On the Cayley Graphs of Completely Simple Semigroups

Abstract

In this paper first we study and characterize the Cayley graph of Rees matrix semigroups (completely simple semigroups). Then we investigate Cayley D-saturated property of these graphs and we study the Cayley D-saturated property of Rees matrix semigroups.

1 Introduction and Preliminaries

Throughout this paper, by a graph we mean a directed graph without multiple arcs, but possibly with loops. Recall that a graph $\Gamma =(V,E)$ is a set $V=V(\Gamma )$ of vertices, together with a binary relation $E=E(\Gamma )$ on V. The elements of E are called the arcs of $\Gamma$.

Let S be a semigroup and let C be a non-empty subset of S. The Cayley graph Cay(S, C) of S relative to C is defined as the graph with vertex set S and arc set E(Cay(S, C)) consisting of those ordered pairs (s, t) such that $cs=t$, for some $c \in C$. The set C is called the connection set of Cay(S, C) (see [4]). Obviously, if C is the empty set, then Cay(S, C) is a null graph, i.e., a graph without arcs. Let ${\mathcal {D}}$ be a finite graph. A semigroup S is said to be Cayley ${\mathcal {D}}$ -saturated with respect to a subset C of S if, for all infinite subsets V of S, there exists a subgraph of Cay(S, C) isomorphic to ${\mathcal {D}}$ with all vertices in V (see [9]).

Cayley graphs of semigroups and groups have been extensively studied, and many interesting results have been obtained (see for example, [1–24]). The Cayley graphs of semigroups are closely related to finite state automata and have many valuable applications (see the survey [10] and Section 2.4 of the book [5]). For more general information on applications of algebraic systems, we refer to the monograph [23].

The Cayley ${\mathcal {D}}$-saturated property of semigroups was first introduced by Kelarev and Quinn in [9] and after proving some fundamental lemmas about this combinatorial property, they studied the Cayley ${\mathcal {D}}$-saturated property of semigroup S with respect to $C=S$. Later, Gao and Yang in [24] continued their works and generalized some of their results. In both of these papers, the graphs are assumed to have no loops. In [16], the authors studied the Cayley graphs with possible loops and they determined the bands S such that S is Cayley ${\mathcal {D}}$-saturated with respect to a subset $C\subseteq Z(S)$. We recall that every completely simple semigroup (0-completely simple semigroup) is isomorphic to a Rees matrix semigroup (0-Rees matrix semigroup) (see [19]).

In this paper, first we study the Cayley graphs of Rees matrix semigroups and we present a characterization of these graphs. Then we use these results to improve our theorems in [18] and we present a characterization of Cayley D-saturated property of Rees matrix semigroups more concretely.

Lemma 1.1

([9, 24]) Let ${\mathcal {D}}$ be a finite graph without loops which is not null, S an infinite semigroup and C a non-empty subset of S. Then the following conditions are equivalent:

• (i) S is Cayley ${\mathcal {D}}$-saturated with respect to C;

• (ii) every infinite set of vertices induces a subgraph of Cay(S, C ) which is not null;

• (iii) every subgraph of Cay(S, C) induced by an infinite subset of S contains $A_\infty$, $D_\infty$, or $K_\infty$.

Lemma 1.2

([18]) Let ${\mathcal {D}}$ be a finite graph with some loops such that ${\mathcal {D}}$ has no cycles of length greater than one. Suppose that S is a semigroup and $C\subseteq S$. Then S is Cayley ${\mathcal {D}}$-saturated with respect to C if and only if for every infinite subset $V\subseteq S$ and each $n\in {\mathbb {N}}$, the subgraph induced by V contains $M_n^L$. Also if ${\mathcal {D}}$ is a finite acyclic graph, then we can replace $M_n^L$ by $M_n$ in the above statement.

Proposition 1.3

([18])

• (i) Let L be a left zero semigroup and R be a right zero semigroup. Let $S=L\times R$ be a rectangular band and $C\subseteq S$. Then Cay(S, C) is a disjoint union of |R| copies of $\Gamma _1=(|L\setminus \pi _1(C)|)K_1+\overrightarrow{K_{|\pi _1(C)|}}$.

• (ii) Let A and B be two sets. If $\Gamma$ is the disjoint union of graphs $\{\Gamma _i\}_{i\in I}$ such that for every $i\in I$, $\Gamma _i=(|B| K_1)+\overrightarrow{K_{|A|}}$, then $\Gamma$ is a Cayley graph of a rectangular band.

2 Main Results

Notation 2.1

For every set X, by $L_X$ we mean the left zero semigroup over the set X.

