On the Independence Number of Cayley Digraphs of Clifford Semigroups

Abstract

Let S be a Clifford semigroup and A a subset of S. We write $Cay(S,A)$ for the Cayley digraph of a Clifford semigroup S relative to A. The (weak, path, weak path) independence number of a graph is the maximum cardinality of an (weakly, path, weakly path) independent set of vertices in the graph. In this paper, we characterize maximal connected subdigraphs of $Cay(S,A)$ and apply these results to determine the (weak, path, weak path) independence number of $Cay(S,A)$.

1. Introduction

In algebraic graph theory, Cayley graphs are an important concept relating semigroup theory and graph theory. One of the appealing subjects in the study of Cayley graphs of semigroups is considering how to apply the results obtained from the Cayley graphs of groups to the case of semigroups.

Let S be a semigroup and A a subset of S. The Cayley digraph $Cay(S,A)$ of a semigroup S relative to A (which is simply called Cayley graph) is defined as the digraph with the vertex set S, and the arc set $E(Cay(S,A))$ consisting of those ordered pairs $(x,y)$ such that $y=xa$ for some $a\in A$, i.e., $E(Cay(S,A))=\{(x,xa)|x\in S,a\in A\}$.

The motivation for considering Clifford semigroups lies in their unique and intriguing algebraic properties; Clifford semigroups represent one of the important types of semigroups, which are a union of groups. These semigroups serve as a natural bridge between the worlds of semigroups and groups, providing an avenue to explore the interplay between these two fundamental algebraic structures. Consequently, it can be inferred that the Cayley digraphs of Clifford semigroups contain the Cayley digraphs of groups.

Investigating the Cayley digraph can yield valuable insights into network optimization and communication protocols. In [1] Heydemann has undertaken a comprehensive examination of diverse classes of Cayley graphs of groups, which have been subject to extensive scrutiny as models for interconnection networks. It subsequently presents a detailed analysis of outcomes and issues pertaining to network routings, with a particular focus on evaluating the loads of nodes and links during the routing processes. Xiao and Parhami [2] explored the Cayley digraphs of groups and their coset graphs concerning subgroups, deriving general results on homomorphisms and broadcasting. Additionally, practical applications were discussed in well-known interconnection networks such as the butterfly network, de Bruijn network, cube-connected cycles network, and shuffle-exchange network. Consequently, these results can be effectively applied to the Cayley digraph of Clifford semigroups by specifying a certain collection of groups.

Numerous papers have undertaken the study of characterizations concerning the Cayley graphs of different types of semigroups (see [3,4,5,6] and their references). Notably, specific conditions delineating the characteristics of Cayley graphs of Clifford semigroups have been provided in reference [7]. Recently, Ilić-Georgijević [8] focused on presenting conditions that precisely characterize the Cayley graphs of a particular group known as homogeneous semigroups. It is important to highlight that this class encompasses, among others, the category of Clifford semigroups.

The independence number is a graph parameter that measures the size of the largest vertex set in a graph that induces no edge. There have been many research topics on the independence numbers of graphs and digraphs. The independence number of finite connected simple graphs was studied by Harant and Schiermeyer [9,10]. They gave lower bounds of the independence number in terms of the order, size and degrees. In [11], Löwenstein et al. proved several tight lower bounds of the order and average degree for the independence number of connected graphs. Some results on the upper and lower bounds of the independence number of graphs have been obtained by many authors (see for examples, refs. [12,13,14,15,16,17]). In their work [18], The authors presented the zero forcing number for specific classes of graphs and digraphs. It is worth noting that in certain classes of digraphs, such as cycles or trees, we observed that the zero forcing number is less than or equal to the independence number. As a result, the zero forcing set demonstrates a relationship with the independent set.

The independent sets are interesting topics in the study of Cayley digraphs. In [19], Panma and Nupo studied independent sets and some generalizations of independent sets such as weakly independent, path independent, and weakly path independent sets in Cayley digraphs of rectangular groups. They gave lower and upper bounds for the independence, weak independence, path independence, and weak path independence numbers by using some algebraic properties of groups.

It is natural to investigate the Cayley digraph of Clifford semigroups and consider how the results from the group case exist. The purpose of this work is to find the independence, weak independence, path independence, and weak path independence numbers of Cayley digraphs of Clifford semigroups by using the properties of groups.

In order to attain these results, our approach initiates with an exploration of the independent sets of small size within the Cayley digraph of the Clifford semigroup. It is noteworthy that the independence number of a graph can be expressed as the summation of the independence numbers of its maximal connected subdigraphs. Building upon this fundamental fact, we progress to the second step, which involves a dedicated focus on characterizing a maximal connected subdigraph of the Cayley digraph (Section 3).

Subsequently, we determine the independence number of the Cayley graphs (Section 4) and the weak independence number of the components (Section 5). In continuation, we define a partial order on the set of all left cosets in all subgroups of the Clifford semigroup, effectively representing a path within the component. This facilitates the determination of the path independence number for any given component (Section 6). Lastly, we delve into the investigation of the weak path independence number (Section 7).

2. Preliminaries

Some basic definitions and relevant notations are presented in this section. We refer to [20] for more information on graph theory and [21] for semigroup theory. All sets mentioned in this paper are assumed to be finite. Because, in this work, all mentioned graphs are directed graphs, we will refer to a directed path as a path for convenience.

Let D be a digraph with a vertex set $V(D)$ and an arc set $E(D)$. The vertices u and v in D are said to be:

-

independent if $(u,v)\notin E(D)$ and $(v,u)\notin D$;

-

weakly independent if $(u,v)\notin E(D)$ or $(v,u)\notin E(D)$;

-

path independent if there is neither a path from u to v nor from v to u;

-

weakly path independent if there are no paths from u to v or no paths from v to u.

The non-empty subset I of $V(D)$ is called an independent (respectively, weakly independent, path independent, weakly path independent) set if any two vertices in I are independent (respectively, weakly independent, path independent, weakly path independent).

The independence (respectively, weak independence, path independence, weak path independence) the number of D is the maximum cardinality among all independent (respectively, weakly independent, path independent, weakly path independent) sets of D.

Let $\alpha (D)$ (respectively, ${\alpha }_w(D)$, ${\alpha }_p(D)$, ${\alpha }_{wp}(D)$) denote the independence (respectively, weak independence, path independence, weak path independence) number of D.

The independent (respectively, weakly independent, path independent, weakly path independent) set of D is called an $\alpha -$set (respectively ${\alpha }_w-$set, ${\alpha }_p-$set, ${\alpha }_{wp}-$set) of D if the cardinality of I is equal to $\alpha (D)$ (respectively, ${\alpha }_w(D)$, ${\alpha }_p(D)$, ${\alpha }_{wp}(D)$).

The digraph D is said to be connected if the underlying graph, obtained by replacing all directed edges of D with undirected edges, is connected. It is said to be strongly connected if there exists a path from u to v and a path from v to u for all $u,v\in V(D)$. It is a well-known result that for any group G and a non-empty subset A of G, the Cayley digraph $Cay(\langle A\rangle ,A)$ is strongly connected where $\langle A\rangle$ is a subgroup of G generated by A. Let $G_1=(V_1,E_1)$, $G_2=(V_2,E_2)$ be digraphs. The union $G_1\cup G_2$ of $G_1$ and $G_2$ is the digraph with vertex set $V_1\cup V_2$ and arc set $E_1\cup E_2$. The disjoint union $G_1{\displaystyle \dot{\bigcup }}{G}_2$ of $G_1$ and $G_2$ is the union of $G_1$ and $G_2$ with $V_1\cap V_2=\varnothing$. In view of [7], we obtain the following helpful lemma.

Lemma 1 

([7]). Let G be a group and $\varnothing \ne A\subseteq G$. Then $Cay(G,A)\cong {\displaystyle \bigcup _{i\in I}^{·}}(V_i,E_i)$ where $I=\{1,2,\dots ,\frac{|G|}{|\langle A\rangle |}\}$ and $(V_i,E_i)\cong Cay(\langle A\rangle ,A)$ for all $i\in I$.

Let $(Y,\le )$ be a partially ordered set and X is a non-empty subset of Y, we say that an element c of Y is a lower bound of X if $c\le y$ for every y in X. A lower bound element c of X is called the greatest lower bound (meet) of X if $b\le c$ for every lower bound b in X. An upper bound and the least upper bound (join) are defined dually. The meet (join) of $\{a,b\}$ will be denoted by $a\wedge b$$(a\vee b)$.

A partially ordered set Y is called a meet (join) semilattice if $x\wedge y$$(x\vee y)\in Y$ for all $x,y\in Y$. A partially ordered set Y is called a semilattice if it is a meet semilattice or a join semilattice. In this work, we suppose that all semilattices are meet semilattices. For join semilattices, the results are proved dually.

An element e of a semigroup S is idempotent if $e^2=e$. An element a of a semigroup S is completely regular if there exists an element $x\in S$ such that $a=axa$ and $ax=xa$.

A semigroup S is completely regular if all its elements are completely regular. A semigroup S is a Clifford semigroup if it is completely regular and all its idempotents commute with all elements of S. It can be readily deduced that if S is a group, then the identity element e is the only idempotent element in S such that $ea=ae$ for all $a\in S$ and every element a is a completely regular because $a=a{a}^{-1}a$ and $a{a}^{-1}=a^{-1}a$ where $a^{-1}$ is an inverse of a. This then implies that every group is a Clifford semigroup.

Let Y be a semilattice and $\{(G_{\beta },{\circ }_{\beta })|\beta \in Y\}$ be a family of groups indexed by Y where $G_{\beta }\cap G_{\lambda }=\varnothing$ for any $\beta \ne \lambda \in Y$. Suppose that, for all $\beta \ge \lambda$ in Y, there exists a group homomorphism $f_{\beta ,\lambda }:G_{\beta }\to G_{\lambda }$ such that

(i) 

for all $\lambda \in Y$, $f_{\lambda ,\lambda }=i{d}_{G_{\lambda }}$ is the identity mapping on $G_{\lambda }$,

(ii) 

$f_{\beta ,\lambda }{f}_{\gamma ,\beta }=f_{\gamma ,\lambda }$ for all $\lambda ,\beta ,\gamma \in Y$ with $\gamma \ge \beta \ge \lambda$,

and the multiplication on $S={\displaystyle \bigcup _{\beta \in Y}}{G}_{\beta }$ is defined for $x\in G_{\beta }$ and $y\in G_{\lambda }$ by $xy=f_{\beta ,\beta \wedge \lambda }(x){\circ }_{\beta \wedge \lambda }{f}_{\lambda ,\beta \wedge \lambda }(y)$.

It is easy to check that $S={\displaystyle {\displaystyle \bigcup _{\beta \in Y}}}{G}_{\beta }$ under that multiplication is a semigroup, and called a strong semilattice of groups. We write $S=[Y;G_{\beta },f_{\beta ,\lambda }]$. For convenience, we will refer to $f_{\beta ,\beta \wedge \lambda }(x){\circ }_{\beta \wedge \lambda }{f}_{\lambda ,\beta \wedge \lambda }(y)$ as $f_{\beta ,\beta \wedge \lambda }(x)f_{\lambda ,\beta \wedge \lambda }(y)$.

In 1995, Howie [21] showed a necessary and sufficient condition for a Clifford semigroup that S is a Clifford semigroup if and only if S is a strong semilattice of groups. Thus, every Clifford semigroup can be written in the form $[Y;G_{\beta },f_{\beta ,\lambda }]$ for some semilattice Y, group $G_{\beta }$ and structure homomorphism $f_{\beta ,\lambda }$. Henceforth, whenever we state that $[Y;G_{\beta },f_{\beta ,\lambda }]$ is a Clifford semigroup, it is to be understood that $G_{\beta }$ is a group for every $\beta \in Y$. Consequently, we will use the term strong semilattice of groups instead of Clifford semigroup.

3. Characterizations of Maximal Connected Subdigraphs in $\mathit{Cay}\mathbf{(}\mathit{S}\mathbf{,}\mathit{A}\mathbf{)}$

Clearly, the independence number of a graph is a summation of the independence numbers of all its maximal connected subdigraphs. Thus we begin this work with the characterization of maximal connected subdigraphs of $Cay(S,A)$.

Hereafter, we let $S=[Y;G_{\beta },f_{\beta ,\lambda }]$ be a Clifford semigroup, $Y^{\prime }=\{\gamma \in Y:G_{\gamma }\cap A\ne \varnothing \}$ and $A_{\beta }=\{f_{\gamma ,\beta }(a_{\gamma }):a_{\gamma }\in G_{\gamma }\cap A,\gamma \ge \beta \}$ where $\varnothing \ne A\subseteq S$. For $X\subseteq S$, let us denote by $[X]$ the subdigraph of $Cay(S,A)$ induced by X.

Since the minimum element of Y exists it follows that in each maximal connected subdigraph of $Cay(S,A)$, there exists $\beta \in Y$ such that $\beta \wedge \gamma =\beta$ for all $\gamma \in Y^{\prime }$. Let $B=\{\beta \in Y:\beta \wedge ({\displaystyle \underset{\gamma \in Y^{\prime }}{\bigwedge }}\gamma )=\beta \}$. Clearly, $(\lambda \wedge ({\displaystyle \underset{\gamma \in Y^{\prime }}{\bigwedge }}\gamma ))\wedge ({\displaystyle \underset{\gamma \in Y^{\prime }}{\bigwedge }}\gamma )=\lambda \wedge ({\displaystyle \underset{\gamma \in Y^{\prime }}{\bigwedge }}\gamma )$ for all $\lambda \in Y$. Then we obtain the following lemma.

Lemma 2. 

$\lambda \wedge ({\displaystyle \underset{\gamma \in Y^{\prime }}{\bigwedge }}\gamma )\in B$ for all $\lambda \in Y$.

By Lemma 2, we obtain for each $\lambda \in Y$ there exists $\beta \in B$ such that $\lambda \wedge ({\displaystyle \underset{\gamma \in Y^{\prime }}{\bigwedge }}\gamma )=\beta$. We then define $Y_{\beta }=\{\lambda \in Y:\lambda \wedge ({\displaystyle \underset{\gamma \in Y^{\prime }}{\bigwedge }}\gamma )=\beta \}$ for all $\beta \in B$.

Lemma 3. 

$\{Y_{\beta }:\beta \in B\}$ is a partition of Y.

