Intersection graphs associated with semigroup acts

The intersection graph Int(A) of an S-act A over a semigroup S is an undirected simple graph whose vertices are non-trivial subacts of A, and two distinct vertices are adjacent if and only if they have a nonempty intersection. In this paper, we study some graph-theoretic properties of Int(A) in connection to some algebraic properties of A. It is proved that the finiteness of each of the clique number, the chromatic number, and the degree of some or all vertices in Int(A) is equivalent to the finiteness of the number of subacts of A. Finally, we determine the clique number of the graphs of certain classes of S-acts.


Introduction and preliminaries
In recent decades, assigning graphs to algebraic structures has opened a new direction to study algebraic properties via graph-theoretic properties and vice versa.Several classes of graphs associated with algebraic structures have been extensively investigated by many authors in the literature (for example, [1-4, 6-8, 10, 12, 19, 21]).One such useful graph is the intersection graph which is important in both theoretical as well as in applications.For an overview of the theory of intersection graphs and important special classes of them, see [14].The idea of studying algebraic properties of algebraic structures via their intersection graphs was initiated by Bosák [5] where the intersection graph of proper subsemigroups of a semigroup was considered.The intersection graph of some classes of ordered semigroups were briefly studied in [16,17].Some papers devoted to intersection graphs derived from other structures such as groups, rings, modules, and lattices have appeared during the years.Inspired by these studies, Rasouli and Tehranian [18] extended the idea to acts over semigroups and verified some elementary properties of the intersection graph Int(A) of non-trivial subacts of an Sact A over a semigroup S.Here we investigate more aspects of this graph and obtain more results.We try to relate the algebraic properties of an S-act to the graph-theoretic properties of its intersection graph.
Throughout S stands for a semigroup unless otherwise stated.Let A be an S-act.Recall from [18] that the intersection graph of A, denoted by Int(A), is an undirected simple graph whose vertices are non-trivial subacts of A and two distinct vertices B and C are adjacent if and only if B ∩ C = ∅.Here we first investigate some properties of the graph Int(A) for an S-act A. Hereby, the interplay between some algebraic properties of an S-act A and graph-theoretic properties of Int(A) is considered.We study some graph-theoretic characters such as clique, chromatic, domination, and the independence numbers in Int(A), and prove the equivalences of the finiteness of the clique number, chromatic number, degree of some (or all) vertices, and the degree of Int(A), that is, the number of non-trivial subacts of A.
It is known that deciding whether a graph is weakly perfect is an NPcomplete problem.A class of weakly perfect intersection graphs of ideals of a finite ring can be found in [15].This result was generalized in [9], where Corollary 4.3 shows the intersection graph of submodules of any finite R-module (where R is any ring) is weakly perfect.In fact, the intersection graph of an intersection-closed family of non-empty subsets of a set is weakly perfect if it has finite clique number.As a consequence, the intersec-tion graph of an S-act A with finite clique number is weakly perfect.This motivates us to determine the clique number of such graphs for some classes of S-acts.Regarding to the fact that each semigroup S can be viewed as an S-act over itself, it is worth noting that all results obtained here are also valid for S and the intersection graph Int(S) of non-trivial left ideals of S.
Let us give a brief account of some definitions about S-acts and graphs needed in the sequel.
