Set-theoretical solutions of the pentagon equation on Clifford semigroups

Abstract

Given a set-theoretical solution of the pentagon equation $s:S\times S\rightarrow S\times S$ on a set S and writing $s(a, b)=(a\cdot b,\, \theta _a(b))$, with $\cdot$ a binary operation on S and $\theta _a$ a map from S into itself, for every $a\in S$, one naturally obtains that $\left( S,\,\cdot \right)$ is a semigroup. In this paper, we focus on solutions defined in Clifford semigroups $\left( S,\,\cdot \right)$ satisfying special properties on the set of all idempotents ${{\,\textrm{E}\,}}(S)$. Into the specific, we provide a complete description of idempotent-invariant solutions, namely, those solutions for which $\theta _a$ remains invariant in ${{\,\textrm{E}\,}}(S)$, for every $a\in S$. Moreover, we construct a family of idempotent-fixed solutions, i.e., those solutions for which $\theta _a$ fixes every element in ${{\,\textrm{E}\,}}(S)$ for every $a\in S$, from solutions given on each maximal subgroup of S.

1 Introduction

If V is a vector space over a field F, a linear map ${\mathcal {S}}: V \otimes V \rightarrow V \otimes V$ is said to be a solution of the pentagon equation on V if it satisfies the relation

$$\begin{aligned} {\mathcal {S}}_{12}{\mathcal {S}}_{13}{\mathcal {S}}_{23}={\mathcal {S}}_{23}{\mathcal {S}}_{12}, \end{aligned}$$

(1)

where ${\mathcal {S}}_{12}={\mathcal {S}}\otimes {{\,\textrm{id}\,}}_V$, ${\mathcal {S}}_{23}={{\,\textrm{id}\,}}_V\otimes \, {\mathcal {S}}$, ${\mathcal {S}}_{13}=({{\,\textrm{id}\,}}_V\otimes \, \Sigma )\,{\mathcal {S}}_{12}\;({{\,\textrm{id}\,}}_V\otimes \, \Sigma )$, with $\Sigma$ the flip operator on $V\otimes V$, i.e., $\Sigma (u\otimes v)=v\otimes u$, for all $u,v\in V$. The pentagon equation arose at first at the beginning of the ’80s in [5] as the Biedenharn–Elliott identity for Wigner 6j-symbols and Racah coefficients in the representation theory for the rotation group. Maillet [21] showed that solutions of the pentagon equation lead to solutions of the tetrahedron equation [32], a generalization of the well-known quantum Yang–Baxter equation [4, 30]. Moreover, in [25, Theorem 3.2], Militaru showed that bijective solutions on finite vector spaces are equivalent to finite Hopf algebras, so the classification of the latter is reduced to the classification of solutions. In the subsequent years, the pentagon equation appeared in literature in several forms with different terminologies according to the specific research areas. We highlight some interesting works as [2, 3, 11, 13, 15,16,17, 22, 25, 27,28,29, 31], just to name a few. For a fuller treatment of some applications in which the pentagon equation appears, we suggest the recent paper by Dimakis and Müller-Hoissen [10] (along with the references therein), where the authors dealt with an infinite family of equations named polygon equations.

As well as Drinfel’d in [12] translated the study of solutions of the Yang–Baxter equation into set-theoretical terms, Kashaev and Sergeev in [19] began the study of the pentagon equation with a set-theoretical approach. Namely, if S is a set, a map $s:S\times S\rightarrow S\times S$ satisfying the following “reversed" relation

$$\begin{aligned} s_{23}s_{13}s_{12}=s_{12}s_{23}, \end{aligned}$$

(2)

where $s_{12}=s\times {{\,\textrm{id}\,}}_S$, $s_{23}={{\,\textrm{id}\,}}_S\times s$, $s_{13}=({{\,\textrm{id}\,}}_S\times \tau )\,s_{12}\,({{\,\textrm{id}\,}}_S\times \tau )$, and $\tau (a,b)=(b,a)$, for all $a,b\in S$, is said to be a set-theoretical solution of the pentagon equation, or briefly solution, on S. If, in particular, s is a solution on a finite set S, then the linear map ${\mathcal {S}}:F^{S \times S} \rightarrow F^{S \times S}$ defined by ${\mathcal {S}}(f)(a,b)= f(s(a,b))$, for all $a,b \in S$, is a solution of (1) on the vector space $F^S$ of all functions from S to F. The problem of classifying all possible solutions to the pentagon equation is a fascinating question that is a long way off from being solved. Nevertheless, some partial results have been obtained.

For their purposes, the authors in [19] investigated only bijective maps. This class of solutions was also studied by Kashaev and Reshetikhin in [18], where it is shown that each symmetrically factorizable Lie group is related to a bijective solution. Among these solutions, a description of all those that are involutive, i.e., $s^2={{\,\textrm{id}\,}}_{S\times S}$, has been recently given by Colazzo, Jespers, and Kubat in [9].

One of the key steps in the classification problem is the fact that every set S admitting a solution s is inherently endowed with a semigroup structure (see [6, Proposition 8]): if we write $s(a, b)=(a\cdot b,\, \theta _a(b))$, then $(S,\cdot )$ becomes a semigroup (hereinafter, we will use juxtaposition for the product in a semigroup). Therefore, the study of solutions (S, s) undoubtedly passes through an exhaustive analysis of the impact of s on the structural description of S. In this vein, in [6, Theorem 15] the authors provide a description of all solutions on a group by means of its normal subgroups. Moreover, in [7], we can find several constructions of solutions defined in the matched product of two semigroups, that is a semigroup including the classical Zappa–Szép product. Furthermore, in [23], the first author studies idempotent solutions, namely, maps satisfying the property $s^2=s$, and describes this kind of solutions on monoids having central idempotents. In light of these results, it is becoming more and more clear that idempotents in a semigroup play a prominent role in the study of solutions.

The next natural step is to try to extend the property of idempotents being central to a wider class of semigroups. In this light, this paper aims to begin with the study of solutions on Clifford semigroups, i.e. inverse semigroups whose idempotent elements are central (see the seminal monographs [20] and [26] for a full treatment on inverse semigroups). Bearing in mind that the behaviour of Clifford semigroups is very close to that of groups, the description of solutions on groups in [6] is of great utility in the problem of classifying all solutions on a Clifford semigroup. More concretely, it is easy to check that every solution on a group G satisfies that $\theta _a(1) = 1$, for every $a\in G$. Therefore, it motivates us to consider both classes of solutions on a Clifford semigroup S such that $\theta _a$, respectively, fixes every idempotent or remains invariant on every idempotent, for every $a\in S$. We call them, respectively, idempotent-fixed and idempotent-invariant solutions.

The main results of this paper are the following. Firstly, we provide a complete description of the first class of solutions on a Clifford semigroup S, which includes that made in the context of groups. To this aim, we introduce the kernel of an arbitrary solution on S, which turns out to be a normal subsemigroup, that is a subsemigroup containing the idempotents and closed by conjugation. Secondly, for the second class, considering that any Clifford semigroup is a union of a family of pairwise disjoint groups $\{G_e\}_{e\in {{\,\textrm{E}\,}}(S)}$, we give a construction of solutions obtained starting from a solution on each group $G_e$.

Finally, we raise some questions aimed at continuing the study of the solutions in this class of semigroups.

2 Preliminaries

This section aims to briefly introduce some basics of set-theoretical solutions of the pentagon equation. Initially, we recall some notions related to Clifford semigroups useful for our purposes. For a more detailed treatment of this topic, we refer the reader to [8] and [20].

2.1 Basics on Clifford semigroups

Recall that S is an inverse semigroup if for each $a\in S$ there exists a unique $a^{-1}\in S$ such that $a = aa^{-1}a$ and $a^{-1} = a^{-1}aa^{-1}$. They hold $(a b)^{-1}=b^{-1} a^{-1}$ and $(a^{-1})^{-1}=a$, for all $a,b \in S$. Moreover, ${{\,\textrm{E}\,}}(S) = \{\,aa^{-1} \ | \ a\in S\,\}$ and one can consider the following natural partial-order relation

$$\begin{aligned} \forall \ e,f\in {{\,\textrm{E}\,}}(S)\qquad e\le f \ \Longleftrightarrow \ e = ef = fe. \end{aligned}$$

An inverse semigroup S is Clifford if $aa^{-1}=a^{-1}a$, for any $a\in S$, or, equivalently, the idempotents are central in the sense that commute with every element in S.