Proof. 

Clearly, $\beta \in Y_{\beta }$ for every $\beta \in B$. Thus $Y_{\beta }\ne \varnothing$ for all $\beta \in B$. By Lemma 2, we obtain ${\displaystyle \bigcup _{\beta \in B}}{Y}_{\beta }=Y$. Now, assume that $\mu \in Y_{\beta }\cap Y_{{\beta }^{\prime }}$. Then $\mu \wedge ({\displaystyle \underset{\gamma \in Y^{\prime }}{\bigwedge }}\gamma )=\beta$ and $\mu \wedge ({\displaystyle \underset{\gamma \in Y^{\prime }}{\bigwedge }}\gamma )={\beta }^{\prime }$ which implies $\beta ={\beta }^{\prime }$. Therefore $\{Y_{\beta }:\beta \in B\}$ is a partition of Y. □

Example 1. 

Let $Y=\{{\lambda }_1,{\lambda }_2,\dots ,{\lambda }_6\}$ be a semilattice with a partial order that represented by the Hasse diagram in Figure 1. For $I=\{1,2,\dots ,6\}$, we let $\{G_{{\lambda }_i}:i\in I\}$ be a family of groups, indexed by the semilattice Y where $G_{{\lambda }_i}={\mathbb{Z}}_4=\{{\overline{0}}_{{\lambda }_i},{\overline{1}}_{{\lambda }_i},{\overline{2}}_{{\lambda }_i},{\overline{3}}_{{\lambda }_i}\}$ is an additive group of integers modulo 4, for all $i\in I$. Let $f_{{\lambda }_i,{\lambda }_j}({\overline{x}}_{{\lambda }_i})={\overline{x}}_{{\lambda }_j}$ for every ${\overline{x}}_{{\lambda }_i}\in G_{{\lambda }_i}$, ${\overline{x}}_{{\lambda }_j}\in G_{{\lambda }_j}$ and $i>j$. Then $S=[Y;G_{{\lambda }_i},f_{{\lambda }_i,{\lambda }_j}]$ is a Clifford semigroup.

If we put $A=\{{\overline{2}}_{{\lambda }_6}\}$ where ${\overline{2}}_{{\lambda }_6}\in G_{{\lambda }_6}$, then we get

(i) 

$Y^{\prime }=\{{\lambda }_6\}$ and $B=\{{\lambda }_1,{\lambda }_2,{\lambda }_4,{\lambda }_6\}$,

(ii) 

$Y_{{\lambda }_1}=\{{\lambda }_1\}$, $Y_{{\lambda }_2}=\{{\lambda }_2,{\lambda }_3\}$, $Y_{{\lambda }_4}=\{{\lambda }_4,{\lambda }_5\}$ and $Y_{{\lambda }_6}=\{{\lambda }_6\}$,

(iii) 

$\{Y_{{\lambda }_1},Y_{{\lambda }_2},Y_{{\lambda }_4},Y_{{\lambda }_6}\}$ is a partition of Y.

Lemma 4. 

$f_{\lambda ,\eta }(g\langle A_{\lambda }\rangle )\subseteq f_{\lambda ,\eta }(g)\langle A_{\eta }\rangle$ for all $\eta ,\lambda \in Y$, such that $\eta \le \lambda$.

Proof. 

Let $h\in f_{\lambda ,\eta }(g\langle A_{\lambda }\rangle )$. Then $h=f_{\lambda ,\eta }(ga)$ for some $a=a_1^{t_1}{a}_2^{t_2}\dots a_m^{t_m}\in \langle A_{\lambda }\rangle$ where $a_i\in A_{\lambda }$ and $t_i\in \mathbb{Z}$ for all $1\le i\le m$. For each $a_i\in A_{\lambda }$, we obtain $a_i=f_{{\gamma }_i,\lambda }(b_{\gamma i})$ for some $b_{\gamma i}\in A\cap G_{{\gamma }_i}$ where $\lambda \le {\gamma }_i$ and we then obtain $f_{\lambda ,\eta }(f_{{\gamma }_i,\lambda }(b_{\gamma i}))=f_{{\gamma }_i,\eta }(b_{\gamma i})\in A_{\eta }$ for all $1\le i\le m$. Consider

$$\begin{array}{cc}\hfill h=f_{\lambda ,\eta }(ga)& =f_{\lambda ,\eta }(g{a}_1^{t_1}{a}_2^{t_2}\dots a_m^{t_m})\hfill \\ \hfill & =f_{\lambda ,\eta }(g)f_{\lambda ,\eta }(f_{{\gamma }_1,\lambda }{(b_{\gamma 1})}^{t_1}{f}_{{\gamma }_2,\lambda }{(b_{\gamma 2})}^{t_2}\dots f_{{\gamma }_m,\lambda }{(b_{\gamma m})}^{t_m})\hfill \\ \hfill & =f_{\lambda ,\eta }(g)f_{\lambda ,\eta }{(f_{{\gamma }_1,\lambda }(b_{\gamma 1}))}^{t_1}{f}_{\lambda ,\eta }{(f_{{\gamma }_2,\lambda }(b_{\gamma 2}))}^{t_2}\dots f_{\lambda ,\eta }{(f_{{\gamma }_m,\lambda }(b_{\gamma m}))}^{t_m}\hfill \\ \hfill & =f_{\lambda ,\eta }(g)f_{{\gamma }_1,\eta }{(b_{\gamma 1})}^{t_1}{f}_{{\gamma }_2,\eta }{(b_{\gamma 2})}^{t_2}\dots f_{{\gamma }_m,\eta }{(b_{\gamma m})}^{t_m}.\hfill \end{array}$$

Hence $h=f_{\lambda ,\eta }(g)f_{{\gamma }_1,\eta }{(b_{\gamma 1})}^{t_1}{f}_{{\gamma }_2,\eta }{(b_{\gamma 2})}^{t_2}\dots f_{{\gamma }_m,\eta }{(b_{\gamma m})}^{t_m}\in f_{\lambda ,\eta }(g)\langle A_{\eta }\rangle$.

Therefore $f_{\lambda ,\eta }(g\langle A_{\lambda }\rangle )\subseteq f_{\lambda ,\eta }(g)\langle A_{\eta }\rangle$. □

Let D be a digraph and $v\in V(D)$. The set of in-neighbors and the set of out-neighbors of a vertex v are defined by $N^{-}(v)=\{u\in V(D):(u,v)\in E(D)\}$ and $N^{+}(v)=\{u\in V(D):(v,u)\in E(D)\}$, respectively. In addition, we use $N(v)=N^{-}(v)\cup N^{+}(v)$, the set of neighbors of the vertex v.

From Lemma 1, we observe that the Cayley digraph $Cay(G_{\lambda },A_{\lambda })\cong {\displaystyle \bigcup _{i\in I}}(V_i,E_i)$ for all $\lambda \in Y$ where $I=\{1,2\dots ,\frac{|G_{\lambda }|}{|\langle A_{\lambda }\rangle |}\}$ and $(V_i,E_i)=[g_i\langle A_{\lambda }\rangle ]\cong Cay(\langle A_{\lambda }\rangle ,A_{\lambda })$ for all $i\in I$ where $g_i\langle A_{\lambda }\rangle \in G_{\lambda }/\langle A_{\lambda }\rangle$ and $[g_i\langle A_{\lambda }\rangle ]$ is an induced subdigraph of $Cay(S,A)$.

Lemma 5. 

Let x be a vertex in $Cay(S,A)$ and let $\beta \in B$. Then $N(x)\subseteq {\displaystyle \bigcup _{\begin{array}{c}g\in g_{\beta }\langle A_{\beta }\rangle \\ \lambda \in Y_{\beta }\end{array}}}{f}_{\lambda ,\beta }^{-1}(g)$ for all $x\in {\displaystyle \bigcup _{\begin{array}{c}g\in g_{\beta }\langle A_{\beta }\rangle \\ \lambda \in Y_{\beta }\end{array}}}{f}_{\lambda ,\beta }^{-1}(g)$.

Proof. 

Let $x\in {\displaystyle \bigcup _{\begin{array}{c}g\in g_{\beta }\langle A_{\beta }\rangle \\ \lambda \in Y_{\beta }\end{array}}}{f}_{\lambda ,\beta }^{-1}(g)$. Then $x\in G_{\lambda }$ for some $\lambda \in Y_{\beta }$ where $f_{\lambda ,\beta }(x)\in g_{\beta }\langle A_{\beta }\rangle$. Thus we let $f_{\lambda ,\beta }(x)=g_{\beta }{a}_1^{t_1}{a}_2^{t_2}\dots a_m^{t_m}$ where $a_i\in A_{\beta }$ and $t_i\in \mathbb{Z}$ for all $1\le i\le m$. Assume that $u\in N(x)$, we shall show that $u\in {\displaystyle \bigcup _{\begin{array}{c}g\in g_{\beta }\langle A_{\beta }\rangle \\ \lambda \in Y_{\beta }\end{array}}}{f}_{\lambda ,\beta }^{-1}(g)$. Consider two cases:

(i) 

Case $u\in N^{+}(x)$, which means $(x,u)\in E(Cay(S,A))$. Thus $u=xa$ for some $a\in A$. Since $A\subseteq S$, we assume that $a\in A\subseteq G_{\gamma }$ for some $\gamma \in Y^{\prime }$. By the definition of multiplication on S, we obtain $u\in G_{\lambda \wedge \gamma }$, Clearly, $\lambda \wedge \gamma \in Y_{\beta }$ because $\lambda \in Y_{\beta }$. Then

$$\begin{array}{cc}\hfill f_{\lambda \wedge \gamma ,\beta }(u)& =f_{\lambda \wedge \gamma ,\beta }(f_{\lambda ,\lambda \wedge \gamma }(x)f_{\gamma ,\lambda \wedge \gamma }(a))\hfill \\ \hfill & =f_{\lambda ,\beta }(x)f_{\gamma ,\beta }(a)\hfill \\ \hfill & =g_{\beta }{a}_1^{t_1}{a}_2^{t_2}\dots a_m^{t_m}{f}_{\gamma ,\beta }(a).\hfill \end{array}$$

Since $f_{\gamma ,\beta }(a)\in A_{\beta }$, we obtain $a_1^{t_1}{a}_2^{t_2}\dots a_m^{t_m}{f}_{\gamma ,\beta }(a)\in \langle A_{\beta }\rangle$.

Hence $f_{\lambda \wedge \gamma ,\beta }(u)=g_{\beta }{a}_1^{t_1}{a}_2^{t_2}\dots a_m^{t_m}{f}_{\gamma ,\beta }(a)\in g_{\beta }\langle A_{\beta }\rangle$. From $\lambda \wedge \gamma \in Y_{\beta }$ and $f_{\lambda \wedge \gamma ,\beta }(u)\in g_{\beta }\langle A_{\beta }\rangle$, we conclude that $u\in {\displaystyle \bigcup _{h\in g_{\beta }\langle A_{\beta }\rangle }}{f}_{\lambda \wedge \gamma ,\beta }^{-1}(h)\subseteq {\displaystyle \bigcup _{\begin{array}{c}g\in g_{\beta }\langle A_{\beta }\rangle \\ \lambda \in Y_{\beta }\end{array}}}{f}_{\lambda ,\beta }^{-1}(g)$.

(ii) 

Case $u\in N^{-}(x)$, which means $(u,x)\in E(Cay(S,A))$. Thus $x=ua$ for some $a\in A\cap G_{\gamma }$ and $\gamma \in Y^{\prime }$. Assume that $u\in G_{\eta }$. Clearly, $\eta \wedge \gamma =\lambda$ because $x\in G_{\lambda }$. First, we will show that $\eta \in Y_{\beta }$. From $\eta \wedge \gamma =\lambda$, we obtain $\eta \wedge ({\displaystyle \underset{{\gamma }^{\prime }\in Y^{\prime }}{\bigwedge }}{\gamma }^{\prime })=(\eta \wedge \gamma )\wedge ({\displaystyle \underset{{\gamma }^{\prime }\in Y^{\prime }}{\bigwedge }}{\gamma }^{\prime })=\lambda \wedge ({\displaystyle \underset{{\gamma }^{\prime }\in Y^{\prime }}{\bigwedge }}{\gamma }^{\prime })=\beta$. Then $\eta \in Y_{\beta }$. From $f_{\lambda ,\beta }(x)=f_{\lambda ,\beta }(f_{\eta ,\lambda }(u)f_{\gamma ,\lambda }(a))=f_{\eta ,\beta }(u)f_{\gamma ,\beta }(a)$, we obtain $f_{\eta ,\beta }(u)=f_{\lambda ,\beta }(x){(f_{\gamma ,\beta }(a))}^{-1}$. Since $f_{\gamma ,\beta }(a)\in A_{\beta }$, we conclude that $f_{\gamma ,\beta }{(a)}^{-1}\in \langle A_{\beta }\rangle$.

Therefore $f_{\eta ,\beta }(u)=f_{\lambda ,\beta }(x){(f_{\gamma ,\beta }(a))}^{-1}\in g_{\beta }\langle A_{\beta }\rangle$ which implies $u\in {\displaystyle \bigcup _{h\in g_{\beta }\langle A_{\beta }\rangle }}{f}_{\eta ,\beta }^{-1}(h)\subseteq {\displaystyle \bigcup _{\begin{array}{c}g\in g_{\beta }\langle A_{\beta }\rangle \\ \lambda \in Y_{\beta }\end{array}}}{f}_{\lambda ,\beta }^{-1}(g)$.

Hence $u\in {\displaystyle \bigcup _{\begin{array}{c}{g}_{\beta }\in g_{\beta }\langle A_{\beta }\rangle \\ \lambda \in Y_{\beta }\end{array}}}{f}_{\lambda ,\beta }^{-1}(g_{\beta })$, as required. □

Lemma 6. 

For each $x\in S$. There exists $\beta \in B$, $\lambda \in Y_{\beta }$ and $g_{\beta }\langle A_{\beta }\rangle \in G_{\beta }/\langle A_{\beta }\rangle$ such that $x\in {\displaystyle \bigcup _{g\in g_{\beta }\langle A_{\beta }\rangle }}{f}_{\lambda ,\beta }^{-1}(g)$.

Proof. 