Notation 2.2

For every graph $\Gamma$ and subsets $A, B\subseteq V(\Gamma )$ by $\Gamma (A, B)$, we mean the graph with vertex set $V(\Gamma )=A\cup B$ and edge set $E(A, B)=\{(a, b)\in E(\Gamma ) | \ a\in A, b\in B\}$.

Theorem 2.3

Let G be a group and $S=M(A, G, B; P)$. A graph $\Gamma$ is isomorphic to the Cayley graph Cay(S, C), where $C\subseteq S$ such that $\pi _1(C)=\{a_0\}$ and $\pi _3(C)=\{b_0\}$ if and only if the graph $\Gamma$ is isomorphic to a disjoint union of |B| isomorphic graphs $\Gamma ^{(b)}$, where $b\in B$, which satisfies

• (i) there exists $C^{\prime }\subseteq G$ such that $\Lambda \cong Cay(G, C^{\prime })$;

• (ii) for every $a\in A\setminus \{a_0\}$, $\Gamma _a$ is isomorphic to Nu(|G|) and there exists $g_a\in G$ such that $\Gamma (V(\Lambda _{a}), V(\Lambda ))\cong L_2\times Cay(G, C^{\prime }g_a)$;

• (iii) there is no arc between $\Lambda _a$ and $\Lambda _{a^{\prime }}$ where $a, a^{\prime }\in A\setminus \{a_0\}$;

• (iv) $V(\Gamma ^{(b)})$ is equal to $\bigcup _{a\in A\setminus \{a_0\}}V(\Lambda _{a})\cup \bigcup V(\Lambda )$ which are mutually disjoint sets.

Proof

$(\Leftarrow )$ Let $b\in B$. Since $\Lambda$ is isomorphic to $Cay(G, C^{\prime })$, we can use this isomorphism to label the vertices of $\Lambda$ by elements of G. In fact for the isomorphism $\mu : \Lambda \rightarrow Cay(G, C^{\prime })$, we label the vertex v by $(a_0, g, b)$, if $\mu (v)=g$. Now let $a\in A\setminus \{a_0\}$. We know that $\Gamma (V(\Lambda _a), V(\Lambda ))\cong L_2\times Cay(G, C^{\prime }g_a)$, for every $a\in A\setminus \{a_0\}$. Suppose that $\theta _a: \Gamma (V(\Lambda _a), V(\Lambda ))\rightarrow L_2\times Cay(G, C^{\prime }g_a)$ is the graph isomorphism. We label the vertex v in $V(\Lambda _a)$ by $(a, g^{-1}_ac^{-1}g, b)$, if the vertex v is adjacent to a vertex w in $V(\Lambda )$ which is labeled by $(a_0, g, b)$ and $\pi _2(\theta _{a}(w))=cg_a\pi _2(\theta _{a}(v))$ where $c\in C^{\prime }$ and $\pi _2: V(L_2)\times V(Cay(G, C^{\prime }g_a))\rightarrow V(Cay(G, C^{\prime }g_a))$ is the natural projection on the second component. By this method, we can label all vertices of $\Gamma$ (note that the out-degree of every vertex $v\in V(\Lambda _{a})$ in $\Gamma (V(\Lambda _a), V(\Lambda ))$ is non-zero). Now let

$$\begin{aligned} P(b, a)= \left\{ \begin{array}{lll} g_a &{}\quad {\text { if }}b=b_c, &{}a\ne a_0\\ e_G &{}\quad {\text { if }}b=b_c, &{} a=a_0 \\ e_G &{}\quad {\text { otherwise.}}&{}\\ \end{array} \right. \end{aligned}$$

Consider $S=M(A, G, B; P)$ and $C=\{a_0\}\times C^{\prime }\times \{b_c\}$. We claim that Cay(S, C) is isomorphic to $\Gamma$ under the function $\varphi : \Gamma \rightarrow Cay(S, C)$ defined as follows: $\varphi (v)=(a, g, b)$, where (a, g, b) is the label of v. Clearly $\varphi$ is one to one and onto. To prove that $\varphi$ is a graph homomorphism, consider $(v, v^{\prime })\in E(\Gamma )$. Suppose that the labels of v and $v^{\prime }$ are (a, g, b) and $(a^{\prime }, g^{\prime }, b^{\prime })$, respectively. Since the connected components of $\Gamma$ are $\{\Gamma ^{(b)}\}_{b\in B}$, we conclude that $b=b^{\prime }$. So we have two cases.

