})=\operatorname{dom}(\overline{v}^{\prime\prime})}$.
Keeping Remark 2.1 in mind, it follows that

$(\overline{u}^{\prime},\overline{u}^{\prime\prime})=\overline{u}^{\prime}\cdot\overline{u}^{\prime\prime}=\overline{u}=\overline{v}=\overline{v}^{\prime}\cdot\overline{v}^{\prime\prime}=(\overline{v}^{\prime},\overline{v}^{\prime\prime}),$

and so $\overline{u}^{\prime}=\overline{v}^{\prime}$ (in
$\operatorname{Sing}(M_{0}^{n})$) and
$\overline{u}^{\prime\prime}=\overline{v}^{\prime\prime}$ (in
$\operatorname{Sing}(\mathcal{I}_{n})$). It then follows from Theorems 5.10
and 5.20 that $u^{\prime}\sim v^{\prime}$ and $u^{\prime\prime}\sim
v^{\prime\prime}$, and so $u\sim u^{\prime}u^{\prime\prime}\sim
v^{\prime}v^{\prime\prime}\sim v$, completing the proof of the theorem. ∎

$=$$k=i\phantom{{}=k}$$j$$\overline{x}$$k=i\phantom{{}=k}$$j$$=$$i$$j$$k$$\overline{x}$$i$$j$$k$
Figure 12: Relation (5.34), picturing only strings $i,j,k$, and considering
the cases $i=k$ (left) and $i\not=k$ (right). (Note that no ordering on
$i,j,k$ is implied.)

In light of relation (5.30), one could remove the generators $e_{i}$
($i\in{\bf n}$) from the presentation
$\textup{{Sgp}}\langle{X_{M\wr\operatorname{Sing}(\mathcal{I}_{n})}}:{R_{M\wr\operatorname{Sing}(\mathcal{I}_{n})}}\rangle$,
replacing each occurrence of $e_{i}$ in the relations by some fixed word of
the form $f_{i,j}f_{j,i}$ ($j\in\\{i\\}^{\complement}$). We note however, that
the resulting set of relations would not be quite as symmetrical as
$R_{M\wr\operatorname{Sing}(\mathcal{I}_{n})}$.

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