Let $x\in S$. Then $x\in G_{\lambda }$ for some $\lambda \in Y$. By Lemma 2, we obtain $\lambda \in Y_{\beta }$ for some $\beta \in B$. Thus $f_{\lambda ,\beta }(x)$ is defined in $G_{\beta }$ which implies there exists $g_{\beta }\langle A_{\beta }\rangle \in G_{\beta }/\langle A_{\beta }\rangle$ and $g^{\prime }\in g_{\beta }\langle A_{\beta }\rangle$ such that $f_{\lambda ,\beta }(x)=g^{\prime }$. Therefore $x\in {\displaystyle \bigcup _{g\in g_{\beta }\langle A_{\beta }\rangle }}{f}_{\lambda ,\beta }^{-1}(g)$. □

Theorem 1. 

Let $\beta \in B$ and $g_{\beta }\langle A_{\beta }\rangle \in G_{\beta }/\langle A_{\beta }\rangle$. Then $[{\displaystyle \bigcup _{\begin{array}{c}g\in g_{\beta }\langle A_{\beta }\rangle \\ \lambda \in Y_{\beta }\end{array}}}{f}_{\lambda ,\beta }^{-1}(g)]$ is a maximal connected subdigraph of $Cay(S,A)$.

Proof. 

Let $\beta \in B$, $g_{\beta }\langle A_{\beta }\rangle \in G_{\beta }/\langle A_{\beta }\rangle$ and $g^{\prime }\in g_{\beta }\langle A_{\beta }\rangle$. We first show that, for each $x\in {\displaystyle \bigcup _{\begin{array}{c}{g}_{\beta }\in g_{\beta }\langle A_{\beta }\rangle \\ \lambda \in Y_{\beta }\end{array}}}{f}_{\lambda ,\beta }^{-1}(g_{\beta })$, there exists a path from x to $g^{\prime }$. Now, let $x\in {\displaystyle \bigcup _{\begin{array}{c}{g}_{\beta }\in g_{\beta }\langle A_{\beta }\rangle \\ \lambda \in Y_{\beta }\end{array}}}{f}_{\lambda ,\beta }^{-1}(g_{\beta })$. Then $x\in G_{\lambda }$ for some $\lambda \in Y_{\beta }$. From $\lambda \in Y_{\beta }$, it follows that there exists $\{{\gamma }_1,{\gamma }_2,\dots ,{\gamma }_t\}\subseteq Y_{\beta }$ such that $\lambda \wedge {\gamma }_1\wedge {\gamma }_2\wedge \dots \wedge {\gamma }_t=\beta$, and there exists $\{a_1,a_2,\dots ,a_t\}\subseteq A$ where $a_i\in G_{{\gamma }_i}$ for all $1\le i\le t$, such that $x,x{a}_1,x{a}_1{a}_2,\dots ,x{a}_1{a}_2\dots a_t$ is a path from x to $x{a}_1{a}_2\dots a_t$. Consider

$$\begin{array}{cc}\hfill x{a}_1{a}_2\dots a_t& =(\dots ((f_{\lambda ,\lambda \wedge {\gamma }_1}(x)f_{{\gamma }_1,\lambda \wedge {\gamma }_1}(a_1))a_2)\dots )a_t\hfill \\ & =(\dots ((f_{\lambda \wedge {\gamma }_1,(\lambda \wedge {\gamma }_1)\wedge {\gamma }_2}(f_{\lambda ,\lambda \wedge {\gamma }_1}(x)f_{{\gamma }_1,\lambda \wedge {\gamma }_1}(a_1))f_{{\gamma }_2,(\lambda \wedge {\gamma }_1)\wedge {\gamma }_2}(a_2))\dots )a_t\hfill \\ & ⋮\hfill \\ & =f_{\lambda \wedge ({\displaystyle \underset{i=1}{\overset{t-1}{\bigwedge }}}{\gamma }_i),\lambda \wedge ({\displaystyle \underset{i=1}{\overset{t-1}{\bigwedge }}}{\gamma }_i)\wedge {\gamma }_t}(\dots (f_{\lambda ,\lambda \wedge {\gamma }_1}(x)f_{{\gamma }_1,\lambda \wedge {\gamma }_1}(a_1))\dots )f_{{\gamma }_t,\lambda \wedge ({\displaystyle \underset{i=1}{\overset{t-1}{\bigwedge }}}{\gamma }_i)\wedge {\gamma }_t}(a_t)\hfill \\ & =f_{\lambda ,\beta }(x)f_{{\gamma }_1,\beta }(a_1)f_{{\gamma }_2,\beta }(a_2)\dots f_{{\gamma }_t,\beta }(a_t)\hfill \\ & \in f_{\lambda ,\beta }(x)\langle A_{\beta }\rangle .\hfill \end{array}$$

Since $[f_{\lambda ,\beta }(x)\langle A_{\beta }\rangle ]\cong Cay(\langle A_{\beta }\rangle ,A_{\beta })$, it follows that there exists a path from $x{a}_1{a}_2\dots a_t$ to $g^{\prime }$. Thus $[{\displaystyle \bigcup _{\begin{array}{c}g\in g_{\beta }\langle A_{\beta }\rangle \\ \lambda \in Y_{\beta }\end{array}}}{f}_{\lambda ,\beta }^{-1}(g)]$ is a connected subdigraph of $Cay(S,A)$. Suppose that there exists $x^{\prime }\in S\backslash {\displaystyle \bigcup _{\begin{array}{c}g\in g_{\beta }\langle A_{\beta }\rangle \\ \lambda \in Y_{\beta }\end{array}}}{f}_{\lambda ,\beta }^{-1}(g)$ such that $x^{\prime }\in N(x)$ for some $x\in {\displaystyle \bigcup _{\begin{array}{c}g\in g_{\beta }\langle A_{\beta }\rangle \\ \lambda \in Y_{\beta }\end{array}}}{f}_{\lambda ,\beta }^{-1}(g)$. By using Lemma 5, we conclude that $x^{\prime }\in {\displaystyle \bigcup _{\begin{array}{c}g\in g_{\beta }\langle A_{\beta }\rangle \\ \lambda \in Y_{\beta }\end{array}}}{f}_{\lambda ,\beta }^{-1}(g)$, a contradiction. Therefore $[{\displaystyle \bigcup _{\begin{array}{c}g\in g_{\beta }\langle A_{\beta }\rangle \\ \lambda \in Y_{\beta }\end{array}}}{f}_{\lambda ,\beta }^{-1}(g)]$ is a maximal connected subdigraph of $Cay(S,A)$, as required. □

From Theorem 1, we have investigated the maximal connected subdigraph of $Cay(S,A)$ and obtained some needed properties. Afterward, we then achieve a characterization for a maximal connected subdigraph of $Cay(S,A)$.

Theorem 2. 

A subdigraph C of $Cay(S,A)$ is a maximal connected subdigraph if and only if $C=[{\displaystyle \bigcup _{\begin{array}{c}g\in g_{\beta }\langle A_{\beta }\rangle \\ \lambda \in Y_{\beta }\end{array}}}{f}_{\lambda ,\beta }^{-1}(g)]$ for some $\beta \in B$ and $g_{\beta }\langle A_{\beta }\rangle \in G_{\beta }/\langle A_{\beta }\rangle$.

Proof. 

Let C be a maximal connected subdigraph of $Cay(S,A)$ and x a vertex of C. By Lemma 6, we obtain $x\in {\displaystyle \bigcup _{g\in g_{\beta }\langle A_{\beta }\rangle }}{f}_{\lambda ,\beta }^{-1}(g)$ for some $\beta \in B$, $\lambda \in Y_{\beta }$ and $g_{\beta }\langle A_{\beta }\rangle \in G_{\beta }/\langle A_{\beta }\rangle$, which means $x\in {\displaystyle \bigcup _{\begin{array}{c}g\in g_{\beta }\langle A_{\beta }\rangle \\ \lambda \in Y_{\beta }\end{array}}}{f}_{\lambda ,\beta }^{-1}(g)$. From Lemma 5 and C is connected, we obtain $V(C)\subseteq {\displaystyle \bigcup _{\begin{array}{c}g\in g_{\beta }\langle A_{\beta }\rangle \\ \lambda \in Y_{\beta }\end{array}}}{f}_{\lambda ,\beta }^{-1}(g)$. Since $[{\displaystyle \bigcup _{\begin{array}{c}{g}_{\beta }\in g_{\beta }\langle A_{\beta }\rangle \\ \lambda \in Y_{\beta }\end{array}}}{f}_{\lambda ,\beta }^{-1}(g_{\beta })]$ is an induced subdigraph and C is maximal connected, we conclude that $C=[{\displaystyle \bigcup _{\begin{array}{c}{g}_{\beta }\in g_{\beta }\langle A_{\beta }\rangle \\ \lambda \in Y_{\beta }\end{array}}}{f}_{\lambda ,\beta }^{-1}(g_{\beta })]$.

Conversely, $C=[{\displaystyle \bigcup _{\begin{array}{c}g\in g_{\beta }\langle A_{\beta }\rangle \\ \lambda \in Y_{\beta }\end{array}}}{f}_{\lambda ,\beta }^{-1}(g)]$ is a maximal connected subdigraph by Theorem 1. □

Example 2. 

From the Clifford semigroup $S=[Y;G_{{\lambda }_i},f_{{\lambda }_i,{\lambda }_j}]$ in an Example 1 and $A=\{{\overline{2}}_{{\lambda }_4},{\overline{1}}_{{\lambda }_6}\}$, the $Cay(S,A)$ can be pictured as Figure 2. In addition, we obtain $B=\{{\lambda }_1,{\lambda }_2,{\lambda }_4\}$ and the following:

(i) 

Consider ${\lambda }_1\in B$. We obtain $[{\displaystyle \bigcup _{\begin{array}{c}g\in g_{{\lambda }_1}\langle A_{{\lambda }_1}\rangle \\ \eta \in Y_{{\lambda }_1}\end{array}}}{f}_{\eta ,{\lambda }_1}^{-1}(g)]=[{\displaystyle \bigcup _{g\in G_{{\lambda }_1}}}{f}_{{\lambda }_1,{\lambda }_1}^{-1}(g)]=[G_{{\lambda }_1}]$.

(ii) 

Consider ${\lambda }_2\in B$.

We obtain $[{\displaystyle \bigcup _{\begin{array}{c}g\in \langle A_{{\lambda }_2}\rangle \\ \eta \in Y_{{\lambda }_2}\end{array}}}{f}_{\eta ,{\lambda }_2}^{-1}(g)]=[({\displaystyle \bigcup _{g\in \langle A_{{\lambda }_2}\rangle }}{f}_{{\lambda }_3,{\lambda }_2}^{-1}(g))\cup ({\displaystyle \bigcup _{g\in \langle A_{{\lambda }_2}\rangle }}{f}_{{\lambda }_2,{\lambda }_2}^{-1}(g))]=$

$[G_{{\lambda }_3}\cup G_{{\lambda }_2}]$.

(iii) 

Consider ${\lambda }_4\in B$.

We obtain $[{\displaystyle \bigcup _{\begin{array}{c}g\in \langle A_{{\lambda }_4}\rangle \\ \eta \in Y_{{\lambda }_4}\end{array}}}{f}_{\eta ,{\lambda }_4}^{-1}(g)]=[({\displaystyle \bigcup _{g\in \langle A_{{\lambda }_4}\rangle }}{f}_{{\lambda }_6,{\lambda }_4}^{-1}(g))\cup ({\displaystyle \bigcup _{g\in \langle A_{{\lambda }_4}\rangle }}{f}_{{\lambda }_5,{\lambda }_4}^{-1}(g))\cup$

$({\displaystyle \bigcup _{g\in \langle A_{{\lambda }_4}\rangle }}{f}_{{\lambda }_4,{\lambda }_4}^{-1}(g))]=[G_{{\lambda }_6}\cup G_{{\lambda }_5}\cup G_{{\lambda }_4}]$.

We see that, $Cay(S,A)$ is the union of three maximal connected subdigraphs which are $[G_{{\lambda }_1}]$, $[G_{{\lambda }_3}\cup G_{{\lambda }_2}]$ and $[G_{{\lambda }_6}\cup G_{{\lambda }_5}\cup G_{{\lambda }_4}]$.

From Theorem 1, we denote by $C_{g_{\beta }}$ the maximal connected subdigraph $[{\displaystyle \bigcup _{\begin{array}{c}g\in g_{\beta }\langle A_{\beta }\rangle \\ \lambda \in Y_{\beta }\end{array}}}{f}_{\lambda ,\beta }^{-1}(g)]$ associate with $\beta \in B$ and $g_{\beta }\langle A_{\beta }\rangle \in G_{\beta }/\langle A_{\beta }\rangle$. It follows easily that $C_{g_{\beta }}=C_{g_{\beta }^{{}^{\prime }}}$ if $g_{\beta }^{\prime }\in g_{\beta }\langle A_{\beta }\rangle$ and $Cay(S,A)\cong {\displaystyle \bigcup _{\begin{array}{c}g\in {\mathcal{A}}_{\beta }\\ \beta \in B\end{array}}}{C}_{g_{\beta }}$ where ${\mathcal{A}}_{\beta }$ is the set of representatives of all left cosets in $G_{\beta }/\langle A_{\beta }\rangle$.

4. Lower and Upper Bounds of the Independence Numbers

In this section, we introduce bounds of the independence number of $Cay(S,A)$. We first denote by $A_{\beta ,\lambda }$ the set of all elements of A in which $G_{\beta }{A}_{\beta ,\lambda }\subseteq G_{\lambda }$, i.e., $A_{\beta ,\lambda }=\{a\in A:G_{\beta }a\subseteq G_{\lambda }\}$. Set $Y_{\beta }^{\prime }:=\{\lambda \in Y\backslash \{\beta \}:\lambda \wedge \gamma =\beta ,\exists \gamma \in Y^{\prime }\}$. For any $\beta \in B$ and $\lambda \in Y_{\beta }$, we define $M_{g_{\beta }\lambda }=\left\{\begin{array}{cc}\{g_{\lambda }\langle A_{\lambda }\rangle \in G_{\lambda }/\langle A_{\lambda }\rangle :f_{\lambda ,\beta }(g_{\lambda })\in g_{\beta }\langle A_{\beta }\rangle \},\hfill & \hfill \mathrm{if}{A}_{\lambda }\ne \varnothing \\ \{g_{\lambda }\langle e_{\lambda }\rangle \in G_{\lambda }/\langle e_{\lambda }\rangle :f_{\lambda ,\beta }(g_{\lambda })\in g_{\beta }\langle A_{\beta }\rangle \},\hfill & \hfill \mathrm{if}{A}_{\lambda }=\varnothing .\end{array}\right.$

Let us denote by $\bigcup M_{g_{\beta }\lambda }$ the union of all sets in $M_{g_{\beta }\lambda }$. We here start with the lower bound of the independence number of $C_{g_{\beta }}$.