Case (i):

Suppose that $a=a^{\prime }$. Since for every $a\in A\setminus \{a_0\}$, the graph $\Lambda _a$ is isomorphic to Nu(|G|) and also because there exists no arc from $V(\Lambda _{a^{\prime }})$ to $V(\Lambda _{a^{\prime \prime }})$, where ${a^{\prime },a^{\prime \prime }}\in A \setminus \{a_0\}$, we have $a=a^{\prime }=a_0$. By our algorithm for labeling the vertices of $\Gamma$, since there exists an arc from v to $v^{\prime }$ in $\Gamma$, there exists an arc from g to $g^{\prime }$ in $Cay(G, C^{\prime })$, and therefore there exists $c\in C^{\prime }$ such that $g^{\prime }=cg$. Hence we have $(a_0, c, b_0)(a_0, g, b)=(a_0, cP(b_0, a_0)g, b)=(a_0, cg, b)=(a_0, g^{\prime }, b)$ and so $\varphi$ preserves the adjacency.

Case (ii):

Suppose that $a\ne a^{\prime }$. Again since for every $a^{\prime }, a^{{\prime }{\prime }}\in A\setminus \{a_0\}$, there exists no arc from $V(\Lambda _{a^{\prime }})$ to $V(\Lambda _{a^{{\prime }{\prime }}})$, we get that $a\in A\setminus \{a_0\}$ and $a^{\prime }\ne a_0$. By our algorithm for labeling the vertices of $\Gamma$, there exists $c\in C^{\prime }$ such that $g^{\prime }=cg_ag$. So we have

$$\begin{aligned} (a_0, c, b_0)(a, g, b)=(a_0, cP(b_0, a)g, b)=(a_0, cg_ag, b)=(a_0, g^{\prime }, b). \end{aligned}$$

Hence $\varphi$ preserves the adjacency in this case, too.

Now we prove that $\varphi$ reflects the adjacency. Suppose that $((a, g, b), (a^{\prime }, g^{\prime }, b^{\prime }))\in E(\Gamma (Cay(S, C)))$. So there exists $(a_0, g_c, b_0)\in C$ such that $(a_0, g_c, b_0)(a, g, b)=(a^{\prime }, g^{\prime }, b^{\prime })$, and hence $(a^{\prime }, g^{\prime }, b^{\prime })=(a_0, g_cP(b_0, a)g, b)$.

• (1) Suppose that $a=a_0$. In this case, we have $(a^{\prime }, g^{\prime }, b^{\prime })=(a_0, g^{\prime }, b)=(a_0, g_cP(b_0, a)g, b)=(a_0, g_cg, b)$. Hence $g^{\prime }=g_cg$, where $g_c\in C^{\prime }$. Since $\Lambda$ is isomorphic to $Cay(G, C^{\prime })$, clearly $(\varphi ^{-1}(a, g, b), \varphi ^{-1}(a^{\prime }, g^{\prime }, b^{\prime }))\in E(\Gamma )$. So in this case, $\varphi$ reflects the adjacency.

• (2) Suppose that $a\ne a_0$. Let $\phi ^{-1}(a, g, b)=v$ and $\phi ^{-1}(a^{\prime }, g^{\prime }, b^{\prime })=u$. In this case, we have $(a^{\prime }, g^{\prime }, b^{\prime })=(a_0, g_c, b_0)(a, g, b)=(a_0, g_cP(b_0, a)g, b)=(a_0, g_cg_ag, b)$. Therefore $b=b^{\prime }$, $a^{\prime }=a_0$, and $g^{\prime }=g_cg_ag$. Again by our algorithm for labeling the vertices, since $\varphi ^{-1}(a, g, b)$ has the label $(a, g, b)=(a, g_a^{-1}g_c^{-1}g^{\prime }, b)$, so there exists an arc from v to u in $\Gamma$ (note that we use (ii) to label v and we use (i) to label u). Therefore $\varphi$ reflects the adjacency in this case, too.

$\square$

Theorem 2.4

Let $S=M(A, G, B; P)$ be a Rees matrix semigroup over a group G with sandwich matrix P and $C\subseteq S$. For each $a_c\in \pi _1(C)$ and each $b^{\prime }\in \pi _3(C)$ let $D_{a_c, b^{\prime }}= \{g\in G | (a_c, g, b^{\prime })\in C\}$ and $F_{a_c}=\{b^{\prime }\in B | \exists g\in G : (a_c, g, b^{\prime })\in C\}$. Also for each $a_c\in \pi _1(C)$ and $a\in A\setminus \{a_c\}$ let $\Gamma (\{a\}\times G\times \{b\}, \{a_c\}\times G\times \{b\})\cong L_2\times Cay(G, \bigcup _{b^{\prime }\in F_{a_c}}D_{a_c, b^{\prime }}P(b^{\prime }, a)).$ If $\Gamma$ is isomorphic to Cay(S, C), then $\Gamma$ is isomorphic to

$$\begin{aligned} \dot{\bigcup }_{b\in B}\left( \bigcup _{a_c\in \pi _1(C)}\left( \Gamma (\{a_c\}\times G\times \{b\})\cup \left( \bigcup _{a\in A\setminus \{a_c\}}\Gamma (\{a\}\times G\times \{b\}, \{a_c\}\times G\times \{b\})\right) \right) \right) \end{aligned}$$