Given a Clifford semigroup S, we introduce the following relations and the properties involved themselves. They are an easy consequence of the fact that all Green’s relations coincide in S and they characterize the structure of S itself. If $a,b\in S$, we define

• 1. $a \le b$ if, and only if, $aa^{-1}\le bb^{-1}$, which is an extension of the natural partial order in S;

• 2. $a \, {\mathcal {R}} \, b$ if, and only if, $a \le b$ and $b \le a$.

It follows that $\le$ is a preorder on S and ${\mathcal {R}}$ is an equivalence relation on S such that

$$\begin{aligned} G_{aa^{-1}}:=[a]_{\mathcal {R}} = \{b \in S\, \mid \, bb^{-1} = aa^{-1}\} \end{aligned}$$

is a group with identity $aa^{-1}$, for every $a \in S$. On the other hand, for all $a, b \in S$,

$$\begin{aligned} a \le b \,\Longleftrightarrow \, \exists \, u \in S\quad a=ub \,\,\vee \,\, a=bu. \end{aligned}$$

(3)

Moreover, $\le$ induces an order relation on the equivalence classes of ${\mathcal {R}}$, namely, for all $e,f\in {{\,\textrm{E}\,}}(S)$, $G_e \le G_f$ if, and only if, $e \le f$. The following theorem describes Clifford semigroups.

Theorem 1

Let S be a Clifford semigroup. Then,

• 1. S is a union of a family of pairwise disjoint groups $\{G_e\}_{e\in {{\,\textrm{E}\,}}(S)}$;

• 2. the map $\varphi _{f,e} :G_f \rightarrow G_e$ given by $\varphi _{f,e}(b) = eb$, for every $b \in G_f$, is a group homomorphism, for all $e,f\in {{\,\textrm{E}\,}}(S)$ such that $e \le f$;

• 3. for all $e,f,g \in {{\,\textrm{E}\,}}(S)$ such that $e \le f \le g$, then $\varphi _{g,e} = \varphi _{f,e}\varphi _{g,f}$.

As a consequence of the previous theorem, the product in Clifford semigroups can be written through the group homomorphisms $\varphi _{e,f}$, namely,

$$\begin{aligned} ab = (efa)(efb) = \varphi _{e, ef}\left( a\right) \varphi _{f, ef}\left( b\right) \in G_{ef}, \end{aligned}$$

for all $a\in G_e$, $b\in G_f$. In particular, for all $a \in S$, $e \in {{\,\textrm{E}\,}}(S)$ such that $a \le e$, one has $ae = ea = a$.

For the sake of completeness, the converse of Theorem 1 is also true.

2.2 Basics on solutions

Kashaev and Sergeev [19] first dealt with solutions from an algebraic point of view. Recently, the study of these solutions has been recovered in [6, 7, 9, 23]. Following the notation introduced in these works, given a set S and a map s from $S \times S$ into itself, we will write

$$\begin{aligned} s(a,b):=(a b, \,\theta _a(b)), \end{aligned}$$

for all $a,b \in S$, where $\theta _a$ is a map from S into itself, for every $a \in S$. Then, s is briefly a solution on S if, and only if, the following conditions hold

$$\begin{aligned} (ab)c&=a(bc) \\ \theta _a(b) \theta _{a b}(c)&=\theta _a(b c) \end{aligned}$$

(P1)

$$\begin{aligned} \theta _{\theta _a(b)}\theta _{a b}&=\theta _b \end{aligned}$$

(P2)

for all $a,b,c \in S$. Thus, the first identity naturally gives rise to a semigroup structure on S, which leads the study of solutions to focus on specific classes of semigroups. When describing solutions, it serves to distinguish those solutions that are not isomorphic.

Definition 2

Let S, T be two semigroups and $s(a,b)=(ab, \theta _a(b))$, $t(u, v)=(uv, \eta _u(v))$ two solutions on S and T, respectively. Then, s and t are isomorphic if there exists an isomorphism $\psi : S \rightarrow T$ such that

$$\begin{aligned} \psi \theta _a(b)=\eta _{\psi \left( a\right) }\psi (b), \end{aligned}$$

(4)

for all $a,b \in S$, or, equivalently, $(\psi \times \psi )s=t( \psi \times \psi )$.

The following are easy examples of solutions used throughout this paper.

Examples 1

 

• 1. Let S be a set and $f, g:S\rightarrow S$ idempotent maps such that $fg=gf$. Then, $s(a,b) = \left( f\left( a\right) ,\, g\left( b\right) \right)$ is a solution on S (cf. [24]).

• 2. Let S be a semigroup and $\gamma \in {{\,\textrm{End}\,}}(S)$ such that $\gamma ^2=\gamma$. Then, the map s given by $s(a,b)=\left( ab,\gamma \left( b\right) \right) ,$ for all $a,b \in S$, is a solution on S (see [6, Examples 2-2.]).

Let us observe that every Clifford semigroup S gives rise to the following solutions

$$\begin{aligned} {\mathcal {I}}(a,b)=(ab, b), \qquad {\mathcal {F}}(a,b)=\left( ab, bb^{-1}\right) , \qquad {\mathcal {E}}(a,b)=(ab, e), \end{aligned}$$

(5)

where $e \in {{\,\textrm{E}\,}}(S)$ is a fixed idempotent of S, belonging to the class of solutions in 2. of Examples 1.

In [1], solutions of (1) are defined on Hilbert spaces in terms of commutative and cocommutative multiplicative unitary operators (see [1, Definition 2.1]). These operators motivate the following classes of solutions in the set-theoretical case.

Definition 3

A solution $s: S \times S \rightarrow S \times S$ is said to be commutative if $s_{12}s_{13}=s_{13}s_{12}$ and cocommutative if $s_{13}s_{23}=s_{23}s_{13}$.

Solutions in Examples 1-1. are both commutative and cocommutative. In [9, Corollary 3.4], it is proved that if s is an involutive solution, i.e., $s^2={{\,\textrm{id}\,}}_{S \times S}$, then s is both commutative and cocommutative.

Convention: In the sequel, we assume that S is a Clifford semigroup and simply write that s is a solution on S instead of $s(a,b) = (ab, \theta _a(b))$, for all $a,b\in S$.

3 Properties of solutions on Clifford semigroups

In this section, we show the existence of a normal subsemigroup associated to any solution s on S. We point out that the properties we proved are consistent with those given in the context of groups [6].

Proposition 4

Let s be a solution on S. Then, the following statements hold:

• 1. $\theta _a\left( a^{-1}\right) =\theta _{aa^{-1}}\left( a\right) ^{-1}$,

• 2. $\theta _a\left( a^{-1}a\right) =\theta _a\left( a^{-1}\right) \theta _a\left( a^{-1}\right) ^{-1}\in {{\,\textrm{E}\,}}(S)$,

• 3. $\theta _{aa^{-1}}=\theta _{\theta _{a^{-1}}\left( aa^{-1}\right) }\theta _{a^{-1}}$,

for every $a \in S$.

Proof

Let $a \in S$. Then, by (P1), we have

$$\begin{aligned} \theta _a\left( a^{-1}\right) \theta _{aa^{-1}}\left( a\right) \theta _a\left( a^{-1}\right)&=\theta _a\left( a^{-1}a\right) \theta _a\left( a^{-1}\right) =\theta _a\left( a^{-1}a\right) \theta _{aa^{-1}a}\left( a^{-1}\right) \\&=\theta _a\left( a^{-1}aa^{-1}\right) =\theta _a\left( a^{-1}\right) \end{aligned}$$

and $\theta _{aa^{-1}}\left( a\right) \theta _a\left( a^{-1}\right) \theta _{aa^{-1}}\left( a\right) =\theta _{aa^{-1}}\left( aa^{-1}\right) \theta _{aa^{-1}}\left( a\right) =\theta _{aa^{-1}}\left( aa^{-1}a\right) =\theta _{aa^{-1}}\left( a\right) ,$ hence $\theta _a\left( a^{-1}\right) =\theta _{aa^{-1}}\left( a\right) ^{-1}$, so 1. is satisfied.