Lemma 7. 

${\displaystyle \sum _{\lambda \in Y_{\beta }}}\alpha ([\bigcup M_{g_{\beta }\lambda }-{\displaystyle \bigcup _{\eta \in Y_{\lambda }^{{}^{\prime }}}}{G}_{\eta }{A}_{\eta ,\lambda }])\le \alpha (C_{g_{\beta }})$.

Proof. 

Let $X_{g_{\beta }\lambda }$ be an $\alpha -$set of $[\bigcup M_{g_{\beta }\lambda }-{\displaystyle \bigcup _{\eta \in Y_{\lambda }^{{}^{\prime }}}}{G}_{\eta }{A}_{\eta ,\lambda }]$. We will show that ${\displaystyle \bigcup _{\lambda \in Y_{\beta }}}{X}_{g_{\beta }\lambda }$ is an independent set of $C_{g_{\beta }}$. Let $u,v\in {\displaystyle \bigcup _{\lambda \in Y_{\beta }}}{X}_{g_{\beta }\lambda }$. This gives $u\in X_{g_{\beta }\eta }$ and $v\in X_{g_{\beta }\lambda }$ for some $\eta ,\lambda \in Y_{\beta }$. Assume that $(u,v)\in E(C_{g_{\beta }})$. Then $v=ua$ for some $a\in A$. We consider two cases:

(i) 

Case $\lambda =\eta$, from $[\bigcup M_{g_{\beta }\lambda }-{\displaystyle \bigcup _{\eta \in Y_{\lambda }^{{}^{\prime }}}}{G}_{\eta }{A}_{\eta ,\lambda }]$ is an induced subdigraph of $Cay(S,A)$ and $(u,v)\in E(C_{g_{\beta }})$, we see that $(u,v)$ is an arc in $[\bigcup M_{g_{\beta }\lambda }-{\displaystyle \bigcup _{\eta \in Y_{\lambda }^{{}^{\prime }}}}{G}_{\eta }{A}_{\eta ,\lambda }]$. Since $u,v\in X_{g_{\beta }\lambda }$ and $X_{g_{\beta }\lambda }$ is an independent set of $[\bigcup M_{g_{\beta }\lambda }-{\displaystyle \bigcup _{\eta \in Y_{\lambda }^{{}^{\prime }}}}{G}_{\eta }{A}_{\eta ,\lambda }]$, we obtain $u,v$ are independent, which is a contradiction.

(ii) 

Case $\lambda \ne \eta$, since $u\in X_{g_{\beta }\eta }\subseteq G_{\eta }$ and $v=ua\in X_{g_{\beta }\lambda }\subseteq G_{\lambda }$, we have $a\in A_{\eta ,\lambda }$. Then we obtain $v=ua\in G_{\eta }{A}_{\eta ,\lambda }\subseteq {\displaystyle \bigcup _{\eta \in Y_{\lambda }^{{}^{\prime }}}}{G}_{\eta }{A}_{\eta ,\lambda }$ which implies $v\notin X_{g_{\beta }\lambda }\subseteq \bigcup M_{g_{\beta }\lambda }-{\displaystyle \bigcup _{\eta \in Y_{\lambda }^{{}^{\prime }}}}{G}_{\eta }{A}_{\eta ,\lambda }$, a contradiction.

Thus we conclude that ${\displaystyle \bigcup _{\lambda \in Y_{\beta }}}{X}_{g_{\beta }\lambda }$ is an independent set of $C_{g_{\beta }}$. Clearly, $X_{g_{\beta }\eta }\cap X_{g_{\beta }\lambda }=\varnothing$ for all $\eta ,\lambda \in Y_{\beta }$ where $\eta \ne \lambda$. Therefore $|{\displaystyle \bigcup _{\lambda \in Y_{\beta }}}{X}_{g_{\beta }\lambda }| ={\displaystyle \sum _{\lambda \in Y_{\beta }}}\alpha ([\bigcup M_{g_{\beta }\lambda }-{\displaystyle \bigcup _{\eta \in Y_{\lambda }^{{}^{\prime }}}}{G}_{\eta }{A}_{\eta ,\lambda }])\le \alpha (C_{g_{\beta }})$. □

The following lemma gives an upper bound of $\alpha (C_{g_{\beta }})$ obtained by using the fact that if $D=(V,E)$ and $D^{\prime }=(V,E^{\prime })$ such that $E^{\prime }\subseteq E$, then $\alpha (D)\le \alpha (D^{\prime })$.

Lemma 8. 

$\alpha (C_{g_{\beta }})\le {\displaystyle \sum _{\lambda \in Y_{\beta }}}\alpha ([\bigcup M_{g_{\beta }\lambda }])$.

Proof. 

We see that $V(C_{g_{\beta }})=V({\displaystyle \bigcup _{\lambda \in Y_{\beta }}}[\bigcup M_{g_{\beta }\lambda }])$ because

$$\begin{array}{cc}\hfill u\in V(C_{g_{\beta }})& \iff u\in {\displaystyle \bigcup _{\begin{array}{c}g\in g_{\beta }\langle A_{\beta }\rangle \\ \lambda \in Y_{\beta }\end{array}}}{f}_{\lambda ,\beta }^{-1}(g)\hfill \\ \hfill & \iff u\in f_{\lambda ,\beta }^{-1}(g)\mathrm{for}\mathrm{some}g\in g_{\beta }\langle A_{\beta }\rangle \mathrm{and}\lambda \in Y_{\beta }\hfill \\ \hfill & \iff f_{\lambda ,\beta }(u)=g\in g_{\beta }\langle A_{\beta }\rangle \hfill \\ \hfill & \iff u\langle A_{\lambda }\rangle \in M_{g_{\beta }\lambda }\hfill \\ \hfill & \iff u\in \bigcup M_{g_{\beta }\lambda }.\hfill \end{array}$$

Next, we let $(u,v)\in E({\displaystyle \bigcup _{\lambda \in Y_{\beta }}}[\bigcup M_{g_{\beta }\lambda }])$. Clearly, $u,v\in V(C_{g_{\beta }})$. Since $[\bigcup M_{g_{\beta }\lambda }]$ and $C_{g_{\beta }}$ are induced subdigraphs of $Cay(S,A)$, we conclude that $(u,v)\in E(C_{g_{\beta }})$. Thus $E({\displaystyle \bigcup _{\lambda \in Y_{\beta }}}[\bigcup M_{g_{\beta }\lambda }])\subseteq E(C_{g_{\beta }})$ and so $\alpha (C_{g_{\beta }})\le {\displaystyle \sum _{\lambda \in Y_{\beta }}}\alpha ([\bigcup M_{g_{\beta }\lambda }])$, as required. □

From $C_{g_{\beta }}$ is a maximal connected subdigraph of $Cay(S,A)$ and $Cay(S,A)\cong {\displaystyle \bigcup _{\begin{array}{c}g\in {\mathcal{A}}_{\beta }\\ \beta \in B\end{array}}}{C}_{g_{\beta }}$, we can directly conclude that

$$\alpha (Cay(S,A))={\displaystyle \sum _{\begin{array}{c}g\in {\mathcal{A}}_{\beta }\\ \beta \in B\end{array}}}\alpha (C_{g_{\beta }}).$$

Consequently, a lower(upper) bound of $\alpha (Cay(S,A))$ can be presented in the form of the summation of lower(upper) bounds of each $\alpha (C_{g_{\beta }})$.

Theorem 3. 

${\displaystyle \sum _{\beta \in B}}({\displaystyle \sum _{\lambda \in Y_{\beta }}}\alpha ([\bigcup M_{g_{\beta }\lambda }-{\displaystyle \bigcup _{\eta \in Y_{\lambda }^{{}^{\prime }}}}{G}_{\eta }{A}_{\eta ,\lambda }]))\le \alpha (Cay(S,A))\le {\displaystyle \sum _{\beta \in B}}({\displaystyle \sum _{\lambda \in Y_{\beta }}}\alpha ([\bigcup M_{g_{\beta }\lambda }]))$.

Any two elements a and b of a partially ordered set $(P,\le )$ are called comparable (incomparable) if either $a\le b$ or $b\le a$ (neither $a\le b$ nor $b\le a$). A subset X of P is called a chain (anti-chain) if any two elements of X are comparable (incomparable).

Here, we establish examples to show that the proposed lower and upper bounds are sharp.

Proposition 1. 

Let $S=[Y;G_{\beta },f_{\lambda ,\eta }]$ be a Clifford semigroup in which consists a chain Y with the maximum and minimum elements, namely $m^{\prime }$ and m, respectively. We put $f_{\lambda ,\eta }(x)=e_{\eta }$ for all $x\in G_{\lambda }$, $\lambda >\eta \in Y$ where $e_{\eta }$ is the identity element of $G_{\eta }$. If $A=\{e_{\lambda }:\lambda \in Y\}$, then $C_{e_m}$ is a maximal connected subdigraph of $Cay(S,A)$ and $\alpha (C_{e_m})={\displaystyle \sum _{\lambda \in Y_m}}\alpha ([\bigcup M_{e_m\lambda }-{\displaystyle \bigcup _{\eta \in Y_{\lambda }^{{}^{\prime }}}}{G}_{\eta }{A}_{\eta ,\lambda }])$.

Proof. 

Let $A=\{e_{\lambda }:\lambda \in Y\}$. Then $Y^{\prime }=Y$. From $B=\{\beta \in Y:\beta \wedge \gamma =\beta \mathrm{for}\mathrm{all}\gamma \in Y^{\prime }\}$, we obtain $B=\{m\}$. Thus $C_{e_m}$ is a maximal connected subdigraph of $Cay(S,A)$. Next, we will show that $\alpha (C_{e_m})={\displaystyle \sum _{\lambda \in Y_m}}\alpha ([\bigcup M_{e_m\lambda }-{\displaystyle \bigcup _{\eta \in Y_{\lambda }^{{}^{\prime }}}}{G}_{\eta }{A}_{\eta ,\lambda }])=1+{\displaystyle \sum _{\lambda >m}}(|G_{\lambda }|-1)$. Consider $I:={\displaystyle \bigcup _{\lambda >m}}(G_{\lambda }\backslash \{e_{\lambda }\})\cup \{e_{m^{\prime }}\}$, we claim that I is a maximal independent set of $C_{e_m}$. Let $u,v\in I$ where $u\ne v$. Then $u\in G_{\lambda }$ and $v\in G_{\eta }$ for some $\lambda ,\eta \in Y\backslash \{m\}$. Assume that $(u,v)\in E(C_{e_m})$. This gives $v=u{e}_{\gamma }$ for some $e_{\gamma }\in A$. By the assumption, we obtain $v=f_{\lambda ,\eta }(u)f_{\gamma ,\eta }(e_{\gamma })=e_{\eta }{e}_{\eta }=e_{\eta }$ where $\eta \ne m^{\prime }$. It contradicts to the fact that $e_{\eta }\notin I$. Thus I is an independent set of $C_{e_m}$. In addition, by $V(C_{e_m})\backslash I=\{e_{\lambda }:\lambda \ne m^{\prime }\}$ and $(e_{\lambda },e_{\eta })\in E(C_{e_m})$ for all $\lambda >\eta \in Y$, we obtain I is a maximal independent set of $C_{e_m}$. Thus $\alpha (C_{e_m})=1+{\displaystyle \sum _{\lambda >m}}(|G_{\lambda }|-1)$.

Now, from the assumption, we have $A_{\lambda }=\{e_{\lambda }\}$ for all $\lambda \in Y$. Thus $M_{e_m\lambda }=\{g_{\lambda }\langle \{e_{\lambda }\}\rangle \in G_{\lambda }/\langle \{e_{\lambda }\}\rangle :f_{\lambda ,\beta }(g_{\lambda })\in e_m\langle \{e_m\}\rangle \}=G_{\lambda }$ for all $\lambda \in Y_m$. From $A_{\eta ,\lambda }=\{e_{\lambda }\}$ and $f_{\eta ,\lambda }(g_{\eta })e_{\lambda }=e_{\lambda }{e}_{\lambda }=e_{\lambda }$ for all $g_{\eta }\in G_{\eta }$, $\eta \in Y_{\lambda }^{\prime }$, we obtain ${\displaystyle \bigcup _{\eta \in Y_{\lambda }^{{}^{\prime }}}}{G}_{\eta }{A}_{\eta ,\lambda }=\{e_{\lambda }\}$ where $\lambda \ne m^{\prime }$ and ${\displaystyle \bigcup _{\eta \in Y_{m^{\prime }}^{{}^{\prime }}}}{G}_{\eta }{A}_{\eta ,m^{\prime }}=\varnothing$. Since $A_{\lambda }=\{e_{\lambda }\}$, $E(Cay(G_{\lambda },A_{\lambda }))=\{(g_{\lambda },g_{\lambda }):g_{\lambda }\in G_{\lambda }\}$. Thus, for each $\lambda \in Y$ where $\lambda \ne m^{\prime }$, the induced subdigraph $[\bigcup M_{e_m\lambda }-{\displaystyle \bigcup _{\eta \in Y_{\lambda }^{{}^{\prime }}}}{G}_{\eta }{A}_{\eta ,\lambda }]$ is the digraph with vertex set $G_{\lambda }\backslash \{e_{\lambda }\}$ and arc set $\{(g_{\lambda },g_{\lambda }):g_{\lambda }\in G_{\lambda }\}$ which implies all vertices in $G_{\lambda }\backslash \{e_{\lambda }\}$ are independent. Hence $\alpha ([\bigcup M_{e_m\lambda }-{\displaystyle \bigcup _{\eta \in Y_{\lambda }^{{}^{\prime }}}}{G}_{\eta }{A}_{\eta ,\lambda }])=|G_{\lambda }|-1$ for all $\lambda \in Y$ where $\lambda \ne m^{\prime }$. From ${\displaystyle \bigcup _{\eta \in Y_{m^{\prime }}^{{}^{\prime }}}}{G}_{\eta }{A}_{\eta ,m^{\prime }}=\varnothing$, we obtainthe induced subdigraph $[\bigcup M_{e_m{m}^{\prime }}-{\displaystyle \bigcup _{\eta \in Y_{m^{\prime }}^{{}^{\prime }}}}{G}_{\eta }{A}_{\eta ,m^{\prime }}]$ is the digraph with vertex set $G_{m^{\prime }}$ and arc set $\{(g_{m^{\prime }},g_{m^{\prime }}):g_{m^{\prime }}\in G_{m^{\prime }}\}$ which implies all vertices in $G_{m^{\prime }}$ are independent. Hence $\alpha ([\bigcup M_{e_m{m}^{\prime }}-{\displaystyle \bigcup _{\eta \in Y_{m^{\prime }}^{{}^{\prime }}}}{G}_{\eta }{A}_{\eta ,m^{\prime }}])=|G_{m^{\prime }}|$ and so ${\displaystyle \sum _{\lambda \in Y_m}}\alpha ([\bigcup M_{e_m\lambda }-{\displaystyle \bigcup _{\eta \in Y_{\lambda }^{{}^{\prime }}}}{G}_{\eta }{A}_{\eta ,\lambda }])=1+{\displaystyle \sum _{\lambda >m}}(|G_{\lambda }|-1)$. Therefore $\alpha (C_{e_m})={\displaystyle \sum _{\lambda \in Y_m}}\alpha ([\bigcup M_{e_m\lambda }-{\displaystyle \bigcup _{\eta \in Y_{\lambda }^{{}^{\prime }}}}{G}_{\eta }{A}_{\eta ,\lambda }])$, as required. □

Example 3. 