Proof

Note that if $(a_c, g_c, b_c)\in C$ and $(a, g, b)\in S$, then $(a_c, g_c, b_c)(a, g, b)=(a_c, g_cP(b_c, a)g, b)$. Therefore if $((a, g, b), (a^{\prime }, g^{\prime }, b^{\prime }))\in E(Cay(S, C))$, then $b=b^{\prime }$, $a^{\prime }\in \pi _1(C)$, and $g_cP(b_c, a)g=g^{\prime }$. By these equalities, we have the following results:

• (a) There exists no arc between vertices (a, g, b) and $(a^{\prime }, g^{\prime }, b^{\prime })$, where $b\ne b^{\prime }$;

• (b) There exists no arc between vertices (a, g, b) and $(a^{\prime }, g^{\prime }, b^{\prime })$, where $a, a^{\prime }\notin \pi _1(C)$;

• (c) For every arc $((a, g, b), (a^{\prime }, g^{\prime }, b))\in E(Cay(S, C))$, we have $a^{\prime }\in \pi _1(C)$ and either $((a, g, b), (a^{\prime }, g^{\prime }, b))\in E(\Gamma (\{a^{\prime }\}\times G\times \{b\}))$ or $((a, g, b), (a^{\prime }, g^{\prime }, b))\in E(\Gamma (\{a\}\times G\times \{b\}, \{a^{\prime }\}\times G\times \{b\}))$.

Note that

$$\begin{aligned} ((a, g, b), (a, g^{\prime }, b))\in & {} E(\Gamma (\{a\}\times G\times \{b\})) \\ {}\Leftrightarrow & {} \exists (a, g_c, b_c)\in C, g_cP(b_c, a)g=g^{\prime }\\\Leftrightarrow & {} (g, g^{\prime })\in E(Cay(G, \cup _{b^{\prime }\in F_a} D_{a, b^{\prime }}P(b^{\prime }, a))), \end{aligned}$$

Also note that if $a\ne a^{\prime }$ and $a^{\prime }\in \pi _1(C)$, then

$$\begin{aligned} ((a, g, b), (a^{\prime }, g^{\prime }, b))\in & {} E(\Gamma (\{a\}\times G\times \{b\}, \{a^{\prime }\}\times G\times \{b\}))\\\Leftrightarrow & {} \exists (a^{\prime }, g_c, b_c)\in C : g_cP(b_c, a)g=g^{\prime }. \end{aligned}$$

Similarly to the proof of the above theorem, we can show that for $a^{\prime }\in \pi _1(C)$ and $a\in A\setminus \{a^{\prime }\}$, we have $\Gamma (\{a\}\times G\times \{b\}, \{a^{\prime }\}\times G\times \{b\})$ is isomorphic to $L_2\times Cay(G, \cup _{b^{\prime }\in F_a}D_{a^{\prime }, b^{\prime }}P(b^{\prime }, a))$. $\square$

Theorem 2.5

A graph $\Gamma$ is isomorphic to the Cayley graph Cay(S, C), where S is a Rees matrix semigroup over a group and $C\subseteq S$ if and only if there exist a group G, two non-empty sets $A, \ B$, and a non-empty subset $X\subseteq A$ such that there exist a partition of $V(\Gamma )$, namely $\{Z_{ab} | a\in A, b\in B\}$ and a family of functions $\{\varphi _{ab}:Z_{ab}\rightarrow G | a\in A, b\in B\}$ which satisfy the following conditions:

• (i) for every $x\in X$ and $b\in B$, there exists a subset $D_{xb}\subseteq G$ and for every $a\in A$ and $b\in B$, there exists $g_{ab}\in G$ such that if $T_x=\{b^{\prime }\in B | D_{xb^{\prime }}\ne \emptyset \}$, then

  • (a) the subgraph induced in $\Gamma$ by $Z_{xb}$ as a subgraph of $\Gamma$ is isomorphic to $Cay(G, \bigcup _{b^{\prime }\in T_x}D_{xb^{\prime }}g_{xb^{\prime }})$ under $\varphi _{xb}$;

