# On unit-regular elements in various monoids of transformations

Mosarof Sarkar Department of Mathematics, Central University of South Bihar,
Gaya, Bihar, India<EMAIL_ADDRESS>and  Shubh N. Singh Department of
Mathematics, Central University of South Bihar, Gaya, Bihar, India
<EMAIL_ADDRESS>
###### Abstract.

Let $X$ be an arbitrary set and let $T(X)$ denote the full transformation
monoid on $X$. We prove that an element of $T(X)$ is unit-regular if and only
if it is semi-balanced. For infinite $X$, we discuss regularity of the
submonoid of $T(X)$ consisting of all injective (resp. surjective)
transformations. For a partition $\mathcal{P}$ of $X$, we characterize unit-
regular elements in the monoid $T(X,\mathcal{P})$, under composition, defined
as

$T(X,\mathcal{P})=\\{f\in T(X)\mid(\forall X_{i}\in\mathcal{P})(\exists
X_{j}\in\mathcal{P})\;X_{i}f\subseteq X_{j}\\}.$

We also characterize (unit-)regular elements in various known submonoids of
$T(X,\mathcal{P})$.

###### Key words and phrases:

Transformation monoids, Partitions, Regular elements, Unit-regular elements.

###### 2010 Mathematics Subject Classification:

20M15, 20M20.

##  1\. Introduction

We assume that the reader is familiar with the basic terminology of algebraic
semigroup theory. An element $x$ of a monoid $M$ is said to be _unit-regular_
in $M$ if there exists a unit $u\in M$ such that $xux=x$; and monoids
consisting entirely of unit-regular elements are called _unit-regular_. The
concept of unit-regularity, which is a stronger form of regularity, first
appeared in the context of rings [12]. There are several natural and
motivating examples of such monoids. The unit-regularity in a monoid has
received wide attention by a number of authors (see, e.g., [4, 5, 6, 7, 15,
16, 19, 20, 32]).

Throughout this paper, let $X$ be an arbitrary nonempty set,
$\mathcal{P}=\\{X_{i}|\;i\in I\\}$ be a partition of $X$, $E$ be the
equivalence relation induced by $\mathcal{P}$, and $T(X)$ be the full
transformation monoid on $X$. It is well-known that $T(X)$ is regular (cf.
[18, p. 63, Exercise 2.6.15]). It is proven in [7, Proposition 5] that $T(X)$
is unit-regular if and only if $X$ is finite. Pei [24, Corollary 2.3]
characterized regular elements in the monoid $T(X,\mathcal{P})$, under
composition, defined as

$\displaystyle T(X,\mathcal{P})$ $\displaystyle=\\{f\in T(X)\mid(\forall
X_{i}\in\mathcal{P})(\exists X_{j}\in\mathcal{P})\;X_{i}f\subseteq X_{j}\\}$
$\displaystyle=\\{f\in T(X)\mid(x,y)\in E\Longrightarrow(xf,yf)\in E\\}.$

Pei [24, Proposition 2.4] also showed that $T(X,\mathcal{P})$ is regular if
and only if $\mathcal{P}$ is trivial. For finite $X$, the authors [28, Theorem
3.4] characterized unit-regular elements in $T(X,\mathcal{P})$. Several
interesting results of $T(X,\mathcal{P})$ and its submonoids have also been
investigated (see. e.g., [1, 2, 3, 8, 9, 10, 11, 13, 14, 21, 22, 23, 24, 25,
26, 29, 30, 31]).

In recent years, the submonoids

$\displaystyle\Sigma(X,\mathcal{P})$ $\displaystyle=\\{f\in
T(X,\mathcal{P})\mid(\forall X_{i}\in\mathcal{P})\;Xf\cap
X_{i}\neq\emptyset\\},$ $\displaystyle\ \Gamma(X,\mathcal{P})$
$\displaystyle=\\{f\in T(X,\mathcal{P})\mid(\forall
X_{i}\in\mathcal{P})(\exists X_{j}\in\mathcal{P})\;X_{i}f=X_{j}\\},$
$\displaystyle\ T_{E^{*}}(X)$ $\displaystyle=\\{f\in T(X)\mid(\forall x,y\in
X)(x,y)\in E\Longleftrightarrow(xf,yf)\in E\\}$

of $T(X,\mathcal{P})$ have also been considered. Moreover, the regular
elements in $\Gamma(X,\mathcal{P})$ and $T_{E^{*}}(X)$ have been characterized
(cf. [28, Theorem 5.5] and [8, Theorem 3.1], respectively). For finite $X$,
the authors [28, Proposition 3.5] characterized unit-regular elements in
$\Sigma(X,\mathcal{P})$ and showed in [28, Proposition 5.8] that every regular
element in $\Gamma(X,\mathcal{P})$ is unit-regular.

Let

$\displaystyle\Omega(X,\mathcal{P})$ $\displaystyle=\\{f\in
T(X,\mathcal{P})\mid(\forall x,y\in X,x\neq y)(x,y)\in E\Longrightarrow xf\neq
yf\\}.$

It is routine to verify that $\Omega(X,\mathcal{P})$ is a submonoid of
$T(X,\mathcal{P})$. Clearly, if $E=\\{(x,x)\mid x\in X\\}$, then
$\Omega(X,\mathcal{P})=T(X)$; and if $E=X\times X$, then
$\Omega(X,\mathcal{P})=\\{f\in T(X)\mid f\mbox{ is injective}\\}$.

The rest of this paper is structured as follows. In the next section, we
provide definitions and notation used throughout the paper. In Section $3$, we
prove that an element of $T(X)$ is unit-regular if and only if it is semi-
balanced. For infinite $X$, we discuss regularity of the submonoids
$\Omega(X)=\\{f\in T(X)\mid f\mbox{ is injective}\\}$ and $\Gamma(X)=\\{f\in
T(X)\mid f\mbox{ is surjective}\\}$ of $T(X)$. In Section $4$, we characterize
unit-regular elements in $T(X,\mathcal{P})$. In Section $5$, we first
characterize regular elements in $\Sigma(X,\mathcal{P})$ and then characterize
unit-regular elements in $\Sigma(X,\mathcal{P})$, $\Gamma(X,\mathcal{P})$, and
$T_{E^{*}}(X)$. In Section $6$, we finally characterize regular and unit-
regular elements in $\Omega(X,\mathcal{P})$.

##  2\. Preliminaries and Notation

Let $X$ be a set. The cardinality of $X$ is denoted by $|X|$. For
$A,B\subseteq X$, we write $A\setminus B$ to denote the set of elements of $A$
that are not in $B$. A _partition_ of $X$ is a collection of nonempty disjoint
subsets, called _blocks_ , whose union is $X$. A partition is called _trivial_
if it has only singleton blocks or a single block. We denote by $\Delta(X)$
the equivalence relation $\\{(x,x)\mid x\in X\\}$ on $X$. A _cross-section_ of
an equivalence relation $E$ on $X$ is a subset of $X$ which intersects every
equivalence class of $E$ in exactly one point. The letter $I$ will be reserved
for an indexing set.

We write maps to the right of their arguments, and compose them accordingly.
The composition of maps will be denoted simply by juxtaposition. Let $f\colon
X\to Y$ be a map. We denote by $Af$ the image of a subset $A\subseteq X$ under
$f$. The preimage of a subset $B\subseteq Y$ under $f$ is denoted by
$Bf^{-1}$. We write $\mbox{dom}(f)$, $\mbox{codom}(f)$, and $\mbox{im}(f)$ for
the domain, codomain, and image of $f$, respectively. The _defect_ of $f$,
denoted by $\mbox{d}(f)$, is defined by
$\mbox{d}(f)=|Y\setminus\mbox{im}(f)|$. The _kernel_ of $f$, denoted by
$\ker(f)$, is an equivalence relation on $X$ defined by $\ker(f)=\\{(x,y)\in
X\times X|\;xf=yf\\}$. We denote by $\pi(f)$ the partition of $X$ induced by
$\ker(f)$, namely $\pi(f)=\\{yf^{-1}|\;y\in Xf\\}$. Note that a map $f$ is
injective if and only if $\ker(f)=\Delta(X)$. We write $T_{f}$ to denote any
cross-section of $\ker(f)$. Note that $|T_{f}|=|\mbox{im}(f)|$. The _collapse_
of $f$, denoted by $\mbox{c}(f)$, is defined by $\mbox{c}(f)=|X\setminus
T_{f}|$ (cf. [17, p. 1356]). If $A\subseteq X$ and $B\subseteq Y$ such that
$Af\subseteq B$, then there is a map $g\colon A\to B$ defined by $xg=xf$ for
all $x\in A$ and, in this case, we say that the map $g$ _is induced by_ $f$.

Let $M$ be a monoid with identity $e$. An element $x\in M$ is called _unit_ if
there exists $y\in M$ such that $xy=yx=e$. The set $U(M)$ of all units in $M$
forms a subgroup of $M$, and is called the _group of units_ of $M$. An element
$x\in M$ is said to be _regular_ in $M$ if there exists $y\in M$ such that
$xyx=x$. Such an element $y$ is called an _inner inverse_ for $x$, and it need
not be unique. Denote $U(x)=\\{y\in U(M)\;|\;x=xyx\\}$. If in addition, the
inner inverse $y$ is a unit, then $x$ is said to be _unit-regular_ in $M$. By
$\mbox{reg}(M)$ and $\mbox{ureg}(M)$, we mean the sets of all regular and
unit-regular elements in $M$, respectively. If $\mbox{reg}(M)=M$ (resp.
$\mbox{ureg}(M)=M)$, we say that the monoid $M$ is _regular_ (resp. _unit-
regular_). Note that $U(M)\subseteq\mbox{ureg}(M)\subseteq\mbox{reg}(M)$, and
if $M$ is unit-regular then any submonoid of $M$ containing $U(M)$ is also
unit-regular.