Let $Y=\{{\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4\}$ be a chain such that ${\lambda }_1\le {\lambda }_2\le {\lambda }_3\le {\lambda }_4$. For $I=\{1,2,3,4\}$, we let $\{G_{{\lambda }_i}:i\in I\}$ be a family of groups, indexed by Y where $G_{{\lambda }_i}={\mathbb{Z}}_4=\{{\overline{0}}_{{\lambda }_i},{\overline{1}}_{{\lambda }_i},{\overline{2}}_{{\lambda }_i},{\overline{3}}_{{\lambda }_i}\}$ is an additive group of integers modulo 4 for all $i\in I$. Let $f_{{\lambda }_i,{\lambda }_j}({\overline{x}}_{{\lambda }_i})={\overline{0}}_{{\lambda }_j}$ for every ${\overline{x}}_{{\lambda }_i}\in G_{{\lambda }_i}$, ${\overline{0}}_{{\lambda }_j}\in G_{{\lambda }_j}$ and $i>j$. Then $S=[Y;G_{{\lambda }_i},f_{{\lambda }_i,{\lambda }_j}]$ is a Clifford semigroup. For $A=\{{\overline{0}}_{{\lambda }_i}:i\in I\}$, $Cay(S,A)$ can be pictured as in Figure 3.

We now determine the independence number of the maximal connected subdigraph $C_{{\overline{0}}_{{\lambda }_1}}$. From $A_{{\lambda }_1}=\{{\overline{0}}_{{\lambda }_1}\}$, we obtain ${\overline{0}}_{{\lambda }_1}\langle A_{{\lambda }_1}\rangle =\{{\overline{0}}_{{\lambda }_1}\}$. Since $Y_{{\lambda }_1}=\{{\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4\}$, we obtain the following.

(i) 

Consider ${\lambda }_4\in Y_{{\lambda }_1}$. We obtain $\alpha ([\bigcup M_{{\overline{0}}_{{\lambda }_1}{\lambda }_4}\backslash {\displaystyle \bigcup _{\eta \in Y_{{\lambda }_4}^{{}^{\prime }}}}{G}_{\eta }{A}_{\eta ,{\lambda }_4}])=\alpha ([\{{\overline{0}}_{{\lambda }_4},{\overline{1}}_{{\lambda }_4},{\overline{2}}_{{\lambda }_4},{\overline{3}}_{{\lambda }_4}\}])=4$.

(ii) 

Consider ${\lambda }_3\in Y_{{\lambda }_1}$. We obtain $\alpha ([\bigcup M_{{\overline{0}}_{{\lambda }_1}{\lambda }_3}\backslash {\displaystyle \bigcup _{\eta \in Y_{{\lambda }_3}^{{}^{\prime }}}}{G}_{\eta }{A}_{\eta ,{\lambda }_3}])=\alpha ([\{{\overline{1}}_{{\lambda }_3},{\overline{2}}_{{\lambda }_3},{\overline{3}}_{{\lambda }_3}\}])=3$.

(iii) 

Consider ${\lambda }_2\in Y_{{\lambda }_1}$. We obtain $\alpha ([\bigcup M_{{\overline{0}}_{{\lambda }_1}{\lambda }_2}\backslash {\displaystyle \bigcup _{\eta \in Y_{{\lambda }_2}^{{}^{\prime }}}}{G}_{\eta }{A}_{\eta ,{\lambda }_2}])=\alpha ([\{{\overline{1}}_{{\lambda }_2},{\overline{2}}_{{\lambda }_2},{\overline{3}}_{{\lambda }_2}\}])=3$.

(iv) 

Consider ${\lambda }_1\in Y_{{\lambda }_1}$. We obtain $\alpha ([\bigcup M_{{\overline{0}}_{{\lambda }_1}{\lambda }_1}\backslash {\displaystyle \bigcup _{\eta \in Y_{{\lambda }_1}^{{}^{\prime }}}}{G}_{\eta }{A}_{\eta ,{\lambda }_1}])=0$.

We see that $\alpha (C_{{\overline{0}}_{{\lambda }_1}})={\displaystyle \sum _{\lambda \in Y_{{\lambda }_1}}}\alpha ([\bigcup M_{{\overline{0}}_{{\lambda }_1}\lambda }-{\displaystyle \bigcup _{\eta \in Y_{\lambda }^{{}^{\prime }}}}{G}_{\eta }{A}_{\eta ,\lambda }])=4+3+3=10$.

Proposition 2. 

Let $S=[Y;G_{\beta },f_{\lambda ,\eta }]$ be a Clifford semigroup in which consists a chain Y with the maximum and minimum elements, namely $m^{\prime }$ and m, respectively. We put $f_{\lambda ,\eta }(x)=e_{\eta }$ for all $x\in G_{\lambda }$, $\lambda >\eta \in Y$ where $e_{\eta }$ is the identity element of $G_{\eta }$. For every $\lambda \in Y$, let $G_{\lambda }=\langle h_{\lambda }\rangle$ for some $h_{\lambda }\in G_{\lambda }$ such that $h_{\lambda }\ne e_{\lambda }$ and $A=\{h_{\lambda }:\lambda \in Y\}$. Then $\alpha (C_{e_m})={\displaystyle \sum _{\lambda \in Y_m}}\alpha ([\bigcup M_{e_m\lambda }])$.

Proof. 

Let $X_{\lambda }$ be an $\alpha -$set of the induced subdigraph $[\bigcup M_{e_m\lambda }]$. We then define a set $X_{\lambda }^{{}^{\prime }}$ by $X_{\lambda }^{{}^{\prime }}:=X_{\lambda }{h}_{\lambda }^{-1}$ if $h_{\lambda }\in X_{\lambda }$, otherwise $X_{\lambda }^{{}^{\prime }}=X_{\lambda }$. Consider $X_{\lambda }^{{}^{\prime }}=X_{\lambda }{h}_{\lambda }^{-1}$. If $h_{\lambda }\in X_{\lambda }^{{}^{\prime }}$, we then obtain $h_{\lambda }=x_{\lambda }{h}_{\lambda }^{-1}$ for some $x_{\lambda }\in X_{\lambda }$. Thus $x_{\lambda }=h_{\lambda }{h}_{\lambda }$, which implies $(h_{\lambda },x_{\lambda })\in E([\bigcup M_{e_m\lambda }])$. It contradicts to the fact that $X_{\lambda }$ is an independent set of $[\bigcup M_{e_m\lambda }]$ and $x_{\lambda },h_{\lambda }\in X_{\lambda }$. Hence $h_{\lambda }\notin X_{\lambda }^{{}^{\prime }}$ for all $\lambda \in Y_m$. Additionally, if $(x_{\lambda }{h}^{-1},y_{\lambda }{h}_{\lambda }^{-1})\in E([\bigcup M_{e_m\lambda }])$, we obtain $y_{\lambda }{h}_{\lambda }^{-1}=(x_{\lambda }{h}^{-1})h_{\lambda }$ which implies $y_{\lambda }=x_{\lambda }{h}_{\lambda }$, a contradiction. Thus $X_{\lambda }^{{}^{\prime }}$ is an independent set of $[\bigcup M_{e_m\lambda }]$ for all $\lambda \in Y$. Now, we claim that ${\displaystyle \bigcup _{\lambda \in Y_m}}{X}_{\lambda }^{{}^{\prime }}$ is an independent set of $C_{e_m}$. Let $x_{\lambda }{h}_{\lambda }^{-1},x_{\eta }{h}_{\eta }^{-1}\in {\displaystyle \bigcup _{\lambda \in Y_m}}{X}_{\lambda }^{{}^{\prime }}$. Assume that $(x_{\lambda }{h}_{\lambda }^{-1},x_{\eta }{h}_{\eta }^{-1})\in E(C_{e_m})$ for some $\lambda ,\eta \in Y_m$ where $\lambda \ne \eta$. Then $x_{\eta }{h}_{\eta }^{-1}=(x_{\lambda }{h}_{\lambda }^{-1})h_{\gamma }$ for some $\gamma \in Y$. Because Y is a chain, $\gamma =\eta$. By the assumption, we obtain

$$\begin{array}{cc}\hfill x_{\eta }{h}_{\eta }^{-1}& =f_{\lambda ,\eta }(x_{\lambda }{h}_{\lambda }^{-1})f_{\eta ,\eta }(h_{\eta })\hfill \\ \hfill & =f_{\lambda ,\eta }(x_{\lambda })f_{\lambda ,\eta }(h_{\lambda }^{-1})f_{\eta ,\eta }(h_{\eta })\hfill \\ \hfill & =e_{\eta }{e}_{\eta }{h}_{\eta }=h_{\eta }.\hfill \end{array}$$

It is contradicts to the fact that $h_{\lambda }\notin X_{\lambda }^{{}^{\prime }}$ for all $\lambda \in Y_m$. Thus ${\displaystyle \bigcup _{\lambda \in Y_m}}{X}_{\lambda }^{{}^{\prime }}$ is an independent set of $C_{e_m}$. Hence we can conclude that $\alpha (C_{e_m})\ge |{\displaystyle \bigcup _{\lambda \in Y_m}}{X}_{\lambda }^{{}^{\prime }}|={\displaystyle \sum _{\lambda \in Y_m}}\alpha ([\bigcup M_{e_m\lambda }])$. By using the fact that $V({\displaystyle \bigcup _{\lambda \in Y_{\beta }}}[\bigcup M_{g_{\beta }\lambda }])=V(C_{e_m})$ and $E({\displaystyle \bigcup _{\lambda \in Y_{\beta }}}[\bigcup M_{g_{\beta }\lambda }])\subseteq E(C_{e_m})$, we obtain $\alpha ({\displaystyle \bigcup _{\lambda \in Y_{\beta }}}[\bigcup M_{g_{\beta }\lambda }])={\displaystyle \sum _{\lambda \in Y_m}}\alpha ([\bigcup M_{e_m\lambda }])\ge \alpha (C_{e_m})$. Therefore $\alpha (C_{e_m})={\displaystyle \sum _{\lambda \in Y_m}}\alpha ([\bigcup M_{e_m\lambda }])$, as required. □

5. Lower and Upper Bounds of the Weak Independence Numbers

In this section, we present the exact value of the weakly independent number of $C_{g_{\beta }}$ that based on the order of $M_{g_{\beta }\lambda }$ and the independence number of $Cay(\langle A_{\lambda }\rangle ,A_{\lambda }^{{}^{\prime }})$ where $A_{\lambda }^{{}^{\prime }}:=A_{\lambda }\backslash \{a\in A_{\lambda }:a^{-1}\notin A_{\lambda }\}$ for all $\lambda \in Y$. We start with some simple bounds for ${\alpha }_w(C_{g_{\beta }})$.

Lemma 9. 

$\alpha (C_{g_{\beta }})\le {\alpha }_w(C_{g_{\beta }})\le |V(C_{g_{\beta }})|$.

Proof. 

From the fact that every independent set is a weakly independent set. We obtain $\alpha (C_{g_{\beta }})\le {\alpha }_w(C_{g_{\beta }})$. For an upper bound, it is obvious that ${\alpha }_w(C_{g_{\beta }})\le |V(C_{g_{\beta }})|$.  □

By the definition of a Cayley digraphs, we have both $(x_{\lambda },y_{\lambda })$ and $(y_{\lambda },x_{\lambda })$ are belong to $E(Cay(\langle A_{\lambda }\rangle ,A_{\lambda }))$ if and only if there exists $a,b\in A_{\lambda }$ such that $y_{\lambda }=x_{\lambda }a$ and $x_{\lambda }=y_{\lambda }b$. Since $G_{\lambda }$ is a group, we obtain $b=a^{-1}$, from that property we can say that, in other words, if $y_{\lambda }=x_{\lambda }a$ and $a^{-1}\notin A_{\lambda }$, then $x_{\lambda }$ and $y_{\lambda }$ are weakly independent. We now construct an example and then obtain the sharpness of the lower and upper bounds in Lemma 9 as follows;

Proposition 3. 

Let $S=[Y;G_{\beta },f_{\lambda ,\eta }]$ be a Clifford semigroup in which consists an ordered set $Y=\{\lambda \}$. Then the following conditions hold:

1. 

if $A=\{h_{\lambda },h_{\lambda }^{-1}\}$, then ${\alpha }_w(C_{h_{\lambda }})=\alpha (C_{h_{\lambda }})$;

2. 

if $|G_{\lambda }| >2$ and $A=\{h_{\lambda }\}$ where $h_{\lambda }\ne h_{\lambda }^{-1}$, then ${\alpha }_w(C_{h_{\lambda }})=|V(C_{h_{\lambda }})|$.