Moreover, by 1., we get $\theta _a\left( a^{-1}a\right) =\theta _a\left( a^{-1}\right) \theta _{aa^{-1}}\left( a\right) =\theta _a\left( a^{-1}\right) \theta _a\left( a^{-1}\right) ^{-1}$, thus $\theta _a\left( a^{-1}a\right)$ is an idempotent of S.

Finally, by (P2), $\theta _{aa^{-1}}=\theta _{\theta _{a^{-1}}\left( aa^{-1}\right) }\theta _{a^{-1}aa^{-1}}=\theta _{\theta _{a^{-1}}\left( aa^{-1}\right) }\theta _{a^{-1}}$, which is our claim. $\square$

Note that the previous result also holds in any inverse semigroup that is not necessarily Clifford.

Now, let us introduce a crucial object in studying solutions on Clifford semigroups.

Definition 5

If s is a solution on S, the following set

$$\begin{aligned} K =\{a\in S\, \mid \, \forall \, e \in {{\,\textrm{E}\,}}(S) \,\, e \le a \Rightarrow \theta _e(a) \in {{\,\textrm{E}\,}}(S)\} \end{aligned}$$

is called the kernel of s.

Consistently with [6, Lemma 13], our aim is to show that K is a normal subsemigroup of the Clifford S, namely, ${{\,\textrm{E}\,}}(S)\subseteq K$ and $a^{-1}Ka \subseteq K$, for every $a \in S$. To this end, we first provide a preliminary result.

Lemma 6

Let s be a solution on S and K the kernel of s. Then, they hold:

• 1. $\theta _a(e) \in {{\,\textrm{E}\,}}(S)$, for all $a\in S$ and $e\in {{\,\textrm{E}\,}}(S)$ such that $a\le e$;

• 2. $\theta _{ea}(k) \in {{\,\textrm{E}\,}}(S)$, for all $a \in S$, $k \in K$, and $e \in {{\,\textrm{E}\,}}(S)$ such that $e \le a$, $e \le k$.

Proof

Let $a \in S$ and $e \in {{\,\textrm{E}\,}}(S)$. If $a \le e$, by (P1), we obtain $\theta _a(e) = \theta _a(e)\theta _{ae}(e) = \theta _a(e)^2$, hence 1. follows. Now, if $k \in K$ and $e \le a$, $e \le k$, then $\theta _e(k) \in {{\,\textrm{E}\,}}(S)$ and, by (P2),

$$\begin{aligned} \theta _{ea}(k)=\theta _{\theta _{a^{-1}}\left( ea\right) }\theta _{a^{-1}ea}(k)=\theta _{\theta _{a^{-1}}\left( ea\right) }\theta _{e}(k). \end{aligned}$$

If we prove that $\theta _{a^{-1}}\left( ea\right) \le \theta _{e}(k)$, by 1., we obtain that $\theta _{ea}(k) \in {{\,\textrm{E}\,}}(S)$. We get

$$\begin{aligned} \theta _{a^{-1}}\left( ea\right)&= \theta _{a^{-1}}\left( eakk^{-1}\right) = \theta _{a^{-1}}\left( ea\right) \theta _{a^{-1}ea}\left( kk^{-1}\right) =\theta _{a^{-1}}\left( ea\right) \theta _{e}\left( kk^{-1}\right) \\&=\theta _{a^{-1}}\left( ea\right) \theta _{e}\left( k\right) \theta _{ek}\left( k^{-1}\right) . \end{aligned}$$

Hence, by (3), $\theta _{a^{-1}}\left( ea\right) \le \theta _{e}\left( k\right)$. Therefore, the claim follows. $\square$

Corollary 7

Let s be a solution on S. If $a,b\in S$ are such that $a\le b$, then $\theta _a(b)\in G_{\theta _a\left( bb^{-1}\right) }$. Moreover, they hold $\theta _a(bb^{-1}) = \theta _a(b)\theta _a(b)^{-1}$ and $\theta _a(b)^{-1} = \theta _{ab}(b^{-1})$.

Proof

If $a,b\in S$ are such that $a\le b$, then $a \le bb^{-1}$ and by Lemma 6-1., $\theta _a(bb^{-1})\in {{\,\textrm{E}\,}}(S)$. Now,

$$\begin{aligned} \theta _a(b) = \theta _a\left( bb^{-1}b\right) = \theta _a\left( bb^{-1}\right) \theta _{abb^{-1}}(b) = \theta _a\left( bb^{-1}\right) \theta _a(b) \end{aligned}$$

and $\theta _{a}\left( bb^{-1}\right) = \theta _a(b)\theta _{ab}\left( b^{-1}\right)$. Thus, by (3), $\theta _a(b) \le \theta _a(bb^{-1})$ and $\theta _a(bb^{-1})\le \theta _a(b)$, i.e. $\theta _a(b)\in G_{\theta _a\left( bb^{-1}\right) }$. In addition, by the equality $\theta _a\left( bb ^{-1}\right) =\theta _a\left( b^{-1}b \right) =\theta _a\left( b^{-1}\right) \theta _{ab^{-1}}\left( b\right)$ and the previous paragraph, it follows that $\theta _a(b)$, $\theta _a(b^{-1})$, and $\theta _a(bb^{-1})$ are in the same group with identity $\theta _a(bb^{-1})$. Moreover, $\theta _a(b)^{-1} = \theta _{ab}\left( b^{-1}\right)$, which completes the proof. $\square$

Theorem 8

Let s be a solution on S. Then, the kernel K of s is a normal subsemigroup of S.

Proof

Initially, by Lemma 6-1., ${{\,\textrm{E}\,}}(S) \subseteq K$. Now, if $k,h \in K$ and $e \in {{\,\textrm{E}\,}}(S)$ are such that $e \le kh$, then $e \le k$ and $e \le h$ and thus, $\theta _e(k), \theta _e(h) \in {{\,\textrm{E}\,}}(S)$. By Lemma 6-2., we obtain that $\theta _{ek}\left( h\right) \in {{\,\textrm{E}\,}}(S)$, and so that $\theta _e\left( kh\right) =\theta _e\left( k\right) \theta _{ek}\left( h\right) \in {{\,\textrm{E}\,}}(S)$.

Now, if $a \in S$, $k \in K$, and $e \in {{\,\textrm{E}\,}}(S)$ are such that $e \le a^{-1}ka$, then $e \le a$, $e \le a^{-1}$, and $e \le k$. Then, $\theta _e(k) \in {{\,\textrm{E}\,}}(S)$. Besides,

$$\begin{aligned} \theta _e\left( a^{-1}ka\right) = \theta _e\left( a^{-1}\right) \theta _{ea^{-1}}(k) \theta _{ea^{-1}k}(a). \end{aligned}$$

By Lemma 6-1., $\theta _e\left( a^{-1}\right) \in {{\,\textrm{E}\,}}(S)$ and, by Lemma 6-2., $\theta _{ea^{-1}}(k)\in {{\,\textrm{E}\,}}(S)$. Furthermore, also $\theta _{ea^{-1}k}(a)\in {{\,\textrm{E}\,}}(S)$. In fact, by (P2),

$$\begin{aligned} \theta _{ea^{-1}k}\left( a\right) = \theta _{\theta _{k^{-1}a}\left( ea^{-1}k\right) }\theta _{k^{-1}aea^{-1}k}\left( a\right) = \theta _{\theta _{k^{-1}a}\left( ea^{-1}k\right) }\theta _{e}\left( a\right) \end{aligned}$$

and, since

$$\begin{aligned} \theta _{k^{-1}a}\left( ea^{-1}k\right)&= \theta _{k^{-1}a}\left( ea^{-1}kaa^{-1}\right) \theta _{k^{-1}a}\left( ea^{-1}k\right) \theta _{k^{-1}aea^{-1}k}\left( aa^{-1}\right) = \\&= \theta _{k^{-1}a}\left( ea^{-1}k\right) \theta _{e}\left( a\right) \theta _{ea}\left( a^{-1}\right) , \end{aligned}$$

we obtain that, by (3), $\theta _{k^{-1}a}\left( ea^{-1}k\right) \le \theta _{e}\left( a\right)$. So, as before, by Lemma 6-1., we obtain $\theta _{ea^{-1}k}\left( a\right) \in E\left( S\right)$. Therefore, the claim follows. $\square$

We conclude the section by describing the commutative and cocommutative solutions on Clifford semigroups. It is easy to check that a solution $s(a,b)=(ab, \theta _a(b))$ is commutative if, and only if,

$$\begin{aligned} acb = abc \end{aligned}$$

(C1)

$$\begin{aligned} \theta _a = \theta _{ab} \end{aligned}$$

(C2)

and s is cocommutative if, and only if,

$$\begin{aligned} a\theta _b(c)=ac \end{aligned}$$

(CC1)

$$\begin{aligned}&\theta _a\theta _{b}=\theta _b\theta _a \end{aligned}$$

(CC2)

for all $a,b,c \in S$.