  • (b) the subgraph $\Gamma (Z_{ab}, Z_{xb})$ where $a\in A{\setminus } \{x\}$ is isomorphic to $L_2\times Cay(G, \bigcup _{b^{\prime }\in T_x}D_{xb^{\prime }}g_{xb^{\prime }})$ under the function $\eta ^{(b)}_{a,x}$ defined by

    $$\begin{aligned} \eta ^{(b)}_{a, x}(v)= \left\{ \begin{array}{ll} (a, \varphi _{ab}(v)), &{}\quad {\text { if }}v\in Z_{ab}\\ (x, \varphi _{xb}(v)) &{}\quad {\text { if }}v\in Z_{xb};\\ \end{array} \right. \end{aligned}$$

• (iii) $\Gamma =\dot{\bigcup }_{b\in B}(\bigcup _{x\in X}(\Gamma (Z_{xb})\cup (\bigcup _{a\in A\setminus \{x\}}\Gamma (Z_{ab}, Z_{xb}))))$.

Proof

$(\Leftarrow )$ Let $S=M(A, G, B; P)$, where for every $a\in A$ and $b\in B$, $P(b, a)=g_{ab}$. We show that $\Gamma$ is isomorphic to Cay(S, C), where $C=\bigcup _{x\in X}(\bigcup _{b\in B}\{x\}\times D_{xb}\times \{b\})$. Let $\varphi : V(\Gamma )\rightarrow V(Cay(S, C))=S$ defined by $\varphi (v)=(a, g, b)$ where $v\in Z_{ab}$ and $\varphi _{ab}(v)=g$. Clearly $\varphi$ is well-defined, one-one and onto. Hence to prove the assertion, we just need to show that $\varphi$ preserves adjacency and non-adjacency. Suppose that $(u, v)\in E(\Gamma )$. By (iii), there exists $x\in X$ such that

$$\begin{aligned} (u, v)\in E\left( \bigcup _{x\in X}\bigg (\Gamma (Z_{xb})\cup \bigg (\bigcup _{a\in A\setminus \{x\}}\Gamma (Z_{ab}, Z_{xb})\bigg )\bigg )\right) . \end{aligned}$$

Therefore $v\in Z_{xb}$ and we have two cases for u.

Case (i) $u\in Z_{xb}$. In this case, clearly by (i) (a), we have

$$\begin{aligned} (\varphi _{xb}(u), \varphi _{xb}(v))\in E\left( Cay\left( G, \bigcup _{b^{\prime }\in T_x}D_{xb^{\prime }}g_{xb^{\prime }}\right) \right) . \end{aligned}$$

Therefore by the definition of Cayley graph Cay(S, C) we have $(\varphi (u), \varphi (v))=((x, \varphi _{xb}(u), b), (x, \varphi _{xb}(v), b))\in E(Cay(S, C))$. Hence $\varphi$ preserves the adjacency in this case.

Case (ii) $u\in Z_{ab}$, for some $a\in A{\setminus } \{x\}$. In this case, by (i) (b), we have

$$\begin{aligned} (\varphi _{ab}(u), \varphi _{xb}(v))=\bigg (\pi _2(\eta ^{(b)}_{a,x}(u)\bigg ), \pi _2(\eta ^{(b)}_{a,x}(v))\in E\bigg (Cay\bigg (G, \bigcup _{b^{\prime }\in T_x}D_{xb^{\prime }}g_{xb^{\prime }}\bigg )\bigg ). \end{aligned}$$

Therefore, in each case, we have

$$\begin{aligned} (\varphi (u), \varphi (v))=((a, \varphi _{ab}(u), b), (x, \varphi _{xb}(v), b))\in E(Cay(S, C)), \end{aligned}$$

and $\varphi$ preserves the adjacency in this case, too.

Finally, we show that $\varphi$ reflects the adjacency. Suppose that $((a, g, b), (a^{\prime }, g^{\prime }, b^{\prime }))\in E(Cay(S, C))$. By the definition of Cay(S, C), we have $b=b^{\prime }$ and $a^{\prime }\in X$. Again we have two cases.

Case (i) If $a=a^{\prime }$, then $(g, g^{\prime })\in E(Cay(G, \bigcup _{b^{{\prime }{\prime }}\in T_a}D_{ab^{{\prime }{\prime }}}g_{ab^{{\prime }{\prime }}}))$. Since by (i) (a), $\varphi _{ab}$ is an isomorphism, we have $(\varphi _{ab}^{-1}(g), \varphi _{ab}^{-1}(g^{\prime }))\in E(\Gamma (Z_{ab}))\subseteq E(\Gamma )$. Therefore, in this case, $(\varphi ^{-1}(a, g, b), \varphi ^{-1}(a, g^{\prime }, b))=(\varphi ^{-1}_{ab}(g), \varphi ^{-1}_{ab}(g^{\prime }))\in E(\Gamma )$ and so $\varphi$ reflects the adjacency.