Let $X$ be a nonempty set. A _selfmap_ on $X$ is a map from $X$ into $X$. A
_permutation_ of $X$ is a bijective selfmap on $X$. We say that a selfmap $f$
on $X$ _preserves_ a partition $\mathcal{P}$ of $X$ if for every
$X_{i}\in\mathcal{P}$, there exists $X_{j}\in\mathcal{P}$ such that
$X_{i}f\subseteq X_{j}$. We denote by $S(X)$ the symmetric group on $X$. We
write $\Omega(X)$ (resp. $\Gamma(X)$) to denote the submonoids of $T(X)$
consisting of all injective (resp. surjective) selfmaps on $X$. Note that the
monoids $T(X)$, $\Omega(X)$, and $\Gamma(X)$ have the symmetric group $S(X)$
as their group of units; and $\Omega(X)=\Gamma(X)=S(X)$ if and only if $X$ is
finite.

We refer the reader to the standard book [18] for additional information from
algebraic semigroup theory.

##  3\. Unit-regular elements in $T(X)$

In this section, we characterize the unit-regular elements in $T(X)$. We then
provide an alternative proof of Proposition 5 in [7] which characterizes the
unit-regularity of $T(X)$. We also characterize the regularity of $\Omega(X)$
and $\Gamma(X)$. We begin by proving the following lemma.

######  Lemma 3.1.

Let $f\colon X\to Y$ and $g\colon Y\to X$ be maps. If $fgf=f$, then
$\mbox{im}(fg)$ is a cross-section of the equivalence relation $\ker(f)$.

###### Proof.

Let $yf^{-1}\in\pi(f)$. It suffices to show that $|(Xf)g\cap yf^{-1}|=1$.
Since $y\in Xf$, there exists $x\in X$ such that $xf=y$. Write $yg=z$. Clearly
$z=yg\in(Xf)g$. Since $fgf=f$, we obtain

$zf=(yg)f=(xf)gf=xf=y.$

It follows that $z\in yf^{-1}$, so $|(Xf)g\cap yf^{-1}|\geq 1$.

Moreover, suppose that $z_{1},z_{2}\in(Xf)g\cap yf^{-1}$. Then
$z_{1}f=y=z_{2}f$. As $z_{1},z_{2}\in(Xf)g$, there exist $y_{1},y_{2}\in Xf$
such that $y_{1}g=z_{1}$ and $y_{2}g=z_{2}$. Since $y_{1},y_{2}\in Xf$, there
exist $x_{1},x_{2}\in X$ such that $x_{1}f=y_{1}$ and $x_{2}f=y_{2}$. Since
$fgf=f$, we obtain

$y_{1}=x_{1}f=(x_{1}f)gf=(y_{1}g)f=z_{1}f=y.$

Similarly, we obtain $y_{2}=y$. It follows that $y_{1}=y_{2}$, so
$z_{1}=z_{2}$. Hence $|(Xf)g\cap yf^{-1}|=1$. This completes the proof. ∎

The following corollary is a direct consequence of Lemma 3.1 by replacing $Y$
with $X$.

######  Corollary 3.2.

Let $f,g\in T(X)$. If $fgf=f$, then $\mbox{im}(fg)$ is a cross-section of the
equivalence relation $\ker(f)$.

Before we begin to prove one of the main theorems, we recall an important
definition from [17].

######  Definition 3.3.

[17, p. 1356] An element $f$ of $T(X)$ is said to be _semi-balanced_ if
$\mbox{c}(f)=\mbox{d}(f)$ .

The following theorem gives a characterization of unit-regular elements in
$T(X)$.

######  Theorem 3.4.

Let $f\in T(X)$. Then $f\in\operatorname{\mbox{ureg}}(T(X))$ if and only if
$f$ is semi-balanced.

###### Proof.

Suppose first that $f\in\operatorname{\mbox{ureg}}(T(X))$. Then there exists
$g\in S(X)$ such that $fgf=f$. Therefore, by Corollary 3.2, the set
$\mbox{im}(fg)$ is a cross-section of the equivalence relation $\ker(f)$.
Write $\mbox{im}(fg)=T_{f}$. Since $g$ is bijective, we obtain

$(X\setminus Xf)g=Xg\setminus(Xf)g=X\setminus T_{f}.$

It follows that $\mbox{c}(f)=|X\setminus T_{f}|=|X\setminus Xf|=\mbox{d}(f)$,
so $f$ is semi-balanced.

Conversely, suppose that $f$ is semi-balanced. Note that $|T_{f}|=|Xf|$ and
$|T_{f}\cap yf^{-1}|=1$ for all $yf^{-1}\in\pi(f)$. Therefore, we choose a
bijection $g_{1}\colon Xf\to T_{f}$ such that $yg_{1}=x$ whenever $xf=y$ and
$x\in T_{f}$. Since $|X\setminus T_{f}|=|X\setminus Xf|$, we arbitrarily
choose any bijection $g_{2}\colon X\setminus Xf\to X\setminus T_{f}$. Using
these two bijections, we now define a map $g\colon X\to X$ by

$yg=\begin{cases}yg_{1}&\text{if $y\in Xf$},\\\ yg_{2}&\text{if $y\in
X\setminus Xf$}.\end{cases}$

Clearly $g\in S(X)$. One can also verify in a routine manner that $fgf=f$.
Hence $f\in\operatorname{\mbox{ureg}}(T(X))$. ∎

The following simple lemma will be useful in the sequel.

######  Lemma 3.5.

Let $f\colon X\to Y$ be a map. Then

  1. (i)

$f$ is injective if and only if $\mbox{c}(f)=0$;

  2. (ii)

$f$ is surjective if and only if $\mbox{d}(f)=0$.

If $X$ is finite, then every element of $T(X)$ is semi-balanced (cf. [17, p.
1356]). However, we have the following lemma.

######  Lemma 3.6.

Every element of $T(X)$ is semi-balanced if and only if $X$ is finite.

###### Proof.

Suppose that every element of $T(X)$ is semi-balanced. Assume, to the
contrary, that $X$ is an infinite set. Choose an arbitrary element $x$ of $X$.
Note that the sets $X$ and $X\setminus\\{x\\}$ have the same cardinality.
Therefore there exists is a bijective map $g\colon X\to X\setminus\\{x\\}$, so
the map $g\colon X\to X$ is injective but not surjective. Then we get
$\mbox{c}(g)=0$ and $\mbox{d}(g)\geq 1$ by Lemma 3.5. This leads to a
contradiction. Hence $X$ is finite.

The converse is immediate. ∎

From Lemma 3.6 and Theorem 3.4, we have the corollary below that characterizes
unit-regularity of $T(X)$.

######  Corollary 3.7.

The monoid $T(X)$ is unit-regular if and only if $X$ is finite.

###### Proof.

Suppose that $T(X)$ is unit-regular. Then, by Theorem 3.4, every element of
$T(X)$ is semi-balanced. Hence $X$ is finite by Lemma 3.6.

Conversely, suppose that $X$ is finite and let $f\in T(X)$. Then $f$ is semi-
balanced by Lemma 3.6. It follows from Theorem 3.4 that
$f\in\operatorname{\mbox{ureg}}(T(X))$. Since $f$ is an arbitrary element, the
monoid $T(X)$ is unit-regular. ∎

If $X$ is a finite set and $f\in T(X)$, the next proposition gives the
cardinality of the set $U(f)$.

######  Proposition 3.8.

Let $X$ be a finite set and $f\in T(X)$. Then

$|U(f)|=\mbox{d}(f)!\displaystyle\prod_{x\in Xf}|xf^{-1}|.$

###### Proof.

Let $f\in T(X)$. Since $X$ is finite, there exists $g\in S(X)$ such that
$fgf=f$ by Corollary 3.7. Then $\mbox{im}(fg)$ is a cross-section of $\ker(f)$
by Corollary 3.2. Write $T_{f}=\mbox{im}(fg)$. Note that $|T_{f}|=|Xf|$ and
$T_{f}$ depends on $g$. Observe that any such bijection $g\in S(X)$ can be
only of the following form:

$yg=\begin{cases}yg_{1}&\text{if $y\in Xf$},\\\ yg_{2}&\text{if $y\in
X\setminus Xf$},\end{cases}$

where $g_{1}\colon Xf\to T_{f}$ is a bijection defined by $yg_{1}=x$ whenever
$xf=y$ and $x\in T_{f}$; and $g_{2}\colon X\setminus Xf\to X\setminus T_{f}$
is any bijection. Note that for every cross-section $T_{f}$ of $\ker(f)$ there
is unique such bijection $g_{1}$ and $\mbox{d}(f)!$ number of such bijections
$g_{2}$.

By the multiplication principle, the total number of possible cross-sections
$T_{f}$ of $\ker(f)$ is $\prod_{y\in Xf}|yf^{-1}|$. Hence

$|U(f)|=\mbox{d}(f)!\;\prod_{y\in Xf}|yf^{-1}|$

by the multiplication principle. ∎

We now characterize regularity of $\Omega(X)$ in the following proposition.

######  Proposition 3.9.

The monoid $\Omega(X)$ is regular if and only if $\Omega(X)=S(X)$.

###### Proof.

Suppose that $\Omega(X)$ is regular. Since $S(X)$ is the group of units of
$\Omega(X)$, we have $S(X)\subseteq\Omega(X)$. For the reverse inclusion, let
$f\in\Omega(X)$. Then $\mbox{c}(f)=0$ by Lemma 3.5(i). Since
$f\in\operatorname{\mbox{reg}}(\Omega(X))$, there exists $g\in\Omega(X)$ such
that $fgf=f$. It follows from Corollary 3.2 that $\mbox{im}(fg)$ is a cross-
section of $\ker(f)$. Since $\mbox{c}(f)=0$ and $\mbox{im}(fg)$ is a cross-
section of $\ker(f)$, it follows that $(Xf)g=X$ and therefore
$\mbox{d}(fg)=0$. Since $g$ is injective, we subsequently have $\mbox{d}(f)=0$
by [17, Theorem 2.2(iv)]. Hence $f$ is surjective by Lemma 3.5(ii) and
consequently $f\in S(X)$.