Proposition 9

Let s be a solution on S. Then,

• 1. s is commutative if, and only if, S is a commutative Clifford semigroup and $\theta _a=\gamma$, for every $a \in S$, with $\gamma \in {{\,\textrm{End}\,}}(S)$ and $\gamma ^2=\gamma$.

• 2. s is cocommutative if, and only if, $\theta _a(b) = b$, for all $a,b \in S$, i.e., $s={\mathcal {I}}$.

Proof

At first, we suppose that $s(a,b)=(ab, \theta _a(b))$ is a commutative solution. Then, by (C1), taking $a=cc^{-1}$, we obtain that S is commutative. Moreover, by (C2), we get $\theta _a=\theta _{ab}=\theta _{ba}=\theta _b$. Hence, $\theta _a=\gamma$, for every $a \in S$, and by the definition of solution we obtain the rest of the claim. The converse trivially follows by 2. in Examples 1.

Now, assume that $s(a,b)=(ab, \theta _a(b))$ is a cocommutative solution. Then, by (CC1), taking $a=cc^{-1}$, we obtain

$$\begin{aligned} cc^{-1}\theta _b(c) = c, \quad \text {for all }b,c \in S.\end{aligned}$$

Set $e_0:= \theta _b(c)\theta _b(c)^{-1}$, it follows that $cc^{-1}\le e_0$. On the other hand, again by (CC1), $e\theta _b(c) = ec$, for every $e \in {{\,\textrm{E}\,}}(S)$. In particular, $\theta _b(c) = e_0 \theta _b(c) = e_0c$. Thus, $e_0 \le cc^{-1}$ and so $e_0 = cc^{-1}$. Therefore, we get $\theta _b(c) = c$, that is our claim. $\square$

4 A description of idempotent-invariant solutions

In this section, we provide a description of a specific class of solutions on a Clifford semigroup, the idempotent-invariant ones, which includes the result contained in [6, Theorem 15].

Definition 10

A solution s on S is said to be idempotent-invariant or $E\left( S\right)$-invariant if it holds the identity

$$\begin{aligned} \theta _a(e) = \theta _a(f), \end{aligned}$$

(6)

for all $a\in S$ and $e,f\in {{\,\textrm{E}\,}}(S)$.

An easy example of ${{\,\textrm{E}\,}}(S)$-invariant solution is ${\mathcal {E}}(a,b) = (ab, e)$ in (5), with $e\in {{\,\textrm{E}\,}}(S)$.

Example 2

Let us consider the commutative Clifford monoid $S =\{1,\,a,\,b\}$ with identity 1 and such that $a^2=a$, $b^2=a$, and $ab=b$. Then, other than the map ${\mathcal {E}}$ in (5), there exists the idempotent-invariant solution $s(a,b)=(ab, \gamma (b))$ with $\gamma : S \rightarrow S$ the map given by $\gamma (1)=\gamma (a)=a$ and $\gamma (b)=b$, which belongs to the class of solutions in 2. of Examples 1.

Next, we show how to construct an idempotent-invariant solution on S starting from a specific congruence on S. Recall that the restriction of a congruence $\rho$ in a Clifford semigroup S to ${{\,\textrm{E}\,}}(S)$ is also a congruence on ${{\,\textrm{E}\,}}(S)$, called the trace of $\rho$ and usually denoted by $\tau ={{\,\textrm{tr}\,}}\rho$ (for more details, see [14, Section 5.3]).

Proposition 11

Let S be a Clifford semigroup, $\rho$ a congruence on S such that $S/\rho$ is a group and ${\mathcal {R}}$ a system of representatives of $S/\rho$. If $\mu : S \rightarrow {\mathcal {R}}$ is a map such that

$$\begin{aligned} \mu \left( ab\right) =\mu \left( a\right) \mu \left( a\right) ^{-1}\mu \left( ab\right) , \quad \text {(the product considered is the operation in { S})} \end{aligned}$$

(7)

for all $a,b \in S$, and $\mu (a) \, \rho \, a$, for every $a \in S$, then the map $s:S\times S\rightarrow S\times S$ given by

$$\begin{aligned} s(a,b)=\left( ab, \mu \left( a\right) ^{-1}\mu \left( ab \right) \right) , \end{aligned}$$

for all $a,b \in S$, is an ${{\,\textrm{E}\,}}(S)$-invariant solution on S.

Proof

Let $a,b,c \in S$. Set $\theta _a(b):=\mu \left( a\right) ^{-1}\mu \left( ab \right)$, by (7), we obtain

$$\begin{aligned} \theta _a(b)\theta _{ab}(c)=\mu \left( a\right) ^{-1}\mu \left( ab \right) \mu \left( ab\right) ^{-1}\mu \left( abc \right) =\mu \left( a\right) ^{-1}\mu \left( abc \right) =\theta _a(bc). \end{aligned}$$

Now, if we compare

$$\begin{aligned} \theta _{\theta _a(b)}\theta _{ab}(c):&= \mu \left( \mu \left( a\right) ^{-1} \mu \left( a b\right) \right) ^{-1}\mu \left( \mu \left( a\right) ^{-1} \mu \left( ab\right) \mu \left( ab\right) ^{-1} \mu \left( abc\right) \right) \\&=\mu \left( \mu \left( a\right) ^{-1} \mu \left( ab\right) \right) ^{-1} \mu \left( \mu \left( a\right) ^{-1} \mu \left( abc\right) \right)&\text {by (7)} \end{aligned}$$

and $\theta _b(c):=\mu (b)^{-1} \mu (bc)$, to get the claim it is enough to show that

$$\begin{aligned} \mu (x)^{-1}\mu (xy) = \mu (y), \end{aligned}$$

for all $x, y \in S$. Indeed, by [14, Proposition 5.3.1], ${{\,\textrm{tr}\,}}\rho = {{\,\textrm{E}\,}}(S) \times {{\,\textrm{E}\,}}(S)$, and so

$$\begin{aligned} \mu (x)^{-1}\mu (xy) \ \rho \ x^{-1}xy \ \rho \ y^{-1}yy \ \rho \ y \ \rho \ \mu (y). \end{aligned}$$

Finally, if $a \in S$ and $e, f \in {{\,\textrm{E}\,}}(S)$, we obtain that

$$\begin{aligned} \mu (ae) \ \rho \ ae \ \rho \ af \rho \ \mu (af), \end{aligned}$$

hence $\mu \left( a e\right) =\mu \left( a f\right)$. Thus, $\theta _a(e)=\mu (a)^{-1}\mu \left( a e\right) =\mu (a)^{-1}\mu \left( a f\right) =\theta _a(f)$. Therefore, the claim follows. $\square$

Our aim is to show that all idempotent invariant solutions can be constructed exactly as in Proposition 11. Firstly, let us collect some useful properties of these maps.