Case (ii) If $a\ne a^{\prime }$, then $(g, g^{\prime })\in E(Cay(G, \bigcup _{b^{\prime }\in T_{a^{\prime }}}D_{a^{\prime }b^{\prime }}g_{a^{\prime }b^{\prime }}))$. Since by (i)(b) we know that the function $\eta ^{(b)}_{a,a^{\prime }}$ is an isomorphism from $\Gamma (Z_{ab}, Z_{a^{\prime }b})$ to $L_2\times Cay(G, \bigcup _{b^{\prime }\in T_{a^{\prime }}}D_{a^{\prime }b^{\prime }}g_{a^{\prime }b^{\prime }})$, we have

$$\begin{aligned} \bigg (\bigg (\eta ^{(b)}_{a, a^{\prime }}\bigg )^{-1}(a, g), \bigg (\eta ^{(b)}_{a, a^{\prime }}\bigg )^{-1}(a^{\prime }, g^{\prime })\bigg ). \end{aligned}$$

So by the definition of $\varphi$ we have

$$\begin{aligned} (\varphi ^{-1}((a,g,b), \varphi ^{-1}(a^{\prime }, g^{\prime }, b))= & {} \\ \left( \varphi ^{-1}_{ab}(g), \varphi ^{-1}_{a^{\prime }b}(g^{\prime })\right)= & {} \left( \left( \eta ^{(b)}_{a,a^{\prime }}\right) ^{-1}(a, g), \left( \eta ^{(b)}_{a,a^{\prime }}\right) ^{-1}(a^{\prime }, g^{\prime })\right) \\\in & {} E(\Gamma (Z_{ab}, Z_{a^{\prime }b}))\subseteq E(\Gamma ). \end{aligned}$$

Hence $\varphi$ reflects the adjacency. Therefore $\Gamma$ is isomorphic to Cay(S, C).

$(\Rightarrow )$ Let $X=\pi _1(C)$, $\varphi$ be the isomorphism from $\Gamma$ to Cay(S, C) and $Z_{ab}=\varphi ^{-1}(\{a\}\times G\times \{b\})$ and $\varphi _{ab}=\pi _2(\varphi |_{Z_{ab}})$. The rest of the proof is clear by Theorem 2.4. $\square$

Theorem 2.6

Let $S=M(A, G, B; P)$ be a Rees matrix semigroup over a group and $C\subseteq S$. The semigroup S is Cayley D-saturated with respect to C if and only if B is finite and we have one of the following conditions.

• (i) G is finite and $A{\setminus } \pi _1(C)$ is a finite set,

  • (a) if D has a cycle which is not a loop, then for every infinite subset $V_1\subseteq A$, there exist an infinite subset $U_1\subseteq V_1$, $g^*\in G$ and $b^*\in B$ such that $U_1\times \{g^*\}\times \{b^*\}\subseteq C$.

  • (b) if D is acyclic, then for every infinite subset $V_1\subseteq A$ and every $n\in \mathbb {N}$ there exist $\{b_{ij} | 1\le i<j\le n\}\subseteq B$ and $v_1, \dots , v_n\in V_1$ such that for every $1\le i<j \le n$, $(v_j, P(b_{ij}, v_i)^{-1}, b_{ij})\in C$;

• (ii) G is infinite and $\pi _1(C)=A$ such that

  • (a) for every $a\in A$, the group G is Cayley D-saturated with respect to $\bigcup _{b^{\prime }\in T_a}M_{a,b^{\prime }}P(b^{\prime }, a)$, where $T_a=\{b^{\prime }\in B | M_{a,b^{\prime }}\ne \emptyset \}$ and $M_{a,b^{\prime }}=\{g\in G | (a, g, b^{\prime })\in C\}$;

  • (b) for every infinite subset V of S such that $\pi _1(V)$ is infinite and $|V\cap (\{a\}\times G\times \{b\})|\le 1$, for every $a\in A$ and $b\in B$, there exists $V^{\prime }_2\subseteq \pi _2(V)$ such that $\pi _2^{-1}(V^{\prime }_2)\cap V=\{(a_i, g_i, b_i) | i\in \mathbb {N}\}$ is infinite and if D has a cycle, then (1) and (2) holds and if D is acyclic, then (1) or (2) holds, where

    • (1) for every $i\in \mathbb {N}$ and every $j\ge i$, we have $(a_j, g_jg_i^{-1}P(b^{\prime }, a_i)^{-1}, b^{\prime })\in C$, for some $b^{\prime }\in T_{a_j}$;

    • (2) for every $i\in \mathbb {N}$ and every $j\ge i$, we have $(a_i, g_ig_j^{-1}P(b^{\prime }, a_j)^{-1}, b^{\prime })\in C$, for some $b^{\prime }\in T_{a_i}$.