The converse is immediate. ∎

The next proposition provides a characterization of regularity of $\Gamma(X)$.

######  Proposition 3.10.

The monoid $\Gamma(X)$ is regular if and only if $\Gamma(X)=S(X)$.

###### Proof.

Suppose that $\Gamma(X)$ is regular. Since $S(X)$ is the group of units of
$\Gamma(X)$, we have $S(X)\subseteq\Gamma(X)$. For the reverse inclusion, let
$f\in\Gamma(X)$. Since $f\in\operatorname{\mbox{reg}}(\Gamma(X))$, there
exists $g\in\Gamma(X)$ such that $fgf=f$. It follows from Corollary 3.2 that
$\mbox{im}(fg)$ is a cross-section of $\ker(f)$. Since $f,g\in\Gamma(X)$, we
obtain $\mbox{im}(fg)=(Xf)g=Xg=X$. But $\mbox{im}(fg)$ is a cross-section of
$\ker(f)$, we therefore have $\mbox{c}(f)=0$. Hence $f$ is injective by Lemma
3.5(i) and consequently $f\in S(X)$.

The converse is immediate. ∎

The following corollary is obvious from Proposition 3.9 and 3.10 .

######  Corollary 3.11.

Let $M=\Omega(X),\;\Gamma(X)$. If $M$ is regular, then $M$ is unit-regular.

##  4\. Unit-regular elements in $T(X,\mathcal{P})$

In this section, we characterize unit-regular elements in $T(X,\mathcal{P})$.
For finite $X$, recall that the authors [28, Theorem 3.4] characterized unit-
regular elements in $T(X,\mathcal{P})$. We begin by considering an important
definition corresponding to a map of $T(X,\mathcal{P})$ from [27].

######  Definition 4.1.

([27, p. 220]) Let $\mathcal{P}=\\{X_{i}|\;i\in I\\}$ be a partition of $X$
and let $f\in T(X,\mathcal{P})$. The _character_ of $f$, denoted by
$\chi^{(f)}$, is a map $\chi^{(f)}\colon I\to I$ defined by

$i\chi^{(f)}=j\mbox{ whenever }X_{i}f\subseteq X_{j}.$

In case of finite $X$, the map $\chi^{(f)}$ has also been studied with the
notation $\overline{f}$ (see, e.g., [1, 10, 11]). The authors gave a
characterization of elements of $\Sigma(X,\mathcal{P})$ and $T_{E^{*}}(X)$ in
terms of $\chi^{(f)}$ in [29, Theorem 3.2] and [29, Theorem 3.4],
respectively.

Note that the monoid $T(I)$ is regular (cf. [18, p. 63, Exercise 2.6.15]), but
need not be unit-regular (cf. [7, Proposition 5]). We now prove the following
lemma.

######  Lemma 4.2.

Let $\mathcal{P}=\\{X_{i}|\;i\in I\\}$ be a partition of $X$ and let $f\in
T(X,\mathcal{P})$. If $f\in\operatorname{\mbox{ureg}}(T(X,\mathcal{P}))$, then
$\chi^{(f)}\in\operatorname{\mbox{ureg}}(T(I))$.

###### Proof.

If $f\in\operatorname{\mbox{ureg}}(T(X,\mathcal{P}))$, then there exists $g\in
S(X,\mathcal{P})$ such that $fgf=f$. Note that $\chi^{(g)}\in S(I)$ by [29,
Theorem 5.8]. We also have from [27, Lemma 2.3] that
$\chi^{(f)}\chi^{(g)}\chi^{(f)}=\chi^{(fgf)}=\chi^{(f)}$. Hence
$\chi^{(f)}\in\operatorname{\mbox{ureg}}(T(I))$. ∎

Before proving Theorem 4.4, we recall a crucial lemma from [29].

######  Lemma 4.3.

[29, Lemma 5.2] Let $\mathcal{P}=\\{X_{i}|\;i\in I\\}$ be a partition of $X$
and let $f\in T(X)$. Then $f\in T(X,\mathcal{P})$ if and only if there exists
a unique family

$B(f,I)=\\{f_{i}|\;f_{i}\mbox{ is induced by }f\mbox{and
}\mbox{dom}(f_{i})=X_{i}\;\forall i\in I\\}.$

The following theorem characterizes unit-regular elements in
$T(X,\mathcal{P})$.

######  Theorem 4.4.

Let $\mathcal{P}=\\{X_{i}|\;i\in I\\}$ be a partition of $X$ and let $f\in
T(X,\mathcal{P})$. Then $f\in\operatorname{\mbox{ureg}}(T(X,\mathcal{P}))$ if
and only if

  1. (1)

$\chi^{(f)}\in\operatorname{\mbox{ureg}}(T(I))$;

  2. (2)

for each $j\in I\chi^{(f)}$ there exists $i\in I$ such that

    1. (i)

$|X_{i}|=|X_{j}|$,

    2. (ii)

$X_{i}f=X_{j}\cap Xf$, and

    3. (iii)

$\mbox{c}(f_{i})=\mbox{d}(f_{i})$ where $f_{i}\in B(f,I)$.

###### Proof.

Suppose first that $f\in\operatorname{\mbox{ureg}}(T(X,\mathcal{P}))$. Then
there exists $g\in S(X,\mathcal{P})$ such that $fgf=f$.

  1. (1)

Lemma 4.2 takes care of this part.

  2. (2)

Let $j\in I\chi^{(f)}$ and let $j\chi^{(g)}=i$.

    1. (i)

Since $g\in S(X,\mathcal{P})$ and $j\chi^{(g)}=i$, we get $|X_{i}|=|X_{j}|$ by
[29, Lemma 3.6(ii)].

    2. (ii)

We first show that $X_{i}f\subseteq X_{j}\cap Xf$. Since $X_{i}f\subseteq Xf$,
it suffices to prove that $X_{i}f\subseteq X_{j}$. Since $j\in I\chi^{(f)}$,
there exists $k\in I$ such that $k\chi^{(f)}=j$ and so $X_{k}f\subseteq X_{j}$
by definition of $\chi^{(f)}$. Since $j\chi^{(g)}=i$, we also have
$X_{j}g=X_{i}$ by definition of $\chi^{(g)}$ and [29, Lemma 3.6]. Then
$X_{k}f=(X_{k}f)gf\subseteq(X_{j}g)f=X_{i}f$. This implies that
$X_{k}f\subseteq X_{j}\cap X_{i}f$. Since $f$ preserves $\mathcal{P}$, we thus
get $X_{i}f\subseteq X_{j}$.

For the reverse inclusion, let $y\in X_{j}\cap Xf$. Since $y\in Xf$, there
exists $x\in X$ such that $xf=y$. Also $y\in X_{j}$ and $j\chi^{(g)}=i$, we
see that $yg\in X_{i}$. Then $y=xf=(xf)gf=(yg)f\in X_{i}f$. Hence $X_{j}\cap
Xf\subseteq X_{i}f$ as required.

    3. (iii)

Since $j\chi^{(g)}=i$, we have from [29, Lemma 3.6] that $X_{j}g=X_{i}$ and so
$X_{j}g_{j}=X_{i}$ where $g_{j}\in B(g,I)$. Also, by (ii), we have
$X_{i}f=X_{j}\cap Xf$ and so $X_{i}f_{i}\subseteq X_{j}$ where $f_{i}\in
B(f,I)$. Since $fgf=f$, we see that $f_{i}g_{j}f_{i}=f_{i}$. Then
$(X_{i}f_{i})g_{j}\subseteq X_{i}$ is a cross-section of $\ker(f_{i})$ by
Lemma 3.1. Write $(X_{i}f_{i})g_{j}=T_{f_{i}}$. Since $g\in S(X,\mathcal{P})$,
the map $g_{j}\in B(g,I)$ is bijective by [29, Theorem 5.8]. Therefore
$(X_{j}\setminus
X_{i}f_{i})g_{j}=X_{j}g_{j}\setminus(X_{i}f_{i})g_{j}=X_{i}\setminus
T_{f_{i}}$ and hence $\mbox{c}(f_{i})=\mbox{d}(f_{i})$.

Conversely, suppose that the given conditions hold. By (2), let
$T_{\chi^{(f)}}\subseteq I$ be a cross-section of $\ker(\chi^{(f)})$ such that
the preimage $j(\chi^{(f)})^{-1}\cap T_{\chi^{(f)}}$ of every element $j\in
I\chi^{(f)}$ satisfies all the three conditions of (2). By (1), let
$\chi^{(g)}\in S(I)$ be the following bijection that satisfies
$\chi^{(f)}\chi^{(g)}\chi^{(f)}=\chi^{(f)}$.

$j\chi^{(g)}=\begin{cases}i\in T_{\chi^{(f)}}&\text{if $j\in I\chi^{(f)}$ and
$i\chi^{(f)}=j$},\\\ i\in I\setminus T_{\chi^{(f)}}&\text{if $j\in I\setminus
I\chi^{(f)}$ and $|X_{i}|=|X_{j}|$}.\end{cases}$

Using the above bijection $\chi^{(g)}$, we now construct a bijection of
$S(X,\mathcal{P})$ whose character is equal to $\chi^{(g)}$.