Lemma 12

Let s be an $E\left( S\right)$-invariant solution on S. Then, the following hold:

• 1. $\theta _e=\theta _{f}$,

• 2. $\theta _{ae} = \theta _a$,

• 3. $\theta _{a}\left( e\right) \in {{\,\textrm{E}\,}}\left( S\right)$,

• 4. $\theta _e\theta _a = \theta _e$,

• 5. $\theta _a\left( b\right) = \theta _a\left( eb\right)$,

• 6. $\theta _e(a)^{-1}=\theta _{ea}\left( a^{-1}\right)$,

for all $e,f \in {{\,\textrm{E}\,}}\left( S\right)$ and $a, b\in S$.

Proof

Let $e, f \in {{\,\textrm{E}\,}}(S)$ and $a, b\in S$.

1.:

Since $\theta _e = \theta _{\theta _f(e)}\theta _{fe} = \theta _{\theta _f(fe)}\theta _{ffe} = \theta _{fe}$ and, similarly $\theta _f=\theta _{ef}$, it yields that $\theta _f = \theta _e$.

2.:

We have that

$$\begin{aligned} \theta _{ae}&=\theta _{\theta _{a^{-1}}(ae)}\theta _{aa^{-1}e}\\&=\theta _{\theta _{a^{-1}}(a)\theta _{a^{-1}a}(e)}\theta _{aa^{-1}}&\ aa^{-1}e\in {{\,\textrm{E}\,}}\left( S\right) \\&=\theta _{\theta _{a^{-1}}(a)\theta _{a^{-1}a}\left( a^{-1}a\right) }\theta _{aa^{-1}}&\text {by (6)}\\&=\theta _{\theta _{a^{-1}}\left( a\right) }\theta _{aa^{-1}}=\theta _a. \end{aligned}$$

3.:

According to 2., it follows that $\theta _{a}\left( e\right) = \theta _{a}\left( ee\right) = \theta _{a}\left( e\right) \theta _{ae}\left( e\right) = \theta _{a}\left( e\right) \theta _{a}\left( e\right)$, i.e., $\theta _{a}\left( e\right) \in {{\,\textrm{E}\,}}\left( S\right)$.

4.:

According to 2., we obtain that $\theta _e = \theta _{\theta _{a}\left( e\right) }\theta _{ae} = \theta _{e}\theta _{ae} = \theta _{e}\theta _{a}$.

5.:

Note that, by 2., $\theta _a\left( b\right) = \theta _a\left( bb^{-1}b\right) = \theta _a\left( bb^{-1}\right) \theta _{abb^{-1}}\left( b\right) = \theta _a\left( e\right) \theta _{ae}\left( b\right) = \theta _a\left( eb\right)$.

6.:

Applying 1., we get $\theta _e\left( a\right) \theta _{ea}\left( a^{-1}\right) \theta _{e}(a)=\theta _e\left( aa^{-1}\right) \theta _{eaa^{-1}}(a)=\theta _e(a)$ and, on the other hand,

$$\begin{aligned} \theta _{ea}\left( a^{-1}\right) \theta _e(a)\theta _{ea}\left( a^{-1}\right){} & {} =\theta _{ea}\left( a^{-1}\right) \theta _e\left( aa^{-1}\right) =\theta _{ea}\left( a^{-1}\right) \theta _{eaa^{-1}}\left( aa^{-1}\right) \\ {}{} & {} =\theta _{ea}\left( a^{-1}\right) . \end{aligned}$$

Therefore, the claim follows. $\square$

To prove the converse of Proposition 11, we need to recall the notion of the congruence pair of inverse semigroups that are Clifford (see [14, p. 155]). Given a Clifford semigroup S, a congruence $\tau$ on ${{\,\textrm{E}\,}}(S)$ is said to be normal if

$$\begin{aligned} \forall \ e,f\in {{\,\textrm{E}\,}}(S) \quad e \ \tau \ f \ \Longrightarrow \ \forall \ a\in S \quad a^{-1}ea\ \tau \ a^{-1}fa. \end{aligned}$$

If K is a normal subsemigroup of S, the pair $(K,\tau )$ is named a congruence pair of S if

$$\begin{aligned} \forall \ a \in S,\ e\in {{\,\textrm{E}\,}}(S) \quad ae\in K \ \ \text {and} \ \ (e, a^{-1}a)\in \tau \ \Longrightarrow \ a\in K. \end{aligned}$$

Given a congruence $\rho$, denoted by ${{\,\textrm{Ker}\,}}\rho$ the union of all the idempotent $\rho$-classes, its properties can be described entirely in terms of ${{\,\textrm{Ker}\,}}\rho$ and ${{\,\textrm{tr}\,}}\rho$.

Theorem 13

(cf. Theorem 5.3.3 in [14]) Let S be an inverse semigroup. If $\rho$ is a congruence on S, then $({{\,\textrm{Ker}\,}}\rho , {{\,\textrm{tr}\,}}\rho )$ is a congruence pair. Conversely, if $(K, \tau )$ is a congruence pair, then

$$\begin{aligned} \rho _{(K, \tau )}=\lbrace (a,b) \in S \times S \, \mid \, \left( a^{-1}a, b^{-1}b\right) \in \tau , \, ab^{-1} \in K\rbrace \end{aligned}$$

is a congruence on S. Moreover, ${{\,\textrm{Ker}\,}}\rho _{(K, \tau )}=K$, ${{\,\textrm{tr}\,}}\rho _{(K, \tau )}=\tau$, and $\rho _{({{\,\textrm{Ker}\,}}\rho , {{\,\textrm{tr}\,}}\rho )}=\rho$.

Lemma 14

Let s be an ${{\,\textrm{E}\,}}\left( S\right)$-invariant solution on S, $\tau = {{\,\textrm{E}\,}}(S)\times {{\,\textrm{E}\,}}(S)$, and K the kernel of s. Then, $\left( K, \tau \right)$ is a congruence pair of S.

Proof

At first, let us observe that the kernel K of s can be written as

$$\begin{aligned} K=\{a\in S\, \mid \, \forall \ e \in {{\,\textrm{E}\,}}(S)\quad \theta _e(a) \in {{\,\textrm{E}\,}}(S)\}. \end{aligned}$$

Now, let $a \in S$ and $e \in {{\,\textrm{E}\,}}(S)$ such that $ae\in K$. To get the claim it is enough to show that if $f\in {{\,\textrm{E}\,}}(S)$, then $\theta _{f}\left( a\right) \in {{\,\textrm{E}\,}}\left( S\right)$, i.e., $a\in K$. By 1. and 5. in Lemma 12, we obtain that

$$\begin{aligned} \theta _f\left( a\right) = \theta _{ef}\left( a\right) = \theta _{ef}\left( ae\right) \in {{\,\textrm{E}\,}}(S), \end{aligned}$$

which is our claim. $\square$

The following result completely describes idempotent-invariant solutions.

Theorem 15

Let s be an ${{\,\textrm{E}\,}}\left( S\right)$-invariant solution on S. Then, the map $\theta _e$ satisfies (7), for every $e \in {{\,\textrm{E}\,}}(S)$, and

$$\begin{aligned} \theta _a(b)=\theta _e(a)^{-1}\theta _e(ab), \end{aligned}$$

for all $a, b\in S$ and $e\in {{\,\textrm{E}\,}}\left( S\right)$. Moreover, there exists the congruence pair $\left( K, \tau \right)$, with K the kernel of S and $\tau ={{\,\textrm{E}\,}}(S) \times {{\,\textrm{E}\,}}(S)$, such that $\theta _e\left( S\right)$ is a system of representatives of the group $S/\rho _{\left( K, \tau \right) }$ and $\left( \theta _e\left( a\right) , a \right) \in \rho _{\left( K, \tau \right) }$, for all $e \in {{\,\textrm{E}\,}}(S)$ and $a\in S$.