Let S be a semigroup.A non-empty set A is said to be a (left) S-act if there is a mapping λ : S × A → A, denoting λ(s, a) by sa, satisfying (st)a = s(ta) and, if S is a monoid with 1, 1a = a, for all a ∈ A and s, t ∈ S.An element θ ∈ A is said to be a fixed element if sθ = θ for all s ∈ S. A non-empty subset B of A is called a subact of A if it is closed under the action, that is, sb ∈ B, for every s ∈ S and b ∈ B. By a non-trivial subact of an S-act A we mean a (non-empty) proper subact of A. The set of all non-trivial subacts of A is denoted by Sub(A).Clearly, S is an S-act with its operation as the action and so subacts of S are exactly the left ideals of S, that is, the non-empty subsets I of S satisfying SI ⊆ I.An element z ∈ S is called a left zero element if zs = z for all s ∈ S. If each element of S is a left zero element, then we say that S is a left zero semigroup.A non-empty S-act is said to be simple if it has no non-trivial subact.A nontrivial subact M of an S-act A is called a minimal subact of A if it properly contains no subact of A. We denote the set of all minimal subacts of A by Min(A).The socle of an S-act A, written as Soc(A), is the union of all minimal subacts of A. A maximal subact of A is a non-trivial subact M for which there is no subact of A properly contained between M and A. The coproduct of a family {A i | i ∈ I} of S-acts, denoted by i∈I A i , is their disjoint union i∈I (A i × {i}) with the action s(a, i) = (sa, i) for every s ∈ S and a ∈ A i , i ∈ I.For more information about S-acts and related notions, the reader is referred to [13].
Let G be a graph with a vertex set V (G).For distinct elements x and y of V (G), an x,y-path (or x − y) is a path with starting vertex x and ending vertex y, and the length of the shortest x, y-path is denoted by d(x, y).If G does not have such a path, then d(x, y)=∞.By the order of G, denoted by |G|, we mean the number of vertices of G.The diameter of G, diam(G), is the supremum of the set {d(x, y) : x, y ∈ V (G), x = y}.The number of vertices which are adjacent to x is called the degree of x and is denoted by deg(x).The girth of a graph is the length of its shortest cycle.A graph with no cycle has infinite girth.A complete graph with n vertices, denoted by K n , is a graph in which every pair of distinct vertices are adjacent.For a graph G let χ(G) denote the chromatic number of G, that is, the minimum number of colors which can be assigned to the vertices of G in such a way that every two adjacent vertices have different colors.A clique of G is a complete subgraph of G and the number of vertices in the largest clique of G, denoted by ω(G), is called the clique number of G.A graph G is called weakly perfect if χ(G) = ω(G).For undefined terms and concepts, one may consult [20].
2 Some properties of the graph Int(A) In this section, we proceed with the study of some facts about the intersection graphs of S-acts.
It is of interest to know whether a graph is an intersection graph of an Sact.This property holds for every complete graph (see [18,Proposition 2.2]).
Here the two classes of bipartite and wheel graphs which are intersection graphs of some S-acts are fully characterized.
A bipartite graph is a graph whose vertices can be partitioned into two sets in such a way that no two vertices within the same set are adjacent.Equivalently, a bipartite graph is a graph that contains no odd-length cycle.A wheel graph of order n ≥ 4, denoted by W n , is a graph formed by connecting a single vertex to all vertices of a cycle.
Theorem 2.1.The following assertions hold: (i) A bipartite graph G is the intersection graph of an S-act if and only if G is one of the graphs K 1 , K 2 , K c 2 and K 1,2 .(ii) The wheel graph W n is the intersection graph of an S-act if and only if n = 4.
Proof.(i) Suppose that G = Int(A) is bipartite where A is an S-act and |G| > 3.So A contains at least four non-trivial subacts.Let also V 1 = {B 1 , B 2 , . . ., B n } and V 2 = {C 1 , C 2 , . . ., C m } be two (non-empty) disjoint partitions of vertices of Int(A).Using [18, Theorem 3.1] and with no loss of generality, one can assume that B 1 , C 1 are adjacent.We claim that m, n ≤ 2. (ii) Let n ≥ 5 and suppose that there exists an S-act A with non-trivial subacts B 0 , B 1 , B 2 , . . ., B n−1 such that the intersection graph Int(A) is the following wheel graph W n : In each case, we get a contradiction.The converse follows from [18, Proposition 2.2].Theorem 2.1(ii) answers the question posed in [18] about the existence of a connected graph with diameter 2 and girth 3 which is the intersection graph of no S-act.In fact, all wheel graphs with at least five vertices satisfy the mentioned condition.
In view of Theorem 2.1(i) and [18, Theorem 4.1(ii)], the following is obtained.