Proof. 

• Let $A=\{h_{\lambda },h_{\lambda }^{-1}\}$. It is easy to check that if $(x_{\lambda },y_{\lambda })\in E(C_{h_{\lambda }})$ then $(y_{\lambda },x_{\lambda })\in E(C_{h_{\lambda }})$. Thus ${\alpha }_w(C_{h_{\lambda }})=\alpha (C_{h_{\lambda }})$.
• Let $A=\{h_{\lambda }\}$. Let $(x_{\lambda },y_{\lambda })\in E(C_{h_{\lambda }})$. Then $y_{\lambda }=x_{\lambda }{h}_{\lambda }$. Assume that $x_{\lambda }=y_{\lambda }{h}_{\lambda }$. Thus $x_{\lambda }{h}_{\lambda }=y_{\lambda }$ which implies $h_{\lambda }=h_{\lambda }^{-1}$, a contradiction. We conclude that if $(x_{\lambda },y_{\lambda })\in E(C_{h_{\lambda }})$ then $(y_{\lambda },x_{\lambda })\notin E(C_{h_{\lambda }})$ for every $x_{\lambda },y_{\lambda }\in V(C_{h_{\lambda }})$ which implies $V(C_{h_{\lambda }})$ is weakly independent set. Therefore ${\alpha }_w(C_{h_{\lambda }})=|V(C_{h_{\lambda }})|$, as required.

 □

Let G be a group and $\varnothing \ne A\subseteq G$. We next present a result on the weak independence number of $Cay(G,A)$ as follows.

Lemma 10. 

Let G be a group and $\varnothing \ne A\subseteq G$. Then ${\alpha }_w(Cay(G,A))={\alpha }_w(Cay(G,A^{{}^{\prime }}))=\alpha (Cay(G,A^{{}^{\prime }}))$ where $A^{{}^{\prime }}:=A\backslash \{a\in A:a^{-1}\notin A\}$.

Proof. 

We first show that ${\alpha }_w(Cay(G,A))={\alpha }_w(Cay(G,A^{{}^{\prime }}))$. Since $A^{{}^{\prime }}\subseteq A$, we obtain $E(Cay(G,A^{{}^{\prime }}))\subseteq E(Cay(G,A))$. Thus ${\alpha }_w(Cay(G,A))\le {\alpha }_w(Cay(G,A^{{}^{\prime }}))$.

Conversely, we let $X^{{}^{\prime }}$ be a weakly independent set of $Cay(G,A^{{}^{\prime }})$ and $x,y\in X^{{}^{\prime }}$. We claim that $X^{{}^{\prime }}$ is a weakly independent set of $Cay(G,A)$. Assume that $(x,y),(y,x)\in E(Cay(G,A))$. Then there exist $a,a^{-1}\in A$ such that $y=xa$ and $x=y{a}^{-1}$. Thus $a,a^{-1}\in A^{{}^{\prime }}$ and it follows that $(x,y),(y,x)\in E(Cay(G,A^{{}^{\prime }}))$, a contradiction, because $X^{{}^{\prime }}$ is a weakly independent set and $x,y\in X^{{}^{\prime }}$. Hence $X^{{}^{\prime }}$ is a weakly independent set of $Cay(G,A)$ and then ${\alpha }_w(Cay(G,A^{{}^{\prime }}))\le {\alpha }_w(Cay(G,A))$. Therefore ${\alpha }_w(Cay(G,A^{{}^{\prime }}))={\alpha }_w(Cay(G,A))$.

Next, we will show that ${\alpha }_w(Cay(G,A^{{}^{\prime }}))=\alpha (Cay(G,A^{{}^{\prime }}))$. Clearly, every independent set is a weakly independent set. Thus $\alpha (Cay(G,A^{{}^{\prime }}))\le {\alpha }_w(Cay(G,A^{{}^{\prime }}))$.

Conversely, by the definition of $A{}^{\prime }$, we obtain $(x,y)\in E(Cay(G,A^{{}^{\prime }}))$ if and only if $(y,x)\in E(Cay(G,A^{{}^{\prime }}))$. Let X be a weakly independent set of $Cay(G,A^{{}^{\prime }})$. We then obtain $x,y$ are independent for every $x,y\in X$, which implies X is an independent set of $Cay(G,A^{{}^{\prime }})$. Hence ${\alpha }_w(Cay(G,A^{{}^{\prime }}))\le \alpha (Cay(G,A^{{}^{\prime }}))$. Thus ${\alpha }_w(Cay(G,A^{{}^{\prime }}))=\alpha (Cay(G,A^{{}^{\prime }}))$. □

By the definition of $M_{g_{\beta }\lambda }$, we then obtain $\left[\bigcup M_{g_{\beta }\lambda }\right]={\displaystyle \bigcup _{i\in I^{\prime }}^{·}}{D}_i$ where $D_i\cong [g_i\langle A_{\lambda }\rangle ]\cong Cay(\langle A_{\lambda }\rangle ,A_{\lambda })$ for all $i\in I^{\prime }$ where $I^{\prime }=\{1,2,\dots ,|M_{g_{\beta }\lambda }|\}$.

According to the above lemma, we consequently obtain ${\alpha }_w(Cay(\langle A_{\lambda }\rangle ,A_{\lambda }))={\alpha }_w(Cay(\langle A_{\lambda }\rangle ,A_{\lambda }^{{}^{\prime }}))=\alpha (Cay(\langle A_{\lambda }\rangle ,A_{\lambda }^{{}^{\prime }}))$.

We now present the exact value of the weakly independent number of $C_{g_{\beta }}$ in the form of the summation of $|M_{g_{\beta }\lambda }|{\alpha }_w(Cay(\langle A_{\lambda }\rangle ,A_{\lambda }))$ for all $\lambda \in Y_{\beta }$.

Lemma 11. 

${\alpha }_w(C_{g_{\beta }})={\displaystyle \sum _{\lambda \in Y_{\beta }}}|M_{g_{\beta }\lambda }|{\alpha }_w(Cay(\langle A_{\lambda }\rangle ,A_{\lambda }))$.

Proof. 

Let X be an ${\alpha }_w-$set of $C_{g_{\beta }}$. We will show that $|X|\le {\displaystyle \sum _{\lambda \in Y_{\beta }}}|M_{g_{\beta }\lambda }|{\alpha }_w(Cay(\langle A_{\lambda }\rangle ,A_{\lambda }))$. From $\left[\bigcup M_{g_{\beta }\lambda }\right]={\displaystyle \bigcup _{i\in I^{\prime }}^{·}}{D}_i$ where $D_i\cong [g_i\langle A_{\lambda }\rangle ]\cong Cay(\langle A_{\lambda }\rangle ,A_{\lambda })$ for all $i\in I^{\prime }=\{1,2,\dots ,|M_{g_{\beta }\lambda }|\}$, we then obtain $|X\cap (\bigcup M_{g_{\beta }\lambda })| ={\displaystyle \sum _{g_{\lambda }\langle A_{\lambda }\rangle \in M_{g_{\beta }\lambda }}}|X\cap g_{\lambda }\langle A_{\lambda }\rangle |$ and $|X\cap g_{\lambda }\langle A_{\lambda }\rangle | \le |X_{g_{\lambda }\langle A_{\lambda }\rangle }|$ for all $g_{\lambda }\langle A_{\lambda }\rangle \in M_{g_{\beta }\lambda }$ where $X_{g_{\lambda }\langle A_{\lambda }\rangle }$ is an ${\alpha }_w-$set of $[g_{\lambda }\langle A_{\lambda }\rangle ]$. Moreover, we conclude that ${\displaystyle \sum _{\begin{array}{c}{g}_{\lambda }\langle A_{\lambda }\rangle \in M_{g_{\beta }\lambda }\\ \lambda \in Y_{\beta }\end{array}}}|X\cap g_{\lambda }\langle A_{\lambda }\rangle | \le {\displaystyle \sum _{\begin{array}{c}{g}_{\lambda }\langle A_{\lambda }\rangle \in M_{g_{\beta }\lambda }\\ \lambda \in Y_{\beta }\end{array}}}|X_{g_{\lambda }\langle A_{\lambda }\rangle }|$. Since $X\subseteq {\displaystyle \bigcup _{\lambda \in Y_{\beta }}}(\bigcup M_{g_{\beta }\lambda })$, we obtain $|X|=|X\cap ({\displaystyle \bigcup _{\lambda \in Y_{\beta }}}(\bigcup M_{g_{\beta }\lambda }))| ={\displaystyle \sum _{\begin{array}{c}{g}_{\lambda }\langle A_{\lambda }\rangle \in M_{g_{\beta }\lambda }\\ \lambda \in Y_{\beta }\end{array}}}|X\cap g_{\lambda }\langle A_{\lambda }\rangle | \le {\displaystyle \sum _{\begin{array}{c}{g}_{\lambda }\langle A_{\lambda }\rangle \in M_{g_{\beta }\lambda }\\ \lambda \in Y_{\beta }\end{array}}}|X_{g_{\lambda }\langle A_{\lambda }\rangle }| ={\displaystyle \sum _{\lambda \in Y_{\beta }}}|M_{g_{\beta }\lambda }|{\alpha }_w(Cay(\langle A_{\lambda }\rangle ,A_{\lambda }))$. Thus ${\alpha }_w(C_{g_{\beta }})=|X|$ $\le {\displaystyle \sum _{\lambda \in Y_{\beta }}}|M_{g_{\beta }\lambda }|{\alpha }_w(Cay(\langle A_{\lambda }\rangle ,A_{\lambda }))$.

Conversely, we know that $\left[\bigcup M_{g_{\beta }\lambda }\right]$ is a disjoint union of $[g_{\lambda }\langle A_{\lambda }\rangle ]$ for all $g_{\lambda }\langle A_{\lambda }\rangle \in M_{g_{\beta }\lambda }$, we obtain ${\displaystyle \bigcup _{g_{\lambda }\langle A_{\lambda }\rangle \in M_{g_{\beta }\lambda }}}{X}_{g_{\lambda }\langle A_{\lambda }\rangle }$ is a weakly independent set of $C_{g_{\beta }}$ where $X_{g_{\lambda }\langle A_{\lambda }\rangle }$ is a weakly independent set of $[g_{\lambda }\langle A_{\lambda }\rangle ]$. From the fact that if $u\in G_{\lambda }$ and $v\in G_{\eta }$ where $\lambda \ne \eta$, then $u,v$ are weakly independent, we obtain ${\displaystyle \bigcup _{\begin{array}{c}{g}_{\lambda }\langle A_{\lambda }\rangle \in M_{g_{\beta }\lambda }\\ \lambda \in Y_{\beta }\end{array}}}{X}_{g_{\lambda }\langle A_{\lambda }\rangle }$ is a weakly independent set of $C_{g_{\beta }}$. Since $[g_{\lambda }\langle A_{\lambda }\rangle ]\cong Cay(\langle A_{\lambda }\rangle ,A_{\lambda })$ for all $g_{\lambda }\langle A_{\lambda }\rangle \in M_{g_{\beta }\lambda }$, we obtain $|X_{g_{\lambda }\langle A_{\lambda }\rangle }| ={\alpha }_w(Cay(\langle A_{\lambda }\rangle ,A_{\lambda }))$ for all $g_{\lambda }\langle A_{\lambda }\rangle \in M_{g_{\beta }\lambda }$ and $\lambda \in Y_{\beta }$. Hence ${\displaystyle \sum _{\begin{array}{c}{g}_{\lambda }\langle A_{\lambda }\rangle \in M_{g_{\beta }\lambda }\\ \lambda \in Y_{\beta }\end{array}}}|X_{g_{\lambda }\langle A_{\lambda }\rangle }| ={\displaystyle \sum _{\lambda \in Y_{\beta }}}|M_{g_{\beta }\lambda }|{\alpha }_w(Cay(\langle A_{\lambda }\rangle ,A_{\lambda }))$ and so ${\alpha }_w(C_{g_{\beta }})\ge {\displaystyle \sum _{\lambda \in Y_{\beta }}}|M_{g_{\beta }\lambda }|{\alpha }_w(Cay(\langle A_{\lambda }\rangle ,A_{\lambda }))$. □

Example 4. 

Let $Y=\{{\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4\}$ be a semilattice with a partial order that represented by the Hasse diagram in Figure 4. For $I=\{1,2,3,4\}$, we let $\{G_{{\lambda }_i}:i\in I\}$ be a family of groups, indexed by the semilattice Y where $G_{{\lambda }_i}={\mathbb{Z}}_4=\{{\overline{0}}_{{\lambda }_i},{\overline{1}}_{{\lambda }_i},{\overline{2}}_{{\lambda }_i},{\overline{3}}_{{\lambda }_i}\}$ is an additive group of integers modulo 4 for all $i\in I$. Let $f_{{\lambda }_i,{\lambda }_j}({\overline{x}}_{{\lambda }_i})={\overline{x}}_{{\lambda }_j}$ for every ${\overline{x}}_{{\lambda }_i}\in G_{{\lambda }_i}$, ${\overline{x}}_{{\lambda }_j}\in G_{{\lambda }_j}$ and $i>j$. Then $S=[Y;G_{{\lambda }_i},f_{{\lambda }_i,{\lambda }_j}]$ is a Clifford semigroup.

Let $A=\{{\overline{2}}_{{\lambda }_2}\}$. Then we picture $Cay(S,A)$ in Figure 5. From $A_{{\lambda }_2}=\{{\overline{2}}_{{\lambda }_2}\}$, then $\langle A_{{\lambda }_2}\rangle =\{{\overline{0}}_{{\lambda }_2},{\overline{2}}_{{\lambda }_2}\}$ which implies $G_{{\lambda }_2}/\langle A_{{\lambda }_2}\rangle =\{\langle A_{{\lambda }_2}\rangle ,{\overline{1}}_{{\lambda }_2}\langle A_{{\lambda }_2}\rangle \}$. We here consider $C_{{\overline{0}}_{{\lambda }_2}}$ and each ${\lambda }_2,{\lambda }_4\in Y_{{\lambda }_2}$ as follows.

(i) 

Consider ${\lambda }_4\in Y_{{\lambda }_2}$.