Proof

First note that by Theorem 2.4, $\Gamma =Cay(S, C)$ is a disjoint union of

$$\begin{aligned} \bigcup _{a_c\in \pi _1(C)}\left( \Gamma (\{a_c\}\times G\times \{b\})\cup \left( \bigcup _{a\in A\setminus \{a_c\}}\Gamma \bigg (\{a\}\times G\times \{b\}, \{a_c\}\times G\times \{b\}\bigg )\right) \right) , \end{aligned}$$

for $b\in B$, so clearly since S is Cayley D-saturated with respect to C, the set B must be finite. We consider two cases for $A{\setminus } \pi _1(C)$.

Case (i) Suppose that $A\ne \pi _1(C)$. Let $a_0\in A{\setminus } \pi _1(C)$. Again by Theorem 2.4, since the subgraph induced in Cay(S, C) by $\{a_0\}\times G\times \{b\}$ for $b\in B$ is a null graph, we get that G is finite. First suppose that D has a cycle. Let $V\subseteq S$ be an infinite subset. Since G and B are finite, there exist $g_0\in G$ and $b_0\in B$ such that $V^{\prime }=(A\times \{g_0\}\times \{b_0\})\cap V$ is infinite. Since S is Cayley D-saturated with respect to C, there exists $U_1=\{u_i | i\in \mathbb {N}\}\subseteq \pi _1(V^{\prime })$ such that the subgraph induced in Cay(S, C) by $U_1\times \{g_0\}\times \{b_0\}$ is isomorphic to $K_{\infty }$. So for every $i, j\in \mathbb {N}$ and $i\ne j$, there exists $(a_c, g_c, b_c)$ such that $(a_c, g_c, b_c)(a_i, g_0, b_0)=(a_j, g_0, b_0)$. Therefore $a_c=a_j$ and $g_cP(b_c, a_i)g_0=g_0$. Hence $(a_j, P(b_c, a_i)^{-1}, b_c)\in C$ for some $b_c\in T_{a_j}=\{b^{\prime }\in B | \exists g\in G : (a_j, g, b^{\prime })\in C\}$. Again since B and G are finite, there exist $b^*\in B$, $g^*\in G$ and an infinite subset $U^{\prime }_1\subseteq U_1$ such that for every $a\in U^{\prime }_1$, we have $(a, g^*, b^*)\in C$. Hence $U^{\prime }_1\times \{g^*\}\times \{b^*\}\subseteq C$.

If D is acyclic, then similarly to the above and using Lemma 1.2, we get the result.

Case (ii) Suppose that $A=\pi _1(C)$. If G is finite, then similarly to the above we get that (i) holds. So suppose that G is infinite. Let $V_2$ be an infinite subset of G. Let $a\in A$ and $b\in B$. Since S is Cayley D-saturated with respect to C, the subgraph induced in Cay(S, C) by $\{a\}\times V_2\times \{b\}$ contains $A_{\infty }, D_{\infty }$, or $K_{\infty }$, by Lemma 1.1. Now by the definition of Cay(S, C), we can easily conclude that the subgraph induced in $Cay(G, \cup _{b^{\prime }\in T_{a}}M_{a,b^{\prime }}g_{ab^{\prime }})$ by $V_2$ contains $A_{\infty }, D_{\infty }$, or $K_{\infty }$. So by Lemma 1.1, G is Cayley D-saturated with respect to $\cup _{b^{\prime }\in T_{a}}M_{ab^{\prime }}g_{ab^{\prime }}$ and (ii) (a) is proved.