Proof

Initially, (7) is satisfied since

$$\begin{aligned} \theta _e(a)^{-1}\theta _e(a)\theta _e(ab)=\theta _e(a)^{-1}\theta _e(a)\theta _e(a)\theta _{ea}(b)=\theta _e(a)\theta _{ea}(b)=\theta _e(ab), \end{aligned}$$

for all $a, b\in S$ and $e\in {{\,\textrm{E}\,}}\left( S\right)$. Besides,

$$\begin{aligned} \theta _a(b)&=\theta _a\left( a^{-1}ab\right)&\text { by Lemma 12-5}.\\&=\theta _a\left( a^{-1}\right) \theta _{aa^{-1}}(ab)\\&=\theta _{aa^{-1}}(a)^{-1}\theta _{aa^{-1}}(ab),&\text { by Proposition 4-1}.\\&= \theta _e(a)^{-1}\theta _e(ab)&\text { by Lemma 12-1}. \end{aligned}$$

for all $a, b\in S$ and $e\in {{\,\textrm{E}\,}}\left( S\right)$. Moreover, by Lemma 14, $\left( K, \tau \right)$ is a congruence pair and so, by Theorem 13, $\rho _{\left( K, \tau \right) }$ is a congruence such that $\tau ={{\,\textrm{tr}\,}}\rho _{\left( K, \tau \right) }$. Besides, by [14, Proposition 5.3.1], since ${{\,\textrm{tr}\,}}\rho _{\left( K, \tau \right) } = {{\,\textrm{E}\,}}(S) \times {{\,\textrm{E}\,}}(S)$, $S/\rho _{\left( K, \tau \right) }$ is a group. Now, let $a\in S$ and $e\in {{\,\textrm{E}\,}}\left( S\right)$ and let us check that $\left( \theta _e\left( a\right) , a \right) \in \rho _{\left( K, \tau \right) }$ by proving that $a^{-1}\theta _e\left( a\right) \in K$, i.e., $\theta _e\left( a^{-1}\theta _e\left( a\right) \right) \in {{\,\textrm{E}\,}}\left( S\right)$. To this end, note that

$$\begin{aligned} \theta _e\left( a^{-1}\theta _e\left( a\right) \right)&= \theta _e\theta _a\left( a^{-1}\theta _e\left( a\right) \right)&\text { by Lemma 12-4}.\\&= \theta _e \left( \theta _a\left( a^{-1}\right) \theta _{aa^{-1}}\theta _e\left( a\right) \right) \\&= \theta _e \left( \theta _a\left( a^{-1}\right) \theta _{aa^{-1}}\left( a\right) \right)&\text { by Lemma 12-4}.\\&= \theta _e \left( \theta _a\left( a^{-1}\right) \theta _{a}\left( a^{-1}\right) ^{-1}\right) ,&\text { by Proposition 4-1}. \end{aligned}$$

hence, by Lemma 12-3., $\theta _e\left( a^{-1}\theta _e\left( a\right) \right) \in {{\,\textrm{E}\,}}\left( S\right)$. Now, let us verify that $\theta _e\left( S\right)$ is a system of representatives of $S/\rho _{\left( K, \tau \right) }$. Clearly, $\theta _e\left( S\right) \ne \emptyset$ since $\theta _e\left( e\right) \in {{\,\textrm{E}\,}}\left( S\right)$. Besides, if $\left( \theta _e\left( b\right) , a \right) \in \rho _{\left( K, \tau \right) }$ we have that $a\,\rho _{\left( K, \tau \right) }\, b$, since $\left( \theta _e\left( a\right) , a \right) \in \rho _{\left( K, \tau \right) }$. Thus, $ab^{-1}\in K$ and so $\theta _e\left( ab^{-1}\right) \in {{\,\textrm{E}\,}}\left( S\right)$. This implies that

$$\begin{aligned} \theta _e\left( b\right)&= \theta _e\left( bb^{-1}\right) \theta _{ebb^{-1}}\left( b\right) \\&= \theta _e\left( bb^{-1}\right) \theta _{\theta _e\left( ab^{-1}\right) }\left( b\right)&\text {by Lemma 12-1}.\\&= \theta _e\theta _e\left( ab^{-1}\right) \theta _{\theta _e\left( ab^{-1}\right) }\theta _{eab^{-1}}\left( b\right)&\text {by (6) and Lemma 12-4}.\\&= \theta _e\left( ab^{-1}\right) \theta _{ab^{-1}}\left( b\right)&\text { by Lemma 12-4}.\\&= \theta _e\left( ab^{-1}\right) \theta _{eab^{-1}}\left( b\right)&\text {by Lemma 12-2. and (P2)}\\&= \theta _e\left( ab^{-1}b\right) \\&= \theta _e\left( a\right) .&\text { by Lemma 12-5}. \end{aligned}$$

Therefore, the claim follows. $\square$

Proposition 16

Let $s(a,b)=(ab, \theta _a(b))$ and $t(u,v)=(uv, \eta _u(v))$ be two ${{\,\textrm{E}\,}}(S)$-invariant solutions on S. Then, s and t are isomorphic if, and only if, there exists an inverse semigroup isomorphism $\psi$ of S such that $\psi \theta _e=\eta _e\psi$, i.e., $\psi$ sends the system of representatives $\theta _e(S)$ into the other one $\eta _e\left( \psi (S)\right)$, for every $e \in {{\,\textrm{E}\,}}(S)$.

Proof

Indeed, making explicit the condition (4), we obtain

for all $a,b \in S$ and $e \in {{\,\textrm{E}\,}}(S)$. Taking $b=a^{-1}$, we get

$$\begin{aligned} \psi \left( \theta _e(a)^{-1}\right)&=\psi \left( \theta _{ea}\left( a^{-1}\right) \right)&\text { by Lemma 12-6}.\\&=\psi \left( \theta _{ea}\left( a^{-1}\right) \theta _e\left( aa^{-1}\right) \right)&\text { by Lemma 12-1}.\\&=\psi \left( \theta _e(a)^{-1}\theta _e\left( aa^{-1}\right) \right)&\text { by Lemma 12-6}.\\&= \eta _e\left( \psi (a)\right) ^{-1}\eta _e\left( \psi \left( aa^{-1}\right) \right)&\text {by Lemma (*)}\\&=\eta _{e\psi (a)}\left( \psi (a)^{-1}\right) \eta _e\left( \psi \left( aa^{-1}\right) \right)&\text { by Lemma 12-6}.\\&=\eta _{e\psi (a)}\left( \psi (a)^{-1}\right)&\text { by Lemma 12-1}.\\&=\eta _e\left( \psi (a)\right) ^{-1}&\text { by Lemma 12-6}. \end{aligned}$$

Hence, since $\psi$ is an inverse semigroup homomorphism, $\psi \theta _e=\eta _e\psi$, for every $e \in {{\,\textrm{E}\,}}(S)$. Thus, the claim follows. $\square$

5 A construction of idempotent-fixed solutions

In this section, we deal with a class of solutions different from the idempotent-invariant ones, what we call idempotent-fixed solutions. Bearing in mind that a Clifford semigroup can be seen as a union of groups satisfying certain properties, it is natural to contemplate whether it is possible or not to construct a global solution in a Clifford semigroup from solutions obtained in each of its groups. In this regard, in the case of idempotent-fixed solutions, we manage to construct a family of solutions obtained by starting from given solutions in each group.

Definition 17

Let s be a solution on S. Then, s is idempotent-fixed or ${{\,\textrm{E}\,}}(S)$-fixed if

$$\begin{aligned} \theta _a(e)=e, \end{aligned}$$

(8)

for all $a \in S$ and $e \in {{\,\textrm{E}\,}}(S)$.

The maps ${\mathcal {I}}(a,b)=(ab,b)$ and ${\mathcal {F}}(a,b)=\left( ab, bb^{-1}\right)$ in (5) are idempotent-fixed solutions on S. Clearly, if S is a Clifford that is not a group, i.e., $|{{\,\textrm{E}\,}}(S)|>1$, then a solution on S can not be both idempotent-fixed and idempotent-invariant.

The next results contained several properties of idempotent-fixed solutions.

Proposition 18

Let s be an idempotent-fixed solution on S. Then, $\theta _e=\theta _e\theta _{ae}$, for all $a \in S$ and $e \in {{\,\textrm{E}\,}}(S)$. In particular, $\theta _e$ is an idempotent map.