Corollary 2.2. For an S-act
An S-act A over a monoid S is called free if it has a basis X, that is, each element a ∈ A is uniquely represented as a = sx for some s ∈ S and x ∈ X.In this case, A ∼ = x∈X S.Moreover, A is isomorphic to the S-act S × X with the action given by s(t, x) = (st, x) for all s, t ∈ S, x ∈ X (see [13]).
If A and B are isomorphic S-acts, then their intersection graphs are clearly isomorphic; and there is an example which shows that this implication is strict (see [18,Example 2.3]).Here some conditions are posed to fill the gap.To this aim, first note the following: Lemma 2.3.Let A be a free S-act with a basis X where S is a group.Then Int(A) ∼ = Int(X), in which X is considered as an S-act with trivial action.
Proof.Using the assumption, A is isomorphic to the S-act S × X.Since S is a group, non-trivial subacts of A (if exist) are of the forms S × Y where Y ⊂ X.Consider the set X as an S-act with trivial action.We claim that the graphs Int(A) and Int(X) are isomorphic.For this, define the map f : (ii) This is trivial.
The next result presents some necessary and sufficient conditions for the graph Int(A) to be complete.
A graph G is said to be r-regular for some non-negative integer r if the degree of each vertex of G is r.Proposition 2.5.Let A be an S-act.
(i) If S contains a left zero element z, then Int(A) is complete if and only if za = za for all a, a ∈ A.
(2) Int(A) is r-regular for some r ∈ N. In the rest of this section, we restrict our attention to verify the existence of a cut vertex and a cut edge in Int(A).
A vertex v in a graph is a cut vertex if the removal of v and all edges with v as an end-point from the graph increases the number of components.A cut edge of a graph is an edge whose deletion (the end-points stay in place) from the graph increases the number of components.
Theorem 2.6.Let every subact of an S-act A contain a minimal subact.Then the graph Int(A) has a cut vertex if and only if Soc(A) is the union of two minimal subacts and is a maximal subact of A. Moreover, in this case, Soc(A) is the unique cut vertex of Int(A).
Proof.Assume that B is a cut vertex in Int(A).Then there exist two nontrivial subacts B 1 , B 2 of A such that there is a path between B 1 and B 2 in Int(A) but no path between them in Int(A) − {B}.Note that Int(A) must be connected because, otherwise, it has no cut vertex by [18,Theorem 3.1].Then, using [18, Theorem 4.1(i)], d(B 1 , B 2 ) = 2 in Int(A) and so B 1 −B−B 2 is a path from B 1 to B 2 .Since B 1 ∩ B and B 2 ∩ B are disjoint non-trivial subacts of A, it follows from the assumption that there exist two (distinct) which is a contradiction.This means that Soc(A) is a maximal subact of A.
For the converse, let This clearly gives that M 1 ⊆ B 1 and M 2 ⊆ B n .We claim that there exists i ∈ {1, . . ., n} such that Soc(A) ⊆ B i , whence Soc(A) = B i , by the maximality of Soc(A), which is a contradiction.To do so, note that if M 1 ⊆ B n , then Soc(A) = M 1 ∪ M 2 ⊆ B n and we are done.Otherwise, M 1 B n .Let 1 ≤ k < n be the greatest positive integer for which M 1 ⊆ B k .We show that Soc(A) ⊆ B k .We have B k ∩ B k+1 = ∅.The choice of k implies that M 1 B k ∩ B k+1 .Then, using the assumption, M 2 ⊆ B k ∩ B k+1 and, hence, Soc(A) = M 1 ∪ M 2 ⊆ B k , as claimed.This completes the proof.Definition 2.7.Let A be an S-act and B, C be subacts of A. We say that C covers B (or C is a cover for B), denoted by B `C, if B ⊂ C and no element in Sub(A) lies strictly between B and C, that is, Lemma 2.8.Let an edge e with end-points B 1 and B 2 be a cut edge in Int(A).Then, without loss of generality, B 1 is a minimal subact of A and B 2 is a maximal subact of A as well as the unique cover for B 1 .
Proof.Since e is a cut edge, there is no path between B 1 and B 2 other than e in Int(A).As B 1 ∩ B 2 = ∅, if B 1 , B 2 and B 1 ∩ B 2 are all distinct, then {a} {a,c} {a,b} {b} (ii) If A is an S-act with trivial action, then Soc(A) = A.This implies that the graph Int(A) has no cut vertex, by Theorem 2.6.