• Since $A=\{{\overline{2}}_{{\lambda }_2}\}$, we have $A_{{\lambda }_4}=\varnothing$. Thus we put $A_{{\lambda }_4}=\{{\overline{0}}_{{\lambda }_4}\}$.
• Then $M_{{\overline{0}}_{{\lambda }_2}{\lambda }_4}=\{g_{{\lambda }_4}\langle {\overline{0}}_{{\lambda }_4}\rangle \in G_{{\lambda }_4}/\langle {\overline{0}}_{{\lambda }_4}\rangle :f_{{\lambda }_4,{\lambda }_2}(g_{{\lambda }_4})\in {\overline{0}}_{{\lambda }_2}\langle A_{{\lambda }_2}\rangle \}=\{{\overline{0}}_{{\lambda }_4}\langle {\overline{0}}_{{\lambda }_4}\rangle ,{\overline{2}}_{{\lambda }_4}\langle {\overline{0}}_{{\lambda }_4}\rangle \}$.
• From $A_{{\lambda }_4}=\{{\overline{0}}_{{\lambda }_4}\}$, we obtain ${\alpha }_w(Cay(\langle A_{{\lambda }_4}\rangle ,A_{{\lambda }_4}))={\alpha }_w(Cay(\{{\overline{0}}_{{\lambda }_4}\},\{{\overline{0}}_{{\lambda }_4}\}))=1$.
• Therefore $|M_{{\overline{0}}_{{\lambda }_2}{\lambda }_4}|{\alpha }_w(Cay(\{{\overline{0}}_{{\lambda }_4}\},\{{\overline{0}}_{{\lambda }_4}\}))=2$.

(ii) 

Consider ${\lambda }_2\in Y_{{\lambda }_2}$.

• We have $M_{{\overline{0}}_{{\lambda }_2}{\lambda }_2}=\{g_{{\lambda }_2}\langle A_{{\lambda }_2}\rangle \in G_{{\lambda }_4}/\langle A_{{\lambda }_4}\rangle :f_{{\lambda }_2,{\lambda }_2}(g_{{\lambda }_2})\in {\overline{0}}_{{\lambda }_2}\langle A_{{\lambda }_2}\rangle \}=\{\langle A_{{\lambda }_2}\rangle \}$.
• From $\langle A_{{\lambda }_2}\rangle =\{{\overline{0}}_{{\lambda }_2},{\overline{2}}_{{\lambda }_2}\}$, we obtain ${\alpha }_w(Cay(\langle A_{{\lambda }_2}\rangle ,A_{{\lambda }_2}))=1$.
• Therefore $|M_{{\overline{0}}_{{\lambda }_2}{\lambda }_2}|{\alpha }_w(Cay(\langle A_{{\lambda }_2}\rangle ,A_{{\lambda }_2}))=1$.

Therefore ${\alpha }_w(C_{{\overline{0}}_{{\lambda }_2}})=|M_{{\overline{0}}_{{\lambda }_2}{\lambda }_4}|{\alpha }_w(Cay(\{{\overline{0}}_{{\lambda }_4}\},\{{\overline{0}}_{{\lambda }_4}\}))+|M_{{\overline{0}}_{{\lambda }_2}{\lambda }_2}|{\alpha }_w(Cay(\langle A_{{\lambda }_2}\rangle ,A_{{\lambda }_2}))=3$. Similarly,

• ${\alpha }_w(C_{{\overline{1}}_{{\lambda }_2}})= |M_{{\overline{1}}_{{\lambda }_2}{\lambda }_4}|{\alpha }_w(Cay(\{{\overline{0}}_{{\lambda }_4}\},\{{\overline{0}}_{{\lambda }_4}\}))+|M_{{\overline{1}}_{{\lambda }_2}{\lambda }_2}|{\alpha }_w(Cay(\langle A_{{\lambda }_2}\rangle ,A_{{\lambda }_2}))=3,$
• ${\alpha }_w(C_{{\overline{0}}_{{\lambda }_1}})= |M_{{\overline{0}}_{{\lambda }_1}{\lambda }_3}|{\alpha }_w(Cay(\{{\overline{0}}_{{\lambda }_3}\},\{{\overline{0}}_{{\lambda }_3}\}))+|M_{{\overline{0}}_{{\lambda }_1}{\lambda }_1}|{\alpha }_w(Cay(\langle A_{{\lambda }_1}\rangle ,A_{{\lambda }_1}))=3,$
• ${\alpha }_w(C_{{\overline{1}}_{{\lambda }_1}})= |M_{{\overline{1}}_{{\lambda }_1}{\lambda }_3}|{\alpha }_w(Cay(\{{\overline{0}}_{{\lambda }_3}\},\{{\overline{0}}_{{\lambda }_3}\}))+|M_{{\overline{1}}_{{\lambda }_1}{\lambda }_1}|{\alpha }_w(Cay(\langle A_{{\lambda }_1}\rangle ,A_{{\lambda }_1}))=3.$

By Lemma 10, we directly obtain the following corollary.

Corollary 1. 

${\alpha }_w(C_{g_{\beta }})={\displaystyle \sum _{\lambda \in Y_{\beta }}}|M_{g_{\beta }\lambda }|{\alpha }_w(Cay(\langle A_{\lambda }\rangle ,A_{\lambda }^{\prime }))={\displaystyle \sum _{\lambda \in Y_{\beta }}}|M_{g_{\beta }\lambda }|\alpha (Cay(\langle A_{\lambda }\rangle ,A_{\lambda }^{\prime }))$.

In summary, we now obtain the weak independence number of $Cay(S,A)$, which is presented in terms of ${\alpha }_w(Cay(\langle A_{\lambda }\rangle ,A_{\lambda }))$ for all $\lambda \in Y_{\beta }$.

Theorem 4. 

${\alpha }_w(Cay(S,A))={\displaystyle \sum _{\begin{array}{c}g\in {\mathcal{A}}_{\beta }\\ \beta \in B\end{array}}}({\alpha }_w(C_{g_{\beta }}))={\displaystyle \sum _{\begin{array}{c}g\in {\mathcal{A}}_{\beta }\\ \beta \in B\end{array}}}({\displaystyle \sum _{\lambda \in Y_{\beta }}}|M_{g_{\beta }\lambda }|{\alpha }_w(Cay(\langle A_{\lambda }\rangle ,A_{\lambda })))$.

Example 5. 

From the Example 4, $Cay(S,A)$ consists of four components, $C_{{\overline{0}}_{{\lambda }_2}}$, $C_{{\overline{1}}_{{\lambda }_2}}$, $C_{{\overline{0}}_{{\lambda }_1}}$ and $C_{{\overline{1}}_{{\lambda }_1}}$. We see that $X=\{{\overline{0}}_{{\lambda }_4},{\overline{1}}_{{\lambda }_4},{\overline{2}}_{{\lambda }_4},{\overline{3}}_{{\lambda }_4},{\overline{0}}_{{\lambda }_2},{\overline{1}}_{{\lambda }_2},{\overline{0}}_{{\lambda }_3},{\overline{1}}_{{\lambda }_3},{\overline{2}}_{{\lambda }_3},{\overline{3}}_{{\lambda }_3},{\overline{0}}_{{\lambda }_1},{\overline{1}}_{{\lambda }_1}\}$ is an ${\alpha }_w-$set of $Cay(S,A)$. Then ${\alpha }_w(Cay(S,A))={\alpha }_w(C_{{\overline{0}}_{{\lambda }_2}})+{\alpha }_w(C_{{\overline{1}}_{{\lambda }_2}})+{\alpha }_w(C_{{\overline{0}}_{{\lambda }_1}})+{\alpha }_w(C_{{\overline{1}}_{{\lambda }_1}})=12=|X|$.

6. The Path Independence Numbers

From the fact that, for every $\lambda \in Y$, $Cay(\langle A_{\lambda }\rangle ,A_{\lambda })$ is strongly connected. By this information, we can conclude that ${\alpha }_p(Cay(\langle A_{\lambda }\rangle ,A_{\lambda }))=1$. However, to find the path independence number of $C_{g_{\beta }}$, we need to consider a path between induced subdigraphs $[g_{\lambda }\langle A_{\lambda }\rangle ]$ and $[h_{\eta }\langle A_{\eta }\rangle ]$ of $Cay(S,A)$ where $g_{\lambda }\langle A_{\lambda }\rangle \in M_{g_{\beta }\lambda }$ and $h_{\eta }\langle A_{\eta }\rangle \in M_{g_{\beta }\eta }$.

We here investigate a relation that indicates all paths between $[g_{\lambda }\langle A_{\lambda }\rangle ]$ and $[h_{\eta }\langle A_{\eta }\rangle ]$ in $C_{g_{\beta }}$. Let ${\tilde{C}}_{g_{\beta }}:={\displaystyle \bigcup _{\lambda \in Y_{\beta }}}{M}_{g_{\beta }\lambda }$ and ${\tilde{Y}}_{\eta }:=\{\lambda \in Y:\lambda \wedge ({\displaystyle \underset{\gamma \in {\mathsf{\Gamma }}_{\lambda }}{\bigwedge }}\gamma )=\eta \mathrm{for}\mathrm{some}{\mathsf{\Gamma }}_{\lambda }\subseteq Y^{\prime }\}\cup \{\eta \}$. Define a relation ∼ on ${\tilde{C}}_{g_{\beta }}$ by $g_{\lambda }\langle A_{\lambda }\rangle \sim h_{\eta }\langle A_{\eta }\rangle$ if and only if $f_{\lambda ,\eta }(g_{\lambda })\in h_{\eta }\langle A_{\eta }\rangle$ and $\lambda \in {\tilde{Y}}_{\eta }$.

Lemma 12. 

$({\tilde{C}}_{g_{\beta }},\sim )$ is a partially ordered set.

Proof. 

Let $\beta \in B$. We shall show that ∼ is a partial order on ${\tilde{C}}_{g_{\beta }}$.

(i) 

Since $f_{\lambda ,\lambda }(g_{\lambda })\in g_{\lambda }\langle A_{\lambda }\rangle$ and $\lambda \in {\tilde{Y}}_{\lambda }$, we then obtain $g_{\lambda }\langle A_{\lambda }\rangle \sim g_{\lambda }\langle A_{\lambda }\rangle$.

(ii) 

Let $g_{\lambda }\langle A_{\lambda }\rangle \sim h_{\eta }\langle A_{\eta }\rangle$ and $h_{\eta }\langle A_{\eta }\rangle \sim g_{\lambda }\langle A_{\lambda }\rangle$. Then $f_{\lambda ,\eta }(g_{\lambda })\in h_{\eta }\langle A_{\eta }\rangle$ and $f_{\eta ,\lambda }(h_{\eta })\in g_{\lambda }\langle A_{\lambda }\rangle$. This gives $\eta \le \lambda$ and $\lambda \le \eta$. From Y is a semilattice and $\lambda ,\eta \in Y$, we obtain $\lambda =\eta$. Since $h_{\eta }\langle A_{\eta }\rangle =h_{\lambda }\langle A_{\lambda }\rangle$ and $f_{\eta ,\lambda }(h_{\eta })=f_{\lambda ,\lambda }(h_{\lambda })\in g_{\lambda }\langle A_{\lambda }\rangle$, it follows that $g_{\lambda }\langle A_{\lambda }\rangle =h_{\eta }\langle A_{\eta }\rangle$.

(ii) 

Let $g_{\lambda }\langle A_{\lambda }\rangle \sim h_{\eta }\langle A_{\eta }\rangle$ and $h_{\eta }\langle A_{\eta }\rangle \sim k_{\kappa }\langle A_{\kappa }\rangle$. Then $f_{\lambda ,\eta }(g_{\lambda })\in h_{\eta }\langle A_{\eta }\rangle$ and $f_{\eta ,\kappa }(h_{\eta })\in k_{\kappa }\langle A_{\kappa }\rangle$. From Lemma 4, we obtain $f_{\eta ,\kappa }(h_{\eta }\langle A_{\eta }\rangle )\subseteq f_{\eta ,\kappa }(h_{\eta })\langle A_{\kappa }\rangle$. Thus $f_{\eta ,\kappa }(h_{\eta })\in f_{\eta ,\kappa }(h_{\eta })\langle A_{\kappa }\rangle$ and so $f_{\eta ,\kappa }(h_{\eta })\langle A_{\kappa }\rangle =k_{\kappa }\langle A_{\kappa }\rangle$. Therefore $f_{\lambda ,\kappa }(g_{\lambda })=f_{\eta ,\kappa }(f_{\lambda ,\eta }(g_{\lambda }))\in f_{\eta ,\kappa }(h_{\eta }\langle A_{\eta }\rangle )\subseteq f_{\eta ,\kappa }(h_{\eta })\langle A_{\kappa }\rangle =k_{\kappa }\langle A_{\kappa }\rangle$.

From $\lambda \in {\tilde{Y}}_{\eta }$ and $\eta \in {\tilde{Y}}_{\kappa }$, there exist two subsets of $Y^{\prime }$, denoted by ${\mathsf{\Gamma }}_{\lambda }$ and ${\mathsf{\Gamma }}_{\eta }$, such that $\lambda \wedge ({\displaystyle \underset{\gamma \in {\mathsf{\Gamma }}_{\lambda }}{\bigwedge }}\gamma )=\eta$ and $\eta \wedge ({\displaystyle \underset{\gamma \in {\mathsf{\Gamma }}_{\eta }}{\bigwedge }}\gamma )=\kappa$. Then $\lambda \wedge ({\displaystyle \underset{\gamma \in {\mathsf{\Gamma }}_{\lambda }\cup {\mathsf{\Gamma }}_{\eta }}{\bigwedge }}\gamma )=\kappa$ and thus $\lambda \in {\tilde{Y}}_{\kappa }$.

Therefore $({\tilde{C}}_{g_{\beta }},\sim )$ is a partially ordered set. □

Lemma 13. 

Let $x\in g_{\lambda }\langle A_{\lambda }\rangle$ and $y\in h_{\eta }\langle A_{\eta }\rangle$ where $\lambda \ne \eta$. Then there is a path from x to y if and only if $g_{\lambda }\langle A_{\lambda }\rangle \sim h_{\eta }\langle A_{\eta }\rangle$.

Proof. 

