Minimal paths in the commuting graphs of semigroups

Abstract

Let S be a finite non-commutative semigroup. The commuting graph of S, denoted G(S), is the graph whose vertices are the non-central elements of S and whose edges are the sets {a, b} of vertices such that a = b and ab = ba. Denote by T (X) the semigroup of full transformations on a finite set X. Let J be any ideal of T (X) such that J is different from the ideal of constant transformations on X. We prove that if |X| ≥ 4, then, with a few exceptions, the diameter of G(J) is 5. On the other hand, we prove that for every positive integer n, there exists a semigroup S such that the diameter of G(S) is n.We also study the left paths in G(S), that is, paths a1 − a2 − · · · − am such that a1 = am and a1ai = amai for all i ∈ {1, . . . , m}. We prove that for every positive integer n ≥ 2, except n = 3, there exists a semigroup whose shortest left path has length n. As a corollary, we use the previous results to solve a purely algebraic old problem posed by B.M. Schein.2010 Mathematics Subject Classification. 05C25, 05C12, 20M20.Let S be a semigroup. We denote by G E (S) the subgraph of G(S) induced by the non-central idempotents of S. The graph G E (S) is said to be the idempotent commuting graph of S. We first determine the diameter of G E (J r ). This approach is justified by the following lemma.Lemma 2.1. Let 2 ≤ r < n and let a, b ∈ J r be such that ab = ba. Suppose a−a 1 −a 2 −· · ·−a k −b (k ≥ 1) is a minimal path in G(J r ) from a to b. Then there are idempotents e 1 , e 2 , . . . , e k ∈ J r such that a − e 1 − e 2 − · · · − e k − b is a minimal path in G(J r ) from a to b.Proof. Since J r is finite, there is an integer p ≥ 1 such that e 1 = a p 1 is an idempotent in J r . Note that e 1 / ∈ Z(J r ) since for any x ∈ X − im(e 1 ), e 1 does not commute with c x ∈ J r , where c x is the constant transformation with im(c x ) = {x}. Since a 1 commutes with a and a 2 , the idempotent e 1 = a p 1 also commutes with a and a 2 , and so a − e 1 − a 2 − · · · − a k − b. Repeating the foregoing argument for a 2 , . . . , a k , we obtain idempotents e 2 , . . . , e k in J r such that a − e 1 − e 2 − · · ·− e k − b. Since the path a − a 1 − a 2 − · · · − a k − b is minimal