Let $j\in I\chi^{(f)}$. Then there exists $i\in T_{\chi^{(f)}}$ such that
$i\chi^{(f)}=j$. By (2), we then have $|X_{i}|=|X_{j}|$, $X_{j}\cap
Xf=X_{i}f$, and $\mbox{c}(f_{i})=\mbox{d}(f_{i})$ where $f_{i}\in B(f,I)$.
Therefore, we can choose a bijection $g_{j}\colon X_{j}\to X_{i}$ such that
$yg_{j}=x$ whenever $x\in T_{f_{i}}$ and $xf_{i}=y$.

Let $j\in I\setminus I\chi^{(f)}$. Then we arbitrarily choose a bijection
$h_{j}\colon X_{j}\to X_{j\chi^{(g)}}$. Using these bijections, we define a
map $g\colon X\to X$ by

$xg=\begin{cases}xg_{j}&\text{if $x\in X_{j}$ where $j\in I\chi^{(f)}$},\\\
xh_{j}&\text{if $x\in X_{j}$ where $j\in I\setminus I\chi^{(f)}$}.\end{cases}$

Since $\chi^{(g)}\in S(I)$ and each map of $B(g,I)$ is bijective, it follows
from [29, Theorem 5.8] that $g\in S(X,\mathcal{P})$. We can also verify in a
routine manner that $fgf=f$. Hence
$f\in\operatorname{\mbox{ureg}}(T(X,\mathcal{P}))$. ∎

##  5\. Unit-regular elements in $\Sigma(X,\mathcal{P})$,
$\Gamma(X,\mathcal{P})$, and $T_{E^{*}}(X)$

In this section, we first give a characterization of a regular element $f$ in
$\Sigma(X,\mathcal{P})$ in term of its character $\chi^{(f)}$. We then
characterize unit-regular elements in $\Sigma(X,\mathcal{P})$,
$\Gamma(X,\mathcal{P})$, and $T_{E^{*}}(X)$, respectively.

######  Proposition 5.1.

Let $\mathcal{P}=\\{X_{i}|\;i\in I\\}$ be a partition of $X$ and let
$f\in\Sigma(X,\mathcal{P})$. Then
$f\in\operatorname{\mbox{reg}}(\Sigma(X,\mathcal{P}))$ if and only if
$\chi^{(f)}\in\operatorname{\mbox{reg}}(\Gamma(I))$.

###### Proof.

Suppose first that $f\in\operatorname{\mbox{reg}}(\Sigma(X,\mathcal{P}))$.
Then there exists $g\in\Sigma(X,\mathcal{P})$ such that $fgf=f$. Therefore

$\chi^{(f)}=\chi^{(fgf)}=\chi^{(f)}\chi^{(g)}\chi^{(f)}$

by [27, Lemma 2.3]. Since $f,g\in\Sigma(X,\mathcal{P})$, we have
$\chi^{(f)},\chi^{(g)}\in\Gamma(I)$ by Theorem [29, Theorem 3.4] and so
$\chi^{(f)}\in\operatorname{\mbox{reg}}(\Gamma(I))$.

Conversely, suppose that $\chi^{(f)}\in\operatorname{\mbox{reg}}(\Gamma(I))$.
Then there exists $h\in\Gamma(I)$ such that
$\chi^{(f)}h\chi^{(f)}=\chi^{(f)}$. Since
$\chi^{(f)}\in\operatorname{\mbox{reg}}(\Gamma(I))$, it follows from
Proposition 3.10 that $\chi^{(f)}\in S(I)$ and so $h=(\chi^{(f)})^{-1}$.

For each $i\in I$, let $x_{i}$ be a fixed element of $X_{i}$. Moreover, for
each $x\in Xf$, let $x^{\prime}$ be a fixed element of $xf^{-1}$. Define a map
$g\colon X\to X$ by

$xg=\begin{cases}x^{\prime}&\text{if $x\in X_{i}\cap Xf$},\\\
x_{i(\chi^{(f)})^{-1}}&\text{if $x\in X_{i}\setminus Xf$}.\end{cases}$

Observe that $g\in T(X,\mathcal{P})$ and $\chi^{(g)}=h\in\Gamma(I)$. Therefore
$g\in\Sigma(X,\mathcal{P})$ by [29, Theorem 3.2]. Since $g$ maps each element
$x$ of $Xf$ to a fixed element $x^{\prime}$ of $xf^{-1}$, we can verify in a
routine manner that $fgf=f$. Hence
$f\in\operatorname{\mbox{reg}}(\Sigma(X,\mathcal{P}))$. ∎

We now have the following corollary from Proposition 3.10 and Proposition 5.1.

######  Corollary 5.2.

Let $\mathcal{P}=\\{X_{i}|\;i\in I\\}$ be a partition of $X$ and let
$f\in\Sigma(X,\mathcal{P})$. If $f\in\mbox{ureg}(\Sigma(X,\mathcal{P}))$, then
$\chi^{(f)}\in S(I)$.

###### Proof.

If $f\in\operatorname{\mbox{ureg}}(\Sigma(X,\mathcal{P}))$, then
$f\in\operatorname{\mbox{reg}}(\Sigma(X,\mathcal{P}))$. Therefore
$\chi^{(f)}\in\operatorname{\mbox{reg}}(\Gamma(I))$ by Proposition 5.1 and so
$\chi^{(f)}\in S(I)$ by Proposition 3.10. ∎

Let us now make the following trivial remark which will be used in the sequel.

######  Remark 5.3.

Let $M$ be a monoid and let $N$ be a submonoid of $M$. Then

  1. (i)

$\mbox{ureg}(N)\subseteq\mbox{ureg}(M)$.

  2. (ii)

if $U(M)\subseteq N$, then $N\cap\mbox{ureg}(M)\subseteq\mbox{ureg}(N)$.

Recall that, for finite $X$, the authors [28, Proposition 3.5] gave a
characterization of unit-regular elements in $\Sigma(X,\mathcal{P})$. The
following proposition characterizes unit-regular elements in
$\Sigma(X,\mathcal{P})$ for arbitrary $X$.

######  Proposition 5.4.

Let $\mathcal{P}=\\{X_{i}|\;i\in I\\}$ be a partition of $X$ and let
$f\in\Sigma(X,\mathcal{P})$. Then $f\in\mbox{ureg}(\Sigma(X,\mathcal{P}))$ if
and only if

  1. (i)

$\chi^{(f)}\in\Omega(I)$,

  2. (ii)

$|X_{i}|=|X_{j}|$ whenever $i\chi^{(f)}=j$, and
$\mbox{c}(f_{i})=\mbox{d}(f_{i})$ where $f_{i}\in B(f,I)$.

###### Proof.

Suppose first that $f\in\mbox{ureg}(\Sigma(X,\mathcal{P}))$.

  1. (i)

Then $\chi^{(f)}\in S(I)$ by Corollary 5.2. Since $S(I)\subseteq\Omega(I)$, it
follows that $\chi^{(f)}\in\Omega(I)$.

  2. (ii)

Let $i\chi^{(f)}=j$. Since $f\in\mbox{ureg}(\Sigma(X,\mathcal{P}))$ and
$\Sigma(X,\mathcal{P})\subseteq T(X,\mathcal{P})$, we have
$f\in\mbox{ureg}(T(X,\mathcal{P}))$ and $\chi^{(f)}\in S(I)$ by Remark 5.3(i)
and by Corollary 5.2, respectively. Then, by Theorem 4.4, we get
$|X_{i}|=|X_{j}|$ and $\mbox{c}(f_{i})=\mbox{d}(f_{i})$ where $f_{i}\in
B(f,I)$.

Conversely, suppose that the given conditions hold. Since
$S(X,\mathcal{P})\subseteq\Sigma(X,\mathcal{P})\subseteq T(X,\mathcal{P})$, by
Remark 5.3(ii), it suffices to show that $f\in\mbox{ureg}(T(X,\mathcal{P}))$.

Since $f\in\Sigma(X,\mathcal{P})$, we have $\chi^{(f)}\in\Gamma(I)$ by [29,
Theorem 3.2]. By (i), we know that $\chi^{(f)}\in\Omega(I)$ and so
$\chi^{(f)}\in\Gamma(I)\cap\Omega(I)=S(I)$. This implies that
$\chi^{(f)}\in\operatorname{\mbox{ureg}}(T(I))$.

Let $i\chi^{(f)}=j$. Then $X_{i}f\subseteq X_{j}$ by definition of
$\chi^{(f)}$. Recall that $\chi^{(f)}\in S(I)$. Therefore $X_{i}f=X_{j}\cap
Xf$. By (ii), we also have $|X_{i}|=|X_{j}|$ and
$\mbox{c}(f_{i})=\mbox{d}(f_{i})$ where $f_{i}\in B(f,I)$. Hence
$f\in\operatorname{\mbox{ureg}}(T(X,\mathcal{P}))$ by Theorem 4.4 as required.
∎

For finite $X$, it is proven in [28, Proposition 5.8] that
$\operatorname{\mbox{reg}}(\Gamma(X,\mathcal{P}))=\operatorname{\mbox{ureg}}(\Gamma(X,\mathcal{P}))$.
The following proposition provides a characterization of unit-regular elements
in $\Gamma(X,\mathcal{P})$ for arbitrary $X$.

######  Proposition 5.5.

Let $\mathcal{P}=\\{X_{i}|\;i\in I\\}$ be a partition of $X$ and let
$f\in\Gamma(X,\mathcal{P})$. Then
$f\in\operatorname{\mbox{ureg}}(\Gamma(X,\mathcal{P}))$ if and only if

  1. (i)

$\chi^{(f)}\in\operatorname{\mbox{ureg}}(T(I))$,

  2. (ii)

for each $j\in I\chi^{(f)}$ there exists $i\in I$ such that the map $f_{i}\in
B(f,I)$ from $X_{i}$ to $X_{j}$ is injective.