Assume that there exists a path from x to y. Then we let $x=v_1,v_2,\dots ,v_n=y$ be a path from x to y where $(v_i,v_{i+1})\in E(Cay(S,A))$ for all $i=1,2,\dots n-1$. It follows that there exist $a_1,a_2,\dots ,a_{n-1}\in A$ and ${\gamma }_1,{\gamma }_2,\dots ,{\gamma }_{n-1}\in Y^{\prime }$ such that $v_{i+1}=v_i{a}_i$ and $a_i\in G_{{\gamma }_i}$ for all $i=1,2,\dots ,n-1$. This gives

$$\begin{array}{cc}\hfill v_n& =(\dots ((v_1{a}_1)a_2)\dots )a_{n-1}\hfill \\ & =(\dots ((x{a}_1)a_2)\dots )a_{n-1}\hfill \\ & =(\dots ((f_{\lambda ,\lambda \wedge {\gamma }_1}(x)f_{{\gamma }_1,\lambda \wedge {\gamma }_1}(a_1))a_2)\dots )a_{n-1}\hfill \\ & =(\dots ((f_{\lambda \wedge {\gamma }_1,(\lambda \wedge {\gamma }_1)\wedge {\gamma }_2}(f_{\lambda ,\lambda \wedge {\gamma }_1}(x)f_{{\gamma }_1,\lambda \wedge {\gamma }_1}(a_1))f_{{\gamma }_2,(\lambda \wedge {\gamma }_1)\wedge {\gamma }_2}(a_2))\dots )a_{n-1}\hfill \\ & ⋮\hfill \\ & =f_{\lambda \wedge ({\displaystyle \underset{i=1}{\overset{n-2}{\bigwedge }}}{\gamma }_i),\lambda \wedge ({\displaystyle \underset{i=1}{\overset{n-2}{\bigwedge }}}{\gamma }_i)\wedge {\gamma }_{n-1}}(\dots (f_{\lambda ,\lambda \wedge {\gamma }_1}(x)f_{{\gamma }_1,\lambda \wedge {\gamma }_1}(a_1))\dots )f_{{\gamma }_{n-1},\lambda \wedge ({\displaystyle \underset{i=1}{\overset{n-2}{\bigwedge }}}{\gamma }_i)\wedge {\gamma }_{n-1}}(a_{n-1})\hfill \\ & =f_{\lambda ,\eta }(x)f_{{\gamma }_1,\eta }(a_1)f_{{\gamma }_2,\eta }(a_2)\dots f_{{\gamma }_{n-1},\eta }(a_{n-1})\hfill \\ & \in f_{\lambda ,\eta }(x)\langle A_{\eta }\rangle .\hfill \end{array}$$

Since $v_n=y\in h_{\eta }\langle A_{\eta }\rangle$, we obtain $f_{\lambda ,\eta }(x)\langle A_{\eta }\rangle =h_{\eta }\langle A_{\eta }\rangle$. From Lemma 4, we obtain $f_{\lambda ,\eta }(g_{\lambda }\langle A_{\lambda }\rangle )=f_{\lambda ,\eta }(x\langle A_{\lambda }\rangle )\subseteq f_{\lambda ,\eta }(x)\langle A_{\eta }\rangle =h_{\eta }\langle A_{\eta }\rangle$ which implies $f_{\lambda ,\eta }(g_{\lambda })\in h_{\eta }\langle A_{\eta }\rangle$. Therefore $g_{\lambda }\langle A_{\lambda }\rangle \sim h_{\eta }\langle A_{\eta }\rangle$.

Conversely, suppose that $g_{\lambda }\langle A_{\lambda }\rangle \sim h_{\eta }\langle A_{\eta }\rangle$. Then $f_{\lambda ,\eta }(g_{\lambda })\in h_{\eta }\langle A_{\eta }\rangle$ and $\lambda \in {\tilde{Y}}_{\eta }$, i.e., there exists ${\mathsf{\Gamma }}_{\lambda }\subseteq Y^{\prime }$ such that $\lambda \wedge ({\displaystyle \underset{\gamma \in {\mathsf{\Gamma }}_{\lambda }}{\bigwedge }}\gamma )=\eta$. Now, we let ${\mathsf{\Gamma }}_{\lambda }=\{{\gamma }_1,{\gamma }_2,\dots ,{\gamma }_n\}$ and $a_1,a_2,\dots a_n\in A$. Then $g_{\lambda },g_{\lambda }{a}_1,g_{\lambda }{a}_1{a}_2,\dots ,g_{\lambda }{a}_1{a}_2\dots a_n$ is a path from $g_{\lambda }$ to $g_{\lambda }{a}_1{a}_2\dots a_n$. Consequently, we conclude that

$$\begin{array}{cc}\hfill g_{\lambda }{a}_1{a}_2\dots a_n& =f_{\lambda ,\lambda \wedge ({\displaystyle \underset{i=1}{\overset{n}{\bigwedge }}}{\gamma }_i)}(g_{\lambda })f_{{\gamma }_1,\lambda \wedge ({\displaystyle \underset{i=1}{\overset{n}{\bigwedge }}}{\gamma }_i)}(a_{{\gamma }_1})f_{{\gamma }_2,\lambda \wedge ({\displaystyle \underset{i=1}{\overset{n}{\bigwedge }}}{\gamma }_i)}(a_{{\gamma }_2})\dots f_{{\gamma }_n,\lambda \wedge ({\displaystyle \underset{i=1}{\overset{n}{\bigwedge }}}{\gamma }_i)}(a_{{\gamma }_n})\hfill \\ & \in f_{\lambda ,\eta }(g_{\lambda })\langle A_{\eta }\rangle .\hfill \end{array}$$

Since $f_{\lambda ,\eta }(g_{\lambda })\in h_{\eta }\langle A_{\eta }\rangle$, we obtain $f_{\lambda ,\eta }(g_{\lambda })\langle A_{\eta }\rangle =h_{\eta }\langle A_{\eta }\rangle$. From the fact that $[g_{\lambda }\langle A_{\lambda }\rangle ]$ and $[h_{\eta }\langle A_{\eta }\rangle ]$ are strongly connected, there exist a path from x to $g_{\lambda }$ and a path from $g_{\lambda }{a}_1{a}_2\dots a_n$ to y. Therefore there exists a path from x to y, as required. □

By using an anti-symmetric property, we can conclude that if $\lambda \ne \eta$ and there is a path from $x\in g_{\lambda }\langle A_{\lambda }\rangle$ to $y\in h_{\eta }\langle A_{\eta }\rangle$, then there is no path from y to x. Now, we are ready to give the path independence number of $C_{g_{\beta }}$.

Lemma 14. 

${\alpha }_p(C_{g_{\beta }})=max\{|X|:Xisananti-chainin({\tilde{C}}_{g_{\beta }},\sim )\}$.

Proof. 

Let $X^{\prime }$ be an ${\alpha }_p-$set of $C_{g_{\beta }}$ and $x,y\in X^{\prime }$. Since $x\in g\langle A_{\lambda }\rangle$ and $y\in h\langle A_{\eta }\rangle$ for some $g\in G_{\lambda }$, $h\in G_{\eta }$, we obtain $x\langle A_{\lambda }\rangle \ne y\langle A_{\eta }\rangle$ because $[x\langle A_{\lambda }\rangle ]$ is strongly connected. By Lemma 13, we have $x\langle A_{\lambda }\rangle \nsim y\langle A_{\eta }\rangle$ and $y\langle A_{\eta }\rangle \nsim x\langle A_{\lambda }\rangle$. Thus $X=\{g\langle A_{\lambda }\rangle \in {\tilde{C}}_{g_{\beta }}:x\in X^{\prime }\}$ is an anti-chain in $({\tilde{C}}_{g_{\beta }},\sim )$ and $|X^{\prime }| =|X|$. Therefore ${\alpha }_p(C_{g_{\beta }})\le \max \{|X|:X\mathrm{is}\mathrm{an}\mathrm{anti}-\mathrm{chain}\mathrm{in}({\tilde{C}}_{g_{\beta }},\sim )\}$.

Conversely, we know that for every path independent set $X^{\prime }$ of $C_{g_{\beta }}$, there exists $X=\{x\langle A_{\lambda }\rangle \in {\tilde{C}}_{g_{\beta }}:x\in X^{\prime }\}$ where X is an anti-chain in $({\tilde{C}}_{g_{\beta }},\sim )$, and $|X^{\prime }|=|X|$. Hence ${\alpha }_p(C_{g_{\beta }})\ge \max \{|X|:X\mathrm{is}\mathrm{an}\mathrm{anti}-\mathrm{chain}\mathrm{in}({\tilde{C}}_{g_{\beta }},\sim )\}$. □

In summary, the path independence number of $Cay(S,A)$ will be obtained in the form of the summation of $\max \{|X|:X\mathrm{is}\mathrm{an}\mathrm{anti}-\mathrm{chain}\mathrm{in}({\tilde{C}}_{g_{\beta }},\sim )\}$ for all $g\in {\mathcal{A}}_{\beta }$ and $\beta \in B$.

Theorem 5. 

${\alpha }_p(Cay(S,A))={\displaystyle \sum _{\begin{array}{c}g\in {\mathcal{A}}_{\beta }\\ \beta \in B\end{array}}}({\alpha }_p(C_{g_{\beta }}))={\displaystyle \sum _{\begin{array}{c}g\in {\mathcal{A}}_{\beta }\\ \beta \in B\end{array}}}max\{|X|:Xisananti-chainin({\tilde{C}}_{g_{\beta }},\sim )\}$.

7. The Weak Path Independence Numbers

In this section, a weakly path independent set of $C_{g_{\beta }}$ of $Cay(S,A)$ is investigated. By the definition of a weakly path independent set and $[g\langle A_{\lambda }\rangle ]\cong Cay(\langle A_{\lambda }\rangle ,A_{\lambda })$ is strongly connected, we can conclude that ${\alpha }_{wp}([g\langle A_{\lambda }\rangle ])=1$ for every induced subdigraph $[g\langle A_{\lambda }\rangle ]$ of $Cay(S,A)$ and $\beta \in B$.

From the fact that, for all $g\in g_{\lambda }\langle A_{\lambda }\rangle$ and $h\in h_{\eta }\langle A_{\eta }\rangle$ where $\lambda \ne \eta$, there is no path from h to g if there is a path from g to h. We here obtain the weak path independence number of $C_{g_{\beta }}$ and $Cay(S,A)$ as follows.

Lemma 15. 

${\alpha }_{wp}(C_{g_{\beta }})=|{\displaystyle \bigcup _{\lambda \in Y_{\beta }}}{M}_{g_{\beta }\lambda }|$.

Proof. 

Let ${\mathcal{M}}_{g_{\beta }\lambda }$ be a set of representatives of all cosets in $M_{g_{\beta }\lambda }$. From the fact that there exists a path either from g to h or from h to g for any different $g,h\in {\displaystyle \bigcup _{\lambda \in Y_{\beta }}}{\mathcal{M}}_{g_{\beta }\lambda }$. Thus ${\displaystyle \bigcup _{\lambda \in Y_{\beta }}}{\mathcal{M}}_{g_{\beta }\lambda }$ is a weakly path independent set of $C_{g_{\beta }}$ and hence ${\alpha }_{wp}(C_{g_{\beta }})\ge |{\displaystyle \bigcup _{\lambda \in Y_{\beta }}}{\mathcal{M}}_{g_{\beta }\lambda }| = |{\displaystyle \bigcup _{\lambda \in Y_{\beta }}}{M}_{g_{\beta }\lambda }|$.

Conversely, let X be an ${\alpha }_{wp}-$set of $C_{g_{\beta }}$. We know that $[g\langle A_{\lambda }\rangle ]$ is strongly connected for every $g\langle A_{\lambda }\rangle \in {\displaystyle \bigcup _{\lambda \in Y_{\beta }}}{M}_{g_{\beta }\lambda }$. Hence $|X|\le |{\displaystyle \bigcup _{\lambda \in Y_{\beta }}}{\mathcal{M}}_{g_{\beta }\lambda }|$. It follows that ${\alpha }_{wp}(C_{g_{\beta }})\le |{\displaystyle \bigcup _{\lambda \in Y_{\beta }}}{\mathcal{M}}_{g_{\beta }\lambda }|=|{\displaystyle \bigcup _{\lambda \in Y_{\beta }}}{M}_{g_{\beta }\lambda }|$ and so ${\alpha }_{wp}(C_{g_{\beta }})=|{\displaystyle \bigcup _{\lambda \in Y_{\beta }}}{M}_{g_{\beta }\lambda }|$, as required. □

Now, the weak path independence number of $Cay(S,A)$ will be obtained in the form of the summation of ${\alpha }_{wp}(C_{g_{\beta }})=|{\displaystyle \bigcup _{\lambda \in Y_{\beta }}}{M}_{g_{\beta }\lambda }|$ as follows.

Theorem 6. 

${\alpha }_{wp}(Cay(S,A))={\displaystyle \sum _{\begin{array}{c}g\in {\mathcal{A}}_{\beta }\\ \beta \in B\end{array}}}({\alpha }_{wp}(C_{g_{\beta }}))={\displaystyle \sum _{\begin{array}{c}g\in {\mathcal{A}}_{\beta }\\ \beta \in B\end{array}}}|{\displaystyle {\displaystyle \bigcup _{\lambda \in Y_{\beta }}}}{M}_{g_{\beta }\lambda }|$.

8. Discussion

In this paper, we conducted an in-depth investigation of the independence numbers of the Cayley digraph of the Clifford semigroup. The study focused on a maximal connected subdigraph to analyze its independent sets. Our findings revealed that the independence number of the entire digraph is influenced by the independence numbers of its maximal connected subdigraphs, providing valuable insights into the structural dependencies within the digraph. Although our research aligned with previous studies on the independence numbers of graphs and digraphs, its unique focus on the Cayley digraph of the Clifford semigroup contributes to the theoretical understanding of this specific mathematical structure. As a stepping stone for future research, our study suggests exploring a subdigraph of the Cayley digraphs, which is smaller than a component, to further deepen the understanding of the independence number. In summary, our investigation sheds light on the independence numbers of the Cayley digraph and opens new avenues for future explorations and potential applications in diverse domains.

9. Conclusions

In conclusion, we obtained the characteristic of a maximal connected subdigraph in Cayley digraphs of Clifford semigroups in Section 3. The lower and upper bounds of the independence and the weak independence numbers of Cayley digraphs of Clifford semigroups are presented in Section 4 and Section 5. Finally, we have achieved the exact values of the path independence and the weak path independence numbers in Section 6 and Section 7, respectively.