Some finiteness conditions
In this section, we study finiteness conditions of some parameters of intersection graphs of S-acts such as clique number and chromatic number.
We say that an S-act A is Artinian (Noetherian) if every descending (ascending) chain of subacts of A terminates.It is clear that every subact of an Artinian S-act contains a minimal subact.Remark 3.1.Consider any S-act A with infinitely many pairwise disjoint non-trivial subacts, say B 1 , B 2 , B 3 , . . . .Then A is neither Noetherian nor Artinian, whence ω(Int(A)) = ∞.Indeed, the infinite strict ascending chain of subacts of A gives an infinite clique in Int(A).Moreover, with no loss of generality, one can assume that the subact ∞ i=1 B i is non-trivial.In this case, we have the infinite strict descending chain of subacts of A which gives another infinite clique in Int(A) with no adjacent vertex with the previous one.As a consequence, if A is a Noetherian or an Artinian S-act, then the set Min(A) is finite.For instance, let {A i } ∞ i=1 be a family of Sacts and In the following, a main result concerning finiteness of clique and chromatic numbers of the graph Int(A) is presented.Theorem 3.2.Let A be an S-act.Then the following are equivalent: Proof.The implications (v) ⇒ (iv) ⇒ (iii) and (ii) ⇒ (i) are trivial.
(i) ⇒ (v) Assume that deg(B) < ∞ for some vertex B in Int(A).Suppose, on the contrary, that |Int(A)| = ∞.Since deg(B) < ∞, there exist infinitely many (pairwise distinct) B i ∈ Sub(A), i ∈ I, for which B i ∩ B = ∅.Thus, B i ∪ B = B j ∪ B for all i = j.Hence, {B i ∪ B} i∈I contains infinitely many vertices of Int(A) that are adjacent to B, which is a contradiction.
(iii) ⇒ (ii) Suppose that there exists a vertex B and an infinite set W = {B i : i ∈ I} such that B i is adjacent to B for all i ∈ I.By the well-known Infinite Ramsey's Theorem, the subgraph of Int(A) induced by W contains either an infinite clique or an infinite set of pairwise disjoint subacts.Both cases yield an infinite clique in Int(A) (see Remark 3.1), so if (iii) holds then (ii) holds.
As a direct consequence of Theorem 3.2, a useful result in semigroup theory is obtained below.

Corollary 3.3. For a semigroup S, we have the following equivalent assertions:
(i) There is a non-trivial left ideal of S intersecting only finitely many left ideals.
(ii) Any non-trivial ideal of S intersects only finitely many left ideals.
(iii) Int(S) is colored by finitely many colors.
(iv) S has finitely many left ideals.
The following example shows that the clique number of the graph Int(A) is not necessarily finite.(i) Let {A i } n i=1 be a family of S-acts and A = n i=1 A i .Then χ(Int(A)) < ∞ if and only if χ(Int(A i )) < ∞ for all i ∈ {1, . . ., n}.
(ii) It follows from (i).
We close this section with some results on the domination number and independence number of the graph Int(A).Let us give some definitions.
Let G be a graph.The (open) neighborhood N (x) of a vertex x ∈ V (G) is the set of vertices which are adjacent to x.For a subset T of vertices, we put , then T is said to be a dominating set.It is clear that every vertex not in a dominating set T is adjacent to a vertex in T .The domination number of G, γ(G), is the minimum cardinality of a dominating set of G.An independent set in a graph is a set of pairwise non-adjacent vertices.The independence number of G, written by α(G), is the maximum size of independent sets.Theorem 3.6.Let A be an Artinian S-act.Then the following assertions hold: (i) Min(A) is an independent as well as a dominating set.
size m, then it follows from the hypothesis that there are distinct subacts C i , C j in W such that they contain a same minimal subact.Thus C i ∩ C j = ∅, which is a contradiction.
(iii) It follows from (i) and (ii).
(iv) If A has only one minimal subact, say M , then {M } is a dominating set and so γ(