Proof

It follows by $\theta _e=\theta _{\theta _a(e)}\theta _{ae}=\theta _e\theta _{ae}$, for all $a \in S$ and $e \in {{\,\textrm{E}\,}}(S)$. Taking $a=e$, we obtain that the map $\theta _e$ is idempotent. $\square$

Proposition 19

Let s be an idempotent-fixed solution on S. Then, the following hold:

• 1. $\theta _a(b) = bb^{-1}\theta _a(b)$,

• 2. $\theta _a\left( b\right) \theta _a\left( b\right) ^{-1} = bb^{-1}$,

• 3. $\theta _a(b) = \theta _{abb^{-1}}(b)$,

for all $a,b \in S$.

Proof

Let $a,b\in S$. Then, $\theta _a\left( b\right) = \theta _a\left( b\right) \theta _{ab}\left( b^{-1}b\right) = \theta _{a}\left( b\right) bb^{-1}$. Moreover, we have that $\theta _a\left( b\right) ^{-1} = \theta _{ab}\left( b^{-1}\right)$ since

$$\begin{aligned} \theta _a\left( b\right) \theta _{ab}\left( b^{-1}\right) \theta _a\left( b\right) = \theta _a\left( bb^{-1}\right) \theta _a\left( b\right) = bb^{-1}\theta _a\left( b\right) = \theta _a\left( b\right) \end{aligned}$$

and

$$\begin{aligned} \theta _{ab}\left( b^{-1}\right) \theta _a\left( b\right) \theta _{ab}\left( b^{-1}\right)&= \theta _{ab}\left( b^{-1}\right) \theta _a\left( bb^{-1}\right) = b^{-1}b\,\theta _{ab}\left( b^{-1}\right) \\&= \theta _{ab}\left( bb^{-1}\right) \theta _{abb^{-1}b}\left( b^{-1}\right) = \theta _{ab}\left( b^{-1}\right) . \end{aligned}$$

It follows that $\theta _a\left( b\right) \theta _a\left( b\right) ^{-1} = \theta _a\left( b\right) \theta _{ab}\left( b^{-1}\right) = \theta _a\left( bb^{-1}\right) = bb^{-1}.$ Finally, by 1., we have that

$$\begin{aligned} \theta _{abb^{-1}}\left( b\right) = bb^{-1}\theta _{abb^{-1}}\left( b\right) = \theta _a\left( bb^{-1}\right) \theta _{abb^{-1}}\left( b\right) = \theta _a\left( b\right) \end{aligned}$$

that completes the proof. $\square$

As a consequence of Proposition 19-1., if s is an idempotent-fixed solution on the Clifford S, it follows that every group in S remains invariant by $\theta _a$, for all $a\in S$. Thus, motivated by the fact that solutions on groups are well-described, it makes sense to provide a method to construct this type of solutions from solutions on each group in S. To this end, the inner structure of a Clifford semigroup makes clear that conditions relating to different solutions on the groups of S must be considered. For instance, Proposition 19-3. shows that $\theta _a\left( b\right) = \theta _{\varphi _{e,f}\left( a\right) }\left( b\right)$, for all $e,f\in {{\,\textrm{E}\,}}(S)$, with $e\ge f$, and all $a\in G_e$, $b\in G_f$. In light of these observations, we provide the following family of idempotent-fixed solutions.

Theorem 20

Let $s^{[e]}(a,b)=\left( ab,\, \theta ^{[e]}_a\left( b\right) \right)$ be a solution on $G_e$, for every $e \in {{\,\textrm{E}\,}}(S)$. Moreover, for all $e, f\in {{\,\textrm{E}\,}}(S)$, let $\epsilon _{e, f}: G_{e} \rightarrow G_{f}$ be maps satisfying

$$\begin{aligned} \epsilon _{e,f}&= \varphi _{e,f}, \ text{ if }\ e \ge f,\end{aligned}$$

(9)

$$\begin{aligned} \theta ^{[h]}_{\epsilon _{ef, h}(ab)}&= \theta ^{[h]}_{\epsilon _{e, h}(a)\epsilon _{f, h}(b)},\end{aligned}$$

(10)

$$\begin{aligned} \epsilon _{f, h}\theta ^{[f]}_{\epsilon _{e, f}(a)}(b)&= \theta ^{[h]}_{\epsilon _{e, h}(a)}\epsilon _{f, h}(b), \end{aligned}$$

(11)

for all $e, f, h\in {{\,\textrm{E}\,}}(S)$ and $a\in G_{e}$ and $b \in G_{f}$, set

$$\begin{aligned} \theta _a(b):= \theta ^{[f]}_{\epsilon _{e, f}(a)}(b), \end{aligned}$$

for all $a \in G_{e}$ and $b \in G_{f}$, then the map $s: S \times S \rightarrow S \times S$ given by $s(a,b)=(ab, \theta _a(b))$ is an idempotent-fixed solution on S.

Proof

Let $e,f,h\in {{\,\textrm{E}\,}}(S)$, $a\in G_{e}$, $b\in G_{f}$, and $c\in G_{h}$. Then, since $s^{[fh]}$ is a solution on $G_{fh}$, we obtain

$$\begin{aligned} \theta _a\left( bc\right)&= \theta _{a}\left( \varphi _{f, fh}\left( b\right) \varphi _{h, fh}\left( c\right) \right) = \theta ^{[fh]}_{\epsilon _{e,fh}\left( a\right) } \left( \varphi _{f,fh}\left( b\right) \varphi _{h, fh}\left( c\right) \right) \\&= \theta ^{[fh]}_{\epsilon _{e,fh}\left( a\right) }\varphi _{f,fh}\left( b\right) \theta ^{[fh]}_{\epsilon _{e,fh}\left( a\right) \varphi _{f,fh}\left( b\right) }\varphi _{f,fh}\left( c\right) . \end{aligned}$$

Besides, we have that

$$\begin{aligned} \theta _a\left( b\right) \theta _{ab}\left( c\right)&= \theta ^{[f]}_{\epsilon _{e,f}\left( a\right) }\left( b\right) \theta ^{[h]}_{\epsilon _{ef,h}\left( ab\right) }\left( c\right) = \varphi _{f,fh}\theta ^{[f]}_{\epsilon _{e,f}\left( a\right) }\left( b\right) \varphi _{h,fh}\theta ^{[h]}_{\epsilon _{ef,h}\left( ab\right) }\left( c\right) . \end{aligned}$$

Hence, noting that, by (10),

$$\begin{aligned} \theta ^{[fh]}_{\epsilon _{e,fh}\left( a\right) }\varphi _{f,fh}\left( b\right) = \theta ^{[fh]}_{\epsilon _{e,fh}\left( a\right) }\epsilon _{f,fh}\left( b\right) = \epsilon _{f,fh}\theta ^{[f]}_{\epsilon _{e,f}\left( a\right) }\left( b\right) = \varphi _{f,fh}\theta ^{[f]}_{\epsilon _{e,f}\left( a\right) }\left( b\right) \end{aligned}$$

and

$$\begin{aligned} \theta ^{[fh]}_{\epsilon _{e,fh}\left( a\right) \varphi _{f,fh}\left( b\right) }\varphi _{f,fh}\left( c\right)&= \theta ^{[fh]}_{\epsilon _{e,fh}\left( a\right) \epsilon _{f,fh}\left( b\right) }\epsilon _{f,fh}\left( c\right) \\&= \theta ^{[fh]}_{\epsilon _{ef,fh}\left( ab\right) }\epsilon _{f,fh}\left( c\right)&\text {by (10)}\\&= \epsilon _{h,fh}\theta ^{[h]}_{\epsilon _{ef,h}\left( ab\right) }\left( c\right)&\text {by (11)}\\&= \varphi _{h,fh}\theta ^{[h]}_{\epsilon _{ef,h}\left( ab\right) }\left( c\right) , \end{aligned}$$

it follows that (P1) is satisfied. In addition,

$$\begin{aligned} \theta _{\theta _a\left( b\right) }\theta _{ab}\left( c\right)&= \theta _{\theta ^{[f]}_{\epsilon _{e,f}\left( a\right) }\left( b\right) }\theta ^{[h]}_{\epsilon _{ef, h}\left( ab\right) }\left( c\right) \\&= \theta ^{[h]}_{\epsilon _{f,h}\theta ^{[f]}_{\epsilon _{e,f}\left( a\right) }\left( b\right) }\theta ^{[h]}_{\epsilon _{ef, h}\left( ab\right) }\left( c\right) \\&= \theta ^{[h]}_{\theta ^{[h]}_{\epsilon _{e,h}\left( a\right) }\epsilon _{f,h}\left( b\right) } \theta ^{[h]}_{\epsilon _{e,h}\left( a\right) \epsilon _{f,h}\left( b\right) }\left( c\right)&\text {by (11) and (10)}\\&= \theta ^{[h]}_{\epsilon _{f,h}\left( b\right) }\left( c\right)&s^{[h]} \text {is a solution on } G_h\\&= \theta _b\left( c\right) , \end{aligned}$$

thus (P2) holds. Finally, by [6, Lemma 11-1.], $\theta _a(f)=\theta ^{[f]}_{\epsilon _{e,f}(a)}(f)=f$ and so s is idempotent-fixed. $\square$