###### Proof.

Suppose first that $f\in\operatorname{\mbox{ureg}}(\Gamma(X,\mathcal{P}))$.
Then $f\in\operatorname{\mbox{ureg}}(T(X,\mathcal{P}))$ by Remark 5.3(i).

  1. (i)

Then Lemma 4.2 takes care of this part.

  2. (ii)

Clearly $f\in\operatorname{\mbox{reg}}(\Gamma(X,\mathcal{P}))$. Hence, by
Theorem [28, Theorem 5.5], for each $j\in I\chi^{(f)}$ there exists $i\in I$
such that the map $f_{i}\in B(f,I)$ from $X_{i}$ to $X_{j}$ is injective.

Conversely, suppose that the given conditions hold. Since
$f\in\Gamma(X,\mathcal{P})$, we know from [28, Remark 5.3] that every map of
$B(f,I)$ is surjective. It follows from (ii) that for each $j\in I\chi^{(f)}$
there exists $i\in I$ such that the map $f_{i}\in B(f,I)$ from $X_{i}$ to
$X_{j}$ is bijective. Therefore, for the preimage $i\in j(\chi^{(f)})^{-1}$,
we get $|X_{i}|=|X_{j}|$, $X_{i}f=X_{j}\cap Xf$, and
$\mbox{c}(f_{i})=\mbox{d}(f_{i})$ where $f_{i}\in B(f,I)$. Thus
$f\in\operatorname{\mbox{ureg}}(T(X,\mathcal{P}))$ by Theorem 4.4. Since
$S(X,\mathcal{P})\subseteq\Gamma(X,\mathcal{P})$, it follows from Remark
5.3(ii) that $f\in\operatorname{\mbox{ureg}}(\Gamma(X,\mathcal{P}))$. ∎

The following proposition characterizes unit-regular elements in
$T_{E^{*}}(X)$.

######  Proposition 5.6.

Let $\mathcal{P}=\\{X_{i}|\;i\in I\\}$ be a partition of $X$ and let $f\in
T_{E^{*}}(X)$. Then $f\in\operatorname{\mbox{ureg}}(T_{E^{*}}(X))$ if and only
if

  1. (i)

$\chi^{(f)}\in\Gamma(I)$,

  2. (ii)

$|X_{i}|=|X_{j}|$ whenever $i\chi^{(f)}=j$, and
$\mbox{c}(f_{i})=\mbox{d}(f_{i})$ where $f_{i}\in B(f,I)$.

###### Proof.

Suppose first that $f\in\operatorname{\mbox{ureg}}(T_{E^{*}}(X))$. Then
$f\in\operatorname{\mbox{ureg}}(T(X,\mathcal{P}))$ by Remark 5.3(i).

  1. (i)

It follows that $\chi^{(f)}\in\operatorname{\mbox{ureg}}(T(I))$ by Lemma 4.2.
Since $f\in T_{E^{*}}(X)$, we have $\chi^{(f)}\in\Omega(I)$ by [29, Theorem
3.4]. Since $S(I)\subseteq\Omega(I)$, it follows from Remark 5.3(ii) that
$\chi^{(f)}\in\operatorname{\mbox{ureg}}(\Omega(I))$ and so $\chi^{(f)}\in
S(I)$ by Proposition 3.9. Hence $\chi^{(f)}\in\Gamma(I)$.

  2. (ii)

Let $i\chi^{(f)}=j$. From (i) , we see that $\chi^{(f)}\in S(I)$. Recall that
$f\in\operatorname{\mbox{ureg}}(T(X,\mathcal{P}))$. Therefore, by Theorem 4.4,
we get $|X_{i}|=|X_{j}|$ and $\mbox{c}(f_{i})=\mbox{d}(f_{i})$ where $f_{i}\in
B(f,I)$.

Conversely, suppose that the given conditions hold. Since
$S(X,\mathcal{P})\subseteq T_{E^{*}}(X)\subseteq T(X,\mathcal{P})$, it
suffices to show that $f\in\operatorname{\mbox{ureg}}(T(X,\mathcal{P}))$.

Since $f\in T_{E^{*}}(X)$, we have $\chi^{(f)}\in\Omega(I)$ by Theorem [29,
Theorem 3.4]. It follows from (i) that
$\chi^{(f)}\in\Gamma(I)\cap\Omega(I)=S(I)$ and so
$\chi^{(f)}\in\operatorname{\mbox{ureg}}(T(I))$.

Again, since $\chi^{(f)}\in S(I)$, we see that $X_{j}\cap Xf=X_{i}f$ whenever
$i\chi^{(f)}=j$. Combining these with condition (ii), Theorem 4.4 concludes
that $f\in\operatorname{\mbox{ureg}}(T(X,\mathcal{P}))$ as required. ∎

##  6\. The Submonoid $\Omega(X,\mathcal{P})$

In this section, we characterize regular and unit-regular elements in
$\Omega(X,\mathcal{}P)$, respectively. Let us first make the following useful
remark.

######  Remark 6.1.

Let $f\in T(X,\mathcal{P})$. Then $f\in\Omega(X,\mathcal{P})$ if and only if
every map in $B(f,I)$ is injective.

The next theorem characterizes regular elements in $\Omega(X,\mathcal{P})$.

######  Theorem 6.2.

Let $\mathcal{P}=\\{X_{i}|\;i\in I\\}$ be a partition of $X$ and let
$f\in\Omega(X,\mathcal{P})$. Then
$f\in\operatorname{\mbox{reg}}(\Omega(X,\mathcal{P}))$ if and only if for
every $j\in I\chi^{(f)}$ there exists $i\in I$ such that the map $f_{i}\in
B(f,I)$ from $X_{i}$ to $X_{j}$ is surjective.

###### Proof.

Suppose first that $f\in\operatorname{\mbox{reg}}(\Omega(X,\mathcal{P}))$, and
let $j\in I\chi^{(f)}$. Then there exists $i\in I$ such that $i\chi^{(f)}=j$
and so $X_{i}f\subseteq X_{j}$ by definition of $\chi^{(f)}$. Then there is a
map $f_{i}\in B(f,I)$ from $X_{i}$ to $X_{j}$. If $f_{i}$ is surjective, then
we are done. Otherwise, suppose that $f_{i}$ is not surjective.

Recall that $f\in\operatorname{\mbox{reg}}(\Omega(X,\mathcal{P}))$. There
exists $g\in\Omega(X,\mathcal{P})$ such that $fgf=f$. Since $j\in
I\chi^{(f)}\subseteq I$, there exists $l\in I\chi^{(g)}$ such that
$j\chi^{(g)}=l$ and so $X_{j}g\subseteq X_{l}$ by definition of $\chi^{(g)}$.
Then there is a map $g_{j}\in B(g,I)$ from $X_{j}$ to $X_{l}$. Recall that
$X_{i}f\subseteq X_{j}$, $X_{j}g\subseteq X_{l}$, and $fgf=f$. We obtain

$X_{i}f=(X_{i}f)gf\subseteq(X_{j}g)f\subseteq X_{l}f.$

Since $f$ preserves $\mathcal{P}$, we get $X_{l}f\subseteq X_{j}$. Therefore
there is a map $f_{l}\in B(f,I)$ from $X_{l}$ to $X_{j}$. We now claim that
$f_{l}$ is surjective.

Assume, to the contrary, that there exists $y\in X_{j}\setminus X_{l}f_{l}$.
Write $yg_{j}=z$ and $zf_{l}=w$. Clearly $z\in X_{l}$, $w\in X_{j}$, and
$y\neq w$. Since $f,g\in\Omega(X,\mathcal{P})$, the maps $g_{j}$ and $f_{l}$
are injective by Remark 6.1 and so $(wg_{j})f_{l}\neq(yg_{j})f_{l}$. But
$(yg_{j})f_{l}=zf_{l}=w$, and

$(wg_{j})f_{l}=(wg_{j})f=(wg)f=(zf_{l})gf=(zf)gf=zf=zf_{l}=w$

which is a contradiction. Hence the map $f_{l}\in B(f,I)$ is surjective as
required.

Conversely, suppose that the given condition holds. By Remark 6.1, we know
that every map in $B(f,I)$ is injective. Then, by hypothesis, for every $j\in
I\chi^{(f)}$ there exists $i\in I$ such that the map $f_{i}\in B(f,I)$ from
$X_{i}$ to $X_{j}$ is bijective. Denote by $h_{j}$ the inverse map of
$f_{i}\colon X_{i}\to X_{j}$. Note that the map $h_{j}\colon X_{j}\to X_{i}$
is bijective. We now define a map $g\colon X\to X$ by

$xg=\begin{cases}xh_{j}&\text{if $x\in X_{j}$ where $j\in I\chi^{(f)}$},\\\
x&\text{otherwise}.\end{cases}$

Clearly $g\in\Omega(X,\mathcal{P})$. We can verify in a routine manner that
$fgf=f$. Hence $f\in\operatorname{\mbox{reg}}(\Omega(X,\mathcal{P}))$. ∎

The following proposition characterizes unit-regular elements in
$\Omega(X,\mathcal{P})$.

######  Proposition 6.3.

Let $\mathcal{P}=\\{X_{i}|\;i\in I\\}$ be a partition of $X$ and let
$f\in\Omega(X,\mathcal{P})$. Then
$f\in\operatorname{\mbox{ureg}}(\Omega(X,\mathcal{P}))$ if and only if

  1. (i)

$\chi^{(f)}\in\operatorname{\mbox{ureg}}(T(I))$, and

  2. (ii)

for each $j\in I\chi^{(f)}$ there exists $i\in I$ such that the map $f_{i}\in
B(f,I)$ from $X_{i}$ to $X_{j}$ is surjective.

###### Proof.

Suppose first that $f\in\operatorname{\mbox{ureg}}(\Omega(X,\mathcal{P}))$.
Then $f\in\operatorname{\mbox{ureg}}(T(X,\mathcal{P}))$ by Remark 5.3(i).