On clique number of Int(A)
As we mentioned in Introduction, it follows from [9, Corollary 4.3] that the intersection graph of an S-act with finite clique number is weakly perfect.In this section, we find the clique number (equivalently, chromatic number) of such graphs for some classes of S-acts.
The intersection graph of an S-act A with countably infinitely many subacts is also weakly perfect.Indeed, using Theorem 3.2, χ( It should be noted that the finiteness of ω(Int(A)) for an S-act A implies that A contains finitely many subacts (see Theorem 3.2) and then each nontrivial subact of A contains a minimal subact.This fact is implicitly used in the proof of the following result.Proof.(i) Let Min(A) = {M 1 , M 2 } and m i = deg(M i ) + 1 which is the number of non-trivial subacts of A containing M i , i ∈ {1, 2} and assume, with no loss of generality, that m 1 ≥ m 2 .We claim that ω(Int(A)) = m 1 .First note that the set Sub(A) can be partitioned to the following (possibly empty) subsets:  (iii) Suppose that |Int(A)| = n.Let Min(A) = {M 1 , M 2 , . . ., M t } and m i be the number of non-trivial subacts of A which contains M i for all i ∈ {1, . . ., t}.First we show that n is even and 2 n.For this, take V i := {B ∈ Sub(A) | M i ⊆ B} and W i := Sub(A) \ V i for every i ∈ {1, . . ., t}.Being S a group implies that the map f : V i → W i given by f (B) = A \ B, for any B ∈ V i , is a one to one correspondence so that 2 n.Take i = 1.The elements of V 1 form a clique of order m 1 in Int(A) and m 1 colors are needed for coloring the elements of V 1 .Moreover, any B ∈ W 1 can be colored by the same color of its The elements of W 1 form a clique in Int(A) and so 2 n−1 − 1 colors are needed for coloring them.Furthermore, any B ∈ W 2 can be colored by the same color of its complement A \ B in W 1 .Thus 2 n−1 − 1 ≤ ω(Int(A)) ≤ χ(Int(A)) = 2 n−1 − 1 and then the assertion holds.
Open Problem 4.2.For every S-act A, if the graph Int(A) has finite clique number, is ω(Int(A)) equal to m + 1, where m is the maximum degree of minimal subacts of A? Theorem 4.1 and the following example give a positive answer to the above problem in some particular cases.A planar graph is one which has a drawing in the plane without edge crossing.A well-known characterization of the planar graphs states that a graph is planar if and only if it contains no subgraph which is a subdivision of K 5 or K 3,3 .Proof.(i) Any infinite increasing chain or decreasing chain of non-trivial subacts of A gives K 5 as a subgraph of Int(A), which contradicts the planarity of Int(A).
(ii) Suppose, on the contrary, that M 1 , M 2 , M 3 , and M 4 are minimal subacts of A. Then the distinct non-trivial subacts M 1 , M 1 ∪ M 2 , M 1 ∪ M 3 , M 1 ∪ M 4 and M 1 ∪ M 2 ∪ M 3 form the subgraph K 5 of Int(A), which is a contradiction.Moreover, using (i), A contains at least one minimal subact.Then the assertion holds.
(iii) In view of Theorem 4.1(ii), it remains to consider the case A = M 1 ∪ M 2 ∪ M 3 in which M 1 , M 2 , and M 3 are three minimal subacts of A, noting that any non-trivial subact of A contains one of them by (i).We claim that A has only seven non-trivial subacts, which are M 1 , M 2 , M 3 , M 1 ∪ M 2 , M 1 ∪ M 3 , M 2 ∪ M 3 , and M 1 ∪ M 2 ∪ M 3 .Indeed, let B be another nontrivial subact of A. Without loss of generality, assume that M 1 ⊂ B. Then the subacts M 1 , M 1 ∪ M 2 , M 1 ∪ M 3 , M 1 ∪ M 2 ∪ M 3 , and B form K 5 , which is a contradiction.Now it is easy to see that ω(Int(A)) = 4. Remark 4.5.If Int(A) is planar, where A is a free S-act over a monoid S, then S is a group or isomorphic to A. Indeed, A = S × X, where X is a non-empty set.On the contrary, suppose that S is a non-group and non-isomorphic to A. Then there exist a non-trivial ideal I of S and distinct elements x 1 , x 2 in X.Then we have the non-trivial subacts B i = S × {x i }, C i = I × {x i }, i = 1, 2 of A. Now it is easily seen that the nontrivial subacts B 1 , C 1 , B 2 ∪ C 1 , B 1 ∪ C 2 , and C 1 ∪ C 2 form the subgraph K 5 of Int(A), which contradicts the planarity of Int(A).