The following is a class of idempotent-fixed solutions on S that can be constructed through Theorem 20 and includes the solutions ${\mathcal {I}}(a,b)=(ab,b)$ and ${\mathcal {F}}(a,b)=\left( ab, bb^{-1}\right)$ in (5).

Example 3

Let $s^{[e]}\left( a,b\right) = \left( ab, \gamma ^{[e]}\left( b\right) \right)$ be the solution on $G_e$ as in 2. of Examples 1 with $\gamma ^{[e]}$ an idempotent endomorphism of $G_e$, for every $e\in {{\,\textrm{E}\,}}(S)$. Assume that for all $e, f \in {{\,\textrm{E}\,}}(S)$, with $e \ge f$, the group homomorphisms $\varphi _{e,f}:G_e\rightarrow G_f$ satisfy $\varphi _{e,f}\gamma ^{[e]} = \gamma ^{[f]}\varphi _{e,f}$. Take $\epsilon _{e, f} = \varphi _{e,f}$ if $e \ge f$ and $\epsilon _{e,f}\left( x\right) := f$, otherwise. Then, conditions (10) and (11) are satisfied. Hence, the map

$$\begin{aligned} s(a,b)=\left( ab, \gamma ^{[f]}(b)\right) , \end{aligned}$$

for all $a \in G_e$ and $b \in G_f$, is a solution on S.

As a consequence of Theorem 20, the following construction provides a subclass of idempotent-fixed solutions in Clifford semigroups in which each group $G_f$ is an epimorphic image of $G_e$, whenever $f \le e$, for all $e,f\in {{\,\textrm{E}\,}}(S)$.

Corollary 21

Let S be a Clifford semigroup such that $\varphi _{e,f}$ is an epimorphism, for all $e,f \in {{\,\textrm{E}\,}}(S)$ with $f \le e$. Let $s^{[e]}(a,b) = \left( ab, \theta _a^{[e]}(b)\right)$ be a solution on $G_e$ and set $N_e:= \prod \limits _{f\le e} \ker \varphi _{e,f}$, for every $e\in {{\,\textrm{E}\,}}(S)$. Suppose that

• 1. $\theta _a^{[e]} = \theta _b^{[e]}$, for all $e\in {{\,\textrm{E}\,}}(S)$ and all $a,b \in G_e$ with $aN_e = bN_e$,

• 2. $\varphi _{e,f}\theta _a^{[e]}(b) = \theta _{\varphi _{e,f}(a)}^{[f]}\varphi _{e,f}(b)$, for all $e,f\in {{\,\textrm{E}\,}}(S)$ with $f \le e$, and all $a,b\in G_e$.

Set $\theta _a(b):= \theta _{b'}^{[f]}(b)$, with $b'\in G_{f}$ such that $\varphi _{f,ef}(b) = \varphi _{e,ef}(a)$, for all $e,f\in {{\,\textrm{E}\,}}(S)$, and all $a\in G_e$, $b \in G_f$. Then, the map $s:S\times S \rightarrow S \times S$ given by $s(a,b) = (ab,\theta _a(b))$ is an idempotent-fixed solution on S.

Proof

Initially, by 1., note that $\theta _a$ is well-defined, for every $a\in S$. Now, let $e,f\in {{\,\textrm{E}\,}}(S)$ and consider $T_{e,f}$ a system of representatives of $\ker \varphi _{f,ef}$ in $G_f$. Since $\varphi _{f,ef}$ is an epimorphism, for every $a \in G_e$, we can define a map $\epsilon _{e,f}(a):= x \in T_{e,f}$, with $\varphi _{e,ef}(a) = \varphi _{f,ef}(x)$. Specifically, in the case that $f \le e$, it follows that $\epsilon _{e,f} = \varphi _{e,f}$. Therefore, for all $e,f\in {{\,\textrm{E}\,}}(S)$ and all $a\in G_e$, $b\in G_f$, it holds $\theta _a(b) = \theta _{\epsilon _{e,f}(a)}^{[f]}(b)$. Note that, by 1., the last equality is independent of the choice of $T_{e,f}$. Moreover, applying properties in Theorem 1 of homomorphisms $\varphi _{e,f}$, for all $e,f\in {{\,\textrm{E}\,}}(S)$ with $f\le e$, and the assumptions, it is a routine computation to check that conditions (10) and (11) of Theorem 20 are satisfied. $\square$

Let us observe that the kernel of an idempotent-fixed solution s can be rewritten as

$$\begin{aligned} K=\{a \in S \, \mid \, \forall \, e \in {{\,\textrm{E}\,}}(S), \, e \le a, \ \theta _e(a)=aa^{-1}\}. \end{aligned}$$

Denoted by $K_e$ the kernel of each solution $s^{[e]}$ on $G_e$, i.e., the normal subgroup

$$\begin{aligned} K_e=\{a \in G_e \, \mid \, \theta ^{[e]}_{e}(a)=e\}. \end{aligned}$$

of $G_e$, we have the following result that clarifies the previous construction in Theorem 2 20 is not a description.

Proposition 22

Let s be an idempotent-fixed solution on S constructed as in Theorem 20 and suppose that $\epsilon _{e,f}(e)=f$, for all $e,f \in {{\,\textrm{E}\,}}(S)$ with $e \le f$. Assume that each $G_e$ admits a solution $s^{[e]}$ and let $K_{e}$ be the kernel of such a map $s^{[e]}$, for every $e \in {{\,\textrm{E}\,}}(S)$. Then, $K = \displaystyle \bigcup _{e\in {{\,\textrm{E}\,}}(S)}\,K_{e}$.

Proof

Indeed, let $a \in K \cap G_e$. Then, we get $e = aa^{-1} = \theta _e(a) = \theta ^{[e]}_e(a)$. Thus, $a \in K_e$. On the other hand, if $a \in K_e$ and $f \in {{\,\textrm{E}\,}}(S)$ is such that $f \le a$, then, since $\epsilon _{e,f}(e)=f$, we obtain $\theta _f(a)=\theta ^{[e]}_{\epsilon _{f,e}(f)}(a)=\theta ^{[e]}_{e}(a)=e$, i.e., $a \in K$. $\square$

In light of the previous discussion, the following question arises.

Question 1

Complete a description of all the idempotent-fixed solutions.

To conclude, we observe that not every solution on S lies in the class of idempotent invariant or idempotent-fixed solutions. Indeed, even in Clifford semigroups of low order, it is possible to construct such an example.

Example 4

Let $S=\{1,\,a,\,b\}$ be the Clifford monoid in Example 2. Then, the maps

$$\begin{aligned}&\theta _1(x)=a, \quad \text { for every }\ x \in S, \\&\theta _a=\theta _b: S \rightarrow S, \quad \text {given by } \,\,\theta _a(1)=1, \, \,\theta _a(a)=\theta _a(b)=a \end{aligned}$$

give rise to a solution on S that is neither idempotent invariant nor idempotent fixed.

Question 2

Find and study other classes of solutions on Clifford semigroups, including, for instance, the map in Example 4.