  1. (i)

Then Lemma 4.2 takes care of this part.

  2. (ii)

Note that $f\in\operatorname{\mbox{reg}}(\Omega(X,\mathcal{P}))$ and hence
Theorem 6.2 takes care of this part.

Conversely, suppose that the given conditions hold. Since
$f\in\Omega(X,\mathcal{P})$, every map in $B(f,I)$ is injective by Remark 6.1.
Then, by hypothesis, for each $j\in I\chi^{(f)}$ there exists $i\in I$ such
that the map $f_{i}\in B(f,I)$ from $X_{i}$ to $X_{j}$ is bijective and so
$|X_{i}|=|X_{j}|$, $X_{i}f=X_{j}\cap Xf$, and $c(f_{i})=d(f_{i})$. Hence
$f\in\operatorname{\mbox{ureg}}(T(X,\mathcal{P}))$ by Theorem 4.4. Since
$S(X,\mathcal{P})\subseteq\Omega(X,\mathcal{P})\subseteq T(X,\mathcal{P})$, we
thus get $f\in\operatorname{\mbox{ureg}}(\Omega(X,\mathcal{P}))$ by Remark
5.3(ii). ∎

## References

  * [1] J. Araújo, W. Bentz, J. D. Mitchell, and C. Schneider.  The rank of the semigroup of transformations stabilising a partition of a finite set.  Mathematical Proceedings of the Cambridge Philosophical Society, 159(2): 339–353, 2015.
  * [2] J. Araújo and J. Konieczny.  Semigroups of transformations preserving an equivalence relation and a cross-section.  Communications in Algebra, 32(5): 1917–1935, 2004.
  * [3] J. Araújo and C. Schneider.  The rank of the endomorphism monoid of a uniform partition.  Semigroup Forum, 78(3): 498–510, 2009.
  * [4] T. S. Blyth and R. McFadden.  Unit orthodox semigroups.  Glasgow Mathematical Journal, 24(1): 39–42, 1983.
  * [5] Y. Chaiya.  Natural partial order and finiteness conditions on semigroups of linear transformations with invariant subspaces.  Semigroup Forum, 99: 579–590, 2019.
  * [6] S. Y. Chen and S. C. Hsieh.  Factorizable inverse semigroups.  Semigroup Forum, 8: 283–297, 1974.
  * [7] H. D’Alarcao.  Factorizable as a finiteness condition.  Semigroup Forum, 20: 281–282, 1980.
  * [8] L. Z. Deng, J. W. Zeng, and B. Xu.  Green’s relations and regularity for semigroups of transformations that preserve double direction equivalence.  Semigroup Forum, 80(3): 416–425, 2010.
  * [9] L. Z. Deng, J. W. Zeng, and T. J. You.  Green’s relations and regularity for semigroups of transformations that preserve order and a double direction equivalence,  Semigroup Forum, 84(1): 59–68, 2012.
  * [10] I. Dolinka and J. East.  Idempotent generation in the endomorphism monoid of a uniform partition.  Communications in Algebra, 44(12): 5179–5198, 2016.
  * [11] I. Dolinka, J. East, and J. D. Mitchell.  Idempotent rank in the endomorphism monoid of a non-uniform partition.  Bulletin of the Australian Mathematical Society, 93(1): 73–91, 2016.
  * [12] G. Ehrlich.  Unit-regular rings.  Portugaliae Mathematica, 27(4): 209–212, 1968.
  * [13] V. H. Fernandes and T. M. Quinteiro.  On the monoids of transformations that preserve the order and a uniform partition.  Communications in Algebra, 39(8): 2798–2815, 2011.
  * [14] V. H. Fernandes and T. M. Quinteiro.  On the ranks of certain monoids of transformations that preserve a uniform partition.  Communications in Algebra, 42(2): 615–636, 2014.
  * [15] J. Fountain.  An introduction to covers for semigroups.  In Semigroups, Algorithms, Automata and Languages, pages 155–194.  World Scientific, Singapore, 2002.
  * [16] J. B. Hickey and M. V. Lawson.  Unit regular monoids.  Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 127(1): 127–143, 1997.
  * [17] , P. M. Higgins, J. M. Howie, and N. Ruškuc.  Generators and factorisations of transformation semigroups.  Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 128(6): 1355–1369, 1998.
  * [18] J. M. Howie.  Fundamentals of Semigroup Theory.  volume 12 of London Mathematical Society Monographs. New Series.  The Clarendon Press, Oxford University Press, New York, 1995.
  * [19] D. B. McAllister.  Some covering and embedding theorems for inverse semigroups  Journal of the Australian Mathematical Society, 22(2): 188–211, 1976.
  * [20] R. B. McFadden.  Unit-regular orthodox semigroups.  Glasgow Mathematical Journal, 25(2): 229–240, 1984.
  * [21] H. Pei.  Equivalences, $\alpha$-semigroups and $\alpha$-congruences.  Semigroup Forum, 49(1): 49–58, 1994.
  * [22] H. Pei.  Some $\alpha$-semigroups inducing certain lattices.  Semigroup Forum, 57: 48–59, 1998.
  * [23] H. Pei.  On the rank of the semigroup ${T_{E}(X)}$.  Semigroup Forum, 70(1): 107–117, 2005.
  * [24] H Pei.  Regularity and Green’s relations for semigroups of transformations that preserve an equivalence.  Communications in Algebra, 33(1): 109–118, 2005.
  * [25] H. Pei and H. Zhou.  Abundant semigroups of transformations preserving an equivalence relation.  Algebra Colloquium, 18(1): 77–82, 2011.
  * [26] H. Pei and D. Zou.  Green’s equivalences on semigroups of transformations preserving order and an equivalence relation.  Semigroup Forum, 71(2): 241–251, 2005.
  * [27] P. Purisang and J. Rakbud.  Regularity of transformation semigroups defined by a partition.  Communications of the Korean Mathematical Society, 31(2): 217–227, 2016.
  * [28] M. Sarkar and S. N. Singh.  Regular, unit-regular, and idempotent elements of semigroups of transformations that preserve a partition.  arXiv:abs/2010.14033, 2020.
  * [29] M. Sarkar and S. N. Singh.  On certain semigroups of transformations that preserve a partition.  Communications in Algebra, 49(1): 331–342, 2021.
  * [30] L. Sun, H. Pei, and Z. Cheng.  Regularity and Green’s relations for semigroups of transformations preserving orientation and an equivalence.  Semigroup Forum, 74(3): 473–486, 2007.
  * [31] L. Sun and J. Sun.  A partial order on transformation semigroups that preserve double direction equivalence relation.  Journal of Algebra and Its Applications, 12(8): 1350041, 2013.
  * [32] Y. Tirasupa.  Factorizable transformation semigroups.  Semigroup Forum, 18: 15–19: 1979.


# Mapping the Design Space of Interactions in Human-AI Text Co-creation Tasks

Zijian Ding  College of Information StudiesUniversity of Maryland, College
ParkUSA and  Joel Chan  College of Information StudiesUniversity of Maryland,
College ParkUSA

(2023)

###### Abstract.

Large Language Models (LLMs) have demonstrated impressive text generation
capabilities, prompting us to reconsider the future of human-AI co-creation
and how humans interact with LLMs. In this paper, we present a spectrum of
content generation tasks and their corresponding human-AI interaction
patterns. These tasks include: 1) fixed-scope content curation tasks with
minimal human-AI interactions, 2) independent creative tasks with precise
human-AI interactions, and 3) complex and interdependent creative tasks with
iterative human-AI interactions. We encourage the generative AI and HCI
research communities to focus on the more complex and interdependent tasks,
which require greater levels of human involvement.

Large Language Models, Analogy, Creativity Support Tools

††copyright: acmcopyright††journalyear: 2023††copyright:
rightsretained††conference: CHI 2023 Workshop on Generative AI and HCI; Apr
28, 2023; Virtual††booktitle: CHI 2023 Workshop on Generative AI and HCI), Apr
28, 2023, Virtual††doi: 10.1145/3411764.3445618

Figure 1. Spectrum of human-AI co-creation tasks and corresponding human
intervention complexity. The upper half describes the spectrum of text-based
co-creation tasks from low to high creativity and complexity; the bottom half
proposes a mapping of the points on these spectrum to human-AI interaction
patterns from the taxonomy in (Cheng et al., 2022).

TODO

##  1\. Introduction

Large Language Models (LLMs), such as Generative Pre-trained Transformer 3
(GPT-3) (Brown et al., 2020), have garnered significant attention from
researchers and practitioners for their ability to generate text content. The
rapid success of ChatGPT111https://chat.openai.com/chat \- reaching 100
million monthly active users just two months after its launch and setting a
record for the fastest-growing consumer application in
history222https://www.reuters.com/technology/chatgpt-sets-record-fastest-
growing-user-base-analyst-note-2023-02-01/ \- highlights not only the
potential and capabilities of Generative AI for producing precise and
personalized text content, but also the critical role of interface and
interaction in communicating with AI. Since ChatGPT is a variant of GPT-3
fine-tuned for conversational tasks, the technical foundation remains similar;
instead, the primary difference seems to be the shift in human-AI interaction
paradigms, from prompt programming, parameter tuning and autocompleting in
OpenAI playground or API333https://platform.openai.com/overview, to
interactive conversations in ChatGPT.

The wide deployment of these language models beyond academic research projects
carries risks, but has also uncovered a broader sense of the potential
applications of these models: from foundation models for traditional NLP
applications such as text classification, summarization, and information
extraction, to generative applications such as  analogy generation and even
complex creative tasks such as  fiction creation. What might this wave of
progress enable for augmenting — rather than automating — human creativity?

To help make sense of this question, in this position paper we sketch out a
possible design space of human-AI co-creation, focusing on the role of humans
and their interactions and collaboration with Generative AI. Specifically, we
synthesize previous research on human-AI interactions in text generation
applications and tasks onto a spectrum of task complexity and creativity: 1)
fixed-scope text curation tasks, 2) atomic creative tasks and 3) complex and
interdependent creative tasks; and propose a mapping of this spectrum to a
taxonomy of existing design patterns of human-AI interaction that can address
the requirements and challenges of each point in the spectrum, as shown in
Figure 1. Finally, we suggest future avenues for further exploring this
spectrum, especially towards more complex and interdependent creative tasks.

##  2\. Types of human-AI interactions for text generation

Our discussion of human-AI interaction for text generation draws on the
taxonomy of five common human-AI interactions proposed by Cheng et al. (Cheng
et al., 2022): 1) guiding model output, 2) selecting or rating model output,
3) post-editing, 4) interactive editing (initiated by AI) and 5) writing with
model assistance (initiated by human).

The former three interaction types do not involve rounds of iterations between
human and AI, referred to as precise human-AI interactions. In contrast, the
latter two types involve rounds of iterations between human and AI, referred
to as iterative human-AI interactions. In the following sections, we discuss
how these interaction types can be usefully mapped to a spectrum of text-based
human-AI co-creation tasks that range from low to high complexity and
creativity.

##  3\. Spectrum of text-based human-AI co-creation tasks

###  3.1. Minimal human-AI interactions in fixed-scope content curation tasks

The first point in our spectrum can be described as fixed-scope content
curation tasks. Previous research has indicated that Large language models
(LLMs) can effectively handle well-defined, content curation tasks such as
text summarization (Dang et al., 2022; Goyal et al., 2022), content refinement
(Lin, 2023), and code explanation (MacNeil et al., 2022a, b). In these tasks,
existing knowledge and information are summarized and presented in a cohesive
manner, but no new knowledge is generated. In these fixed-scope tasks,
advanced LLM models such as GPT-3 DaVinci have already produced satisfactory
results with no human intervention on the outputs (Brown et al., 2020).
According to a study conducted by Clark et al. (Clark et al., 2021), text
generated by GPT-3 exhibited such a high degree of linguistic sophistication
that it was almost impossible for human evaluators to discern whether it had
been authored by a machine or a human, and carefully designed frameworks are
required to scrutinize different types of human and machine errors and
determine the authorship of a piece of text (Dou et al., 2022). The
aforementioned evidence serves as a testament to the high quality of machine-
generated text, raising the possibility that in the future, the amount of
human involvement required for content curation tasks could be significantly
reduced.

###  3.2. Precise human-AI interactions in atomic creative tasks

The second point in our spectrum can be described as atomic creative tasks. By
creative, we mean outputs that are both novel and useful (Sawyer, 2012; Runco
and Jaeger, 2012), including generating analogies / metaphors / analogous
design concepts (Webb et al., 2022; Zhu and Luo, 2022; Bhavya et al., 2022;
Lee et al., 2022), slogans (Clark et al., 2018) and inspirations for
tweetorials (short technical explanations on Twitter) (Gero et al., 2022). The
broad coverage of existing knowledge by LLMs such as GPT-3 can help to create
novel connections, known as ”creative leaps” (Chan et al., 2017; Siangliulue
et al., 2015; Noy et al., 2012). However, generating truly creative,
inspiring, and insightful content often requires domain-specific knowledge,
including subtle and implicit knowledge, which may not be present in the
training data for LLMs. To compensate for this lack of knowledge, LLMs must be
guided by carefully crafted prompts and examples, and their outputs must be
selected or edited by humans to ensure their quality. In other words, these
specific creative tasks demand precise human-AI interactions.

For instance, when generating analogous problems, classic analogous problems
created or selected by humans, such as Duncker and Lees’ (Duncker, 1945)
radiation problem, can be applied to guide LLMs in the analogy generation
process. The generated analogies must then be selected or rated, even post-
edited by humans to avoid any biases, illegal, or inappropriate content.  In
some recent experiments to explore the current performance of LLMs on atomic
creative tasks, we generated 120 analogous problems with GPT-3 text-
davinci-002 model and the Duncker and Lees’ analogous problem for one-shot
learning, and asked participants to use them to reformulate the original
problem (Ding et al., 2023). Our results showed that the AI-generated
analogous problems were frequently perceived as helpful (with a median
helpfulness rating of 4 out of 5) and led to observable changes in problem
formulation in approximately 80% of cases. However, we also found that up to
25% of the outputs were potentially harmful, mostly due to potentially
upsetting content that was not biased or toxic. Our findings demonstrate the
potential of using LLMs for atomic creative tasks, but also highlight the need
for human intervention. Below is an example of the LLM-generated analogous
problem and how participant used the analogy to stimulate reformulation of the
original problem:

> Original problem
>
> Stakeholder: owners of travel agency
>
> Context: the restriction of pandemic has been mitigated and people are
> willing to travel again
>
> Goal: reopen their traveling business
>
> Obstacle: cannot find enough employees because people have left the travel
> industry during the pandemic
>
> GPT-3 generated analogous problem
>
> Stakeholder: a farmer
>
> Context: the restriction of the use of pesticides has been mitigated
>
> Goal: use pesticides to increase crop yield
>
> Obstacle: the farmer cannot afford to buy pesticides
>
> Participant’s response to reformulation question
>
> It’s not that the travel agency can’t find employees, it’s that they can’t
> afford to pay employees to work for them after being closed for so long,
> thus causing a feedback loop of: not enough employees -¿ less money -¿cant
> afford to hire employees -¿ not enough employees.

###  3.3. Iterative human-AI interactions in complex and interdependent
creative tasks

Our third and final point in the spectrum can be described as complex and
interdependent creative tasks. These are larger scale creative tasks
containing a set of subtasks interdependent to each other, such as
storytelling (Yuan et al., 2022; Roemmele, 2016; Singh et al., 2022; Berkovsky
et al., 2018; Freiknecht and Effelsberg, 2020; Yang et al., [n.d.]; Cao,
2023). Those tasks go beyond just the combination of atomic creative tasks and
cannot be decomposed into atomic creative tasks. Those tasks require not only
domain-specific knowledge but also the ability to plan, reason, delve into
ideas, and retain context over time in order to generate new and coherent
concepts, knowledge and stories.

For these tasks, we argue that humans and large language models (LLMs) must
work closely together and iteratively refine the text content, and different
types of interactions are needed depending on the iteration stages. The co-
creation of a story, for example, may involve a series of iterations of
guidance, selection, and post-editing by both humans and LLMs (Yuan et al.,
2022). And fixed-scope text curation tasks and specific creative tasks can
serve as building blocks for more complex and comprehensive creative tasks,
such as including a literature review for scientific paper writing. Expert
human review, justification, and post-editing are crucial to ensure the
originality and logic of the AI-generated content and its alignment with other
elements.

##  4\. Future directions for text-based human-AI co-creation

Our belief and hope is that iterative human-AI interactions in complex and
interdependent creative tasks will become a focus of future research on human-
AI text co-creation, due to their complexity, potential, and the need for
intensive human-AI interaction.  Current LLM-powered tools show potential for
supporting that vision of human-AI collaboration in creative tasks, but there
is still room for improvement in certain areas. For example, chatbots like
ChatGPT are capable of supporting multiple rounds of interactions but
currently only offer one interaction paradigm - guiding the model output. This
limitation may reduce the efficiency and overall experience of creative
writing with the tool. Within the context of human-AI co-writing, Yuan et al.
(Yuan et al., 2022) also focused primarily on exploring various formats of
guiding model outputs, such as continuation, elaboration, story seeding, and
infilling. While their work sheds important light on the collaborative aspects
of story writing, there is potential for artificial intelligence to play an
even more proactive role through interactive selecting and editing paradigms.
There is also a difficult set of challenges around evaluation. For example,
NLP progress has benefited substantially from well-defined benchmarks. This
can work well for accelerating progress for fixed-scope content curation
tasks, but is a poor fit for atomic and complex and interdependent creative
tasks with no single “correct” reference output. Crowdsourced human
evaluations may also index only surface-level linguistic coherence vs. more
substantive dimensions of quality without more specific (task-specific)
instructions or domain expertise: for example, Clark et al (Clark et al.,
2021) reported that crowd workers mostly relied on form vs. content heuristics
to make their judgments about human-likeness of LLM-generated text. Community-
based and participatory research methods may be needed to address these
challenges. A final challenge concerns the question of how to integrate domain
knowledge and expertise. For example while novice users may have the most to
gain from human-AI co-creation tools, they may need domain expertise to
effectively control the outputs generated by AI. We believe a promising
direction is to explore the design of co-creation tools that integrate the
generative strengths of LLMs with sources of domain knowledge (e.g.,
heuristics, design patterns, knowledge bases, access to other peers and
experts for feedback/validation); for example, the new Bing search
tool444https://www.bing.com/new integrates question-answering and
summarization loops with API calls to knowledge bases. We believe these open
problems are difficult but tractable, and look forward to exploring solutions
to these problems with the human-AI interaction community.

## References

  * (1)
  * Berkovsky et al. (2018) Shlomo Berkovsky, Yoshinori Hijikata, Jun Rekimoto, Margaret Burnett, Mark Billinghurst, and Aaron Quigley. 2018.  How Novelists Use Generative Language Models: An Exploratory User Study. In _23rd International Conference on Intelligent User Interfaces_. ACM, Tokyo Japan.
  * Bhavya et al. (2022) Bhavya Bhavya, Jinjun Xiong, and Chengxiang Zhai. 2022\.  Analogy Generation by Prompting Large Language Models: A Case Study of InstructGPT.  http://arxiv.org/abs/2210.04186 arXiv:2210.04186 [cs].
  * Brown et al. (2020) Tom B. Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, Sandhini Agarwal, Ariel Herbert-Voss, Gretchen Krueger, Tom Henighan, Rewon Child, Aditya Ramesh, Daniel M. Ziegler, Jeffrey Wu, Clemens Winter, Christopher Hesse, Mark Chen, Eric Sigler, Mateusz Litwin, Scott Gray, Benjamin Chess, Jack Clark, Christopher Berner, Sam McCandlish, Alec Radford, Ilya Sutskever, and Dario Amodei. 2020.  Language Models are Few-Shot Learners.  _arXiv:2005.14165 [cs]_ (June 2020).  http://arxiv.org/abs/2005.14165 00030 arXiv: 2005.14165.
  * Cao (2023) Chen Cao. 2023.  Leveraging Large Language Model and Story-Based Gamification in Intelligent Tutoring System to Scaffold Introductory Programming Courses: A Design-Based Research Study.  http://arxiv.org/abs/2302.12834 arXiv:2302.12834 [cs].
  * Chan et al. (2017) Joel Chan, Pao Siangliulue, Denisa Qori McDonald, Ruixue Liu, Reza Moradinezhad, Safa Aman, Erin T. Solovey, Krzysztof Z. Gajos, and Steven P. Dow. 2017.  Semantically Far Inspirations Considered Harmful?: Accounting for Cognitive States in Collaborative Ideation. In _Proceedings of the 2017 ACM SIGCHI Conference on Creativity and Cognition_ _(C &C ’17)_. ACM, New York, NY, USA, 93–105.  https://doi.org/10.1145/3059454.3059455
  * Cheng et al. (2022) Ruijia Cheng, Alison Smith-Renner, Ke Zhang, Joel Tetreault, and Alejandro Jaimes-Larrarte. 2022\.  Mapping the Design Space of Human-AI Interaction in Text Summarization. In _Proceedings of the 2022 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies_. Association for Computational Linguistics, Seattle, United States, 431–455.  https://doi.org/10.18653/v1/2022.naacl-main.33
  * Clark et al. (2021) Elizabeth Clark, Tal August, Sofia Serrano, Nikita Haduong, Suchin Gururangan, and Noah A. Smith. 2021\.  All That’s ’Human’ Is Not Gold: Evaluating Human Evaluation of Generated Text.  _arXiv:2107.00061 [cs]_ (July 2021).  http://arxiv.org/abs/2107.00061 00008 arXiv: 2107.00061.
  * Clark et al. (2018) Elizabeth Clark, Anne Spencer Ross, Chenhao Tan, Yangfeng Ji, and Noah A. Smith. 2018\.  Creative Writing with a Machine in the Loop: Case Studies on Slogans and Stories. In _23rd International Conference on Intelligent User Interfaces_. ACM, Tokyo Japan, 329–340.  https://doi.org/10.1145/3172944.3172983
  * Dang et al. (2022) Hai Dang, Karim Benharrak, Florian Lehmann, and Daniel Buschek. 2022.  Beyond Text Generation: Supporting Writers with Continuous Automatic Text Summaries.  https://doi.org/10.1145/3526113.3545672 arXiv:2208.09323 [cs].
  * Ding et al. (2023) Zijian Ding, Arvind Srinivasan, Stephen MacNeil, and Joel Chan. 2023.  Fluid Transformers and Creative Analogies: Exploring Large Language Models’ Capacity for Augmenting Cross-Domain Analogical Creativity.  _arXiv preprint arXiv:2302.12832_ (2023).
  * Dou et al. (2022) Yao Dou, Maxwell Forbes, Rik Koncel-Kedziorski, Noah Smith, and Yejin Choi. 2022.  Is GPT-3 Text Indistinguishable from Human Text? Scarecrow: A Framework for Scrutinizing Machine Text. In _Proceedings of the 60th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)_. Association for Computational Linguistics, Dublin, Ireland, 7250–7274.  https://doi.org/10.18653/v1/2022.acl-long.501
  * Duncker (1945) Karl Duncker. 1945\.  On problem-solving.  _Psychological Monographs_ 58, 5 (1945), i–113.  https://doi.org/10.1037/h0093599
  * Freiknecht and Effelsberg (2020) Jonas Freiknecht and Wolfgang Effelsberg. 2020.  Procedural Generation of Interactive Stories using Language Models. In _International Conference on the Foundations of Digital Games_. ACM, Bugibba Malta, 1–8.  https://doi.org/10.1145/3402942.3409599
  * Gero et al. (2022) Katy Ilonka Gero, Vivian Liu, and Lydia Chilton. 2022\.  Sparks: Inspiration for Science Writing using Language Models. In _Designing Interactive Systems Conference_. ACM, Virtual Event Australia, 1002–1019.  https://doi.org/10.1145/3532106.3533533
  * Goyal et al. (2022) Tanya Goyal, Junyi Jessy Li, and Greg Durrett. 2022.  News Summarization and Evaluation in the Era of GPT-3.  http://arxiv.org/abs/2209.12356 arXiv:2209.12356 [cs].
  * Lee et al. (2022) Mina Lee, Megha Srivastava, Amelia Hardy, John Thickstun, Esin Durmus, Ashwin Paranjape, Ines Gerard-Ursin, Xiang Lisa Li, Faisal Ladhak, Frieda Rong, Rose E. Wang, Minae Kwon, Joon Sung Park, Hancheng Cao, Tony Lee, Rishi Bommasani, Michael Bernstein, and Percy Liang. 2022\.  Evaluating Human-Language Model Interaction.  http://arxiv.org/abs/2212.09746 arXiv:2212.09746 [cs] version: 2.
  * Lin (2023) Zhicheng Lin. 2023\.  _Why and how to embrace AI such as ChatGPT in your academic life_.  preprint. PsyArXiv.  https://doi.org/10.31234/osf.io/sdx3j
  * MacNeil et al. (2022a) Stephen MacNeil, Andrew Tran, Arto Hellas, Joanne Kim, Sami Sarsa, Paul Denny, Seth Bernstein, and Juho Leinonen. 2022a.  Experiences from Using Code Explanations Generated by Large Language Models in a Web Software Development E-Book.  http://arxiv.org/abs/2211.02265 arXiv:2211.02265 [cs].
  * MacNeil et al. (2022b) Stephen MacNeil, Andrew Tran, Dan Mogil, Seth Bernstein, Erin Ross, and Ziheng Huang. 2022b.  Generating Diverse Code Explanations using the GPT-3 Large Language Model. In _Proceedings of the 2022 ACM Conference on International Computing Education Research - Volume 2_. ACM, Lugano and Virtual Event Switzerland, 37–39.  https://doi.org/10.1145/3501709.3544280
  * Noy et al. (2012) Lior Noy, Yuval Hart, Natalie Andrew, Omer Ramote, Avi Mayo, and Uri Alon. 2012\.  A quantitative study of creative leaps, Mary Maher, Kristian Hammond, Alison Pease, {}Rafael Pérez, Dan Ventura, and Geraint Wiggins (Eds.). 72–76.  http://computationalcreativity.net/iccc2012/wp-content/uploads/2012/05/072-Noy.pdf
  * Roemmele (2016) Melissa Roemmele. 2016\.  Writing Stories with Help from Recurrent Neural Networks.  _Proceedings of the AAAI Conference on Artificial Intelligence_ 30, 1 (March 2016).  https://doi.org/10.1609/aaai.v30i1.9810
  * Runco and Jaeger (2012) Mark A. Runco and Garrett J. Jaeger. 2012.  The Standard Definition of Creativity.  _Creativity Research Journal_ 24, 1 (Jan. 2012), 92–96.  https://doi.org/10.1080/10400419.2012.650092
  * Sawyer (2012) R. Keith Sawyer. 2012\.  _Explaining creativity: the science of human innovation_ (2nd ed.).  Oxford University Press, New York.
  * Siangliulue et al. (2015) Pao Siangliulue, Joel Chan, Krzysztof Gajos, and Steven P. Dow. 2015.  Providing timely examples improves the quantity and quality of generated ideas. In _Proceedings of the ACM Conference on Creativity and Cognition_.  https://doi.org/10.1145/2757226.2757230
  * Singh et al. (2022) Nikhil Singh, Guillermo Bernal, Daria Savchenko, and Elena L. Glassman. 2022.  Where to Hide a Stolen Elephant: Leaps in Creative Writing with Multimodal Machine Intelligence.  _ACM Transactions on Computer-Human Interaction_ (Feb. 2022), 3511599\.  https://doi.org/10.1145/3511599
  * Webb et al. (2022) Taylor Webb, Keith J. Holyoak, and Hongjing Lu. 2022\.  Emergent Analogical Reasoning in Large Language Models.  http://arxiv.org/abs/2212.09196 arXiv:2212.09196 [cs].
  * Yang et al. ([n.d.]) Daijin Yang, Yanpeng Zhou, Zhiyuan Zhang, and Toby Jia-Jun Li. [n.d.].  AI as an Active Writer: Interaction strategies with generated text in human-AI collaborative fiction writing.  ([n. d.]).
  * Yuan et al. (2022) Ann Yuan, Andy Coenen, Emily Reif, and Daphne Ippolito. 2022\.  Wordcraft: Story Writing With Large Language Models. In _27th International Conference on Intelligent User Interfaces_. ACM, Helsinki Finland, 841–852.  https://doi.org/10.1145/3490099.3511105
  * Zhu and Luo (2022) Q. Zhu and J. Luo. 2022\.  Generative Pre-Trained Transformer for Design Concept Generation: An Exploration.  _Proceedings of the Design Society_ 2 (May 2022), 1825–1834.  https://doi.org/10.1017/pds.2022.185 Publisher: Cambridge University Press.