Theorem 2 . 4 .
Let A and B be free S-acts and Int(A) ∼ = Int(B).Then A ∼ = B under each of the conditions (i) S is a group, (ii) S has only finitely many left ideals, and A and B have finite bases.Proof.(i) Assume that X and Y are bases of free S-acts A and B, respectively.Using Lemma 2.3, Int(A) ∼ = Int(X) and Int(B) ∼ = Int(Y ), where X and Y are considered as S-acts with trivial actions.It follows from the assumption that Int(X) ∼ = Int(Y ) and then 2 |X| − 2 = |Sub(X)| = |Sub(Y )| = 2 |Y | − 2. This implies that |X| = |Y | and hence A ∼ = B.

( 3 )
|Min(A)| = 1.Proof.(i) Let Int(A) be a complete graph and a, a ∈ A. Consider the two subacts B = Sa and B = Sa of A. If B, B = A, then B ∩ B = ∅ whence sa = s a for some s, s ∈ S.This gives that za = (zs )a = z(s a ) = z(sa) = (zs)a = za.If B = A or B = A, then the assertion clearly holds.For the converse, it suffices to note that for all non-trivial subacts B, B of A, zb = zb ∈ B ∩ B for all b ∈ B, b ∈ B .(ii) We need only to show the non-trivial direction (2) ⇒ (1): If Int(A) is not complete, then |Min(A)| > 1.Let M 1 , M 2 be minimal subacts of A. Using the hypothesis and [18, Theorem 3.1], Int(A) is connected and hence d(M 1 , M 2 ) = 2 by [18, Theorem 4.1(i)].Thus there exists a non-trivial subact B of A such that M 1 − B − M 2 is a path between M 1 and M 2 .It is clear that each vertex of Int(A) which is adjacent to M 1 and not equal to B is also adjacent to B. This gives that deg(B) > deg(M 1 ), which is a contradiction.

Example 3 . 4 .Proposition 3 . 5 .
(i) Take the monoid S = (N ∞ , min) where n < ∞ for all n ∈ N. The non-trivial ideals of S are exactly the principal ones ↓ k = {x ∈ N ∞ | x ≤ k}, k ∈ N, and its only non-principal ideal N (see[11, Remark 4]).Note that ↓ m ⊂↓ n if and only if m < n for every m, n ∈ N. Therefore, the graph Int(A) is complete with ω(Int(A)) = ∞.(ii)Consider the semigroup S = (N, +).It is easily seen that non-trivial ideals of S are exactly the sets N k = {k + 1, k + 2, . ..}where k ∈ N; and N m ⊂ N n if and only if n < m for every m, n ∈ N. Then Int(A) is complete with ω(Int(A)) = ∞.The following statements are satisfied:

Moreover, |W 1 |
+ |W 3 | = m 1 and |W 2 | + |W 3 | = m 2 .The elements of W 1 induce a clique of order |W 1 | in Int(A) which are colored by |W 1 | colors.On the other hand, for any B 1 ∈ W 1 and B 2 ∈ W 2 , B 1 ∩ B 2 = ∅ and it follows from m 1 ≥ m 2 that |W 1 | ≥ |W 2 |.Therefore, the elements of W 2 forming a clique of order |W 2 | in Int(A) are colored by |W 2 | colors of the elements of W 1 .This means that the elements of W 1 ∪ W 2 are colored by exactly |W 1 | colors.Also the elements of W 3 are colored by |W 3 | colors different from that of W 1 because the elements of W 3 form a clique of order |W 3 | and B 1 ∩ B 3 = ∅ for any B 1 ∈ W 1 and B 3 ∈ W 3 .Furthermore, it is clear that the elements of W 1 ∪ W 3 induce the largest clique in Int(A).Consequently, 3 and M 2 ∪M 3 are all of the non-trivial subacts of A. To this end, consider another non-trivial subact B of A and M 1 ⊂ B, say.This gives that eitherB ∩ M 2 = ∅ or B ∩ M 3 = ∅.So either M 2 ⊆ B or M 3 ⊆ B. Suppose, without loss of generality, that M 2 ⊆ B and M 3 B. Then B ∪ M 3 = A and B ∩ M 3 = ∅.This implies that B = M 1 ∪ M 2 ,which is a contradiction.Hence, it is clear that ω(Int(A)) = 3.

Example 4 . 6 .
(i) The converse of Proposition 4.4(ii) is not true in general.For this, see Example 3.4(i) where |Min(A)| = 1 and Int(A) is not planar.