Suppose that V is an infinite subset of S. Since S is Cayley D-saturated with respect to C by Lemma 1.1, the subgraph induced in Cay(S, C) by V contains $A_{\infty }$, $D_{\infty }$, or $K_{\infty }$. Let $V^{\prime }=\{(a_i, g_i, b_i) | i\in \mathbb {N}\}$ be the vertices of the subgraph which is isomorphic to $A_{\infty }$, $D_{\infty }$, or $K_{\infty }$. Now note that for every $i, j\in \mathbb {N}$, we have an arc form $(a_i, g_i, b_i)$ to $(a_j, g_j, b_j)$ if and only if there exists $(a_c, g_c, b_c)\in C$ such that

$$\begin{aligned} (a_j, g_j, b_j)=(a_c, g_c, b_c)(a_i, g_i, b_i)=(a_c, g_cP(b_c, a_i)g_i, b_i). \end{aligned}$$

(1)

So we have $a_c=a_j$, $g_cP(b_c, a_i)g_i=g_j$ and $b_i=b_j$. Hence $g_c=g_jg_i^{-1}P(b_c, a_i)^{-1}$ and thus $(a_j, g_jg_i^{-1}P(b_c, a_i)^{-1}, b_c)\in C$, where $b_c\in T_{a_j}$. Therefore by the definitions of $A_{\infty }, D_{\infty }$, or $K_{\infty }$ easily we get that (1) and (2) hold if D has a cycle and (1) or (2) holds, otherwise.

$(\Leftarrow )$ Suppose that V is an infinite subset of S. If $\pi _1(V)$ is finite, then there exists $b_0\in B$ and $a\in \pi _1(V)$ such that $V^{\prime }=(\{a\}\times G\times \{b_0\})\cap V$ is infinite (note that B is finite). Let $V^{\prime }_2=\pi _2(V^{\prime })$. By (ii)(a), the subgraph induced in $Cay(G, \cup _{b^{\prime }\in T_{a}}M_{ab^{\prime }}P(b^{\prime }, a))$ by $V^{\prime }_2$ contains $A_{\infty }, D_{\infty }$, or $K_{\infty }$ by Lemma 1.1. Let $U^{\prime }_2$ be the set of vertices of this subgraph which is isomorphic to $A_{\infty }, D_{\infty }$, or $K_{\infty }$. Now by the definition of Cay(S, C), it is straightforward to see that the subgraph induced in Cay(S, C) by $\{a\}\times U^{\prime }_2\times \{b_0\}$ is isomorphic to the subgraph induced in $Cay(G, \cup _{b^{\prime }\in T_a}M_{a,b^{\prime }}P(b^{\prime }, a))$ by $U^{\prime }_2$ (note that if $g^{\prime }=g_cP(b^{\prime }, a)g$ for some $g_c\in M_{a,b^{\prime }}$, then $(a, g^{\prime }, b_0)=(a, g_c, b^{\prime })(a, g, b_0)$). Thus the subgraph induced in Cay(S, C) by V contains $A_{\infty }, D_{\infty }$, or $K_{\infty }$ in this case. If $\pi _1(V)$ is infinite, then we have two cases. If there exists $a_0\in \pi _1(V)$ such that $(\{a_0\}\times G\times B)\cap V$ is infinite, then there exists $b_0\in B$ such that $(\{a_0\}\times G\times \{b_0\})\cap V$ is infinite (note that B is finite) and similarly to the above, we can show that the subgraph induced in Cay(S, C) by V contains $A_{\infty }, D_{\infty }$, or $K_{\infty }$. Hence suppose that for every $a\in A$, the set $V\cap (\{a\}\times G\times \{b\})$ is finite. Without loss of generality, we can suppose that for every $a\in A$ and $b\in B$, the set $V\cap (\{a\}\times G\times \{b\})$ has at most one element. Now note that by (b), there exists $V^{\prime }_2\subseteq \pi _2(V)$ such that $V^{\prime }=\pi _2^{-1}(V^{\prime }_2)\cap V=\{(a_i, g_i, b_i) | i\in \mathbb {N}\}$ is infinite and fulfills at least in one of the conditions (1) or (2). First suppose that $V^{\prime }$ satisfies  (1). Since B is finite, without loss of generality we can suppose that there exists $b^*\in B$ such that $V^{\prime }\subseteq A\times G\times \{b^*\}$ and so $V^{\prime }=\{(a_i, g_i, b^*) | i\in \mathbb {N}\}$. Now note that by (1), for every $i\in \mathbb {N}$ and $j\ge i$, the subgraph induced in Cay(S, C) by $V^{\prime }$ contains $A_{\infty }$. Similarly we can show that if $V^{\prime }$ satisfies (2), then the subgraph induced by $V^{\prime }$ contains $D_{\infty }$. Also if it satisfies both of conditions in (b), then the subgraph induced in Cay(S, C) by $V^{\prime }$ contains $K_{\infty }$. The rest of the proof is clear by the definitions of $A_{\infty }$, $D_{\infty }$, and $K_{\infty }$ (just note that $K_{\infty }$ contains both $A_{\infty }$ and $D_{\infty }$) and S is Cayley D-saturated with respect to C by Lemma 1.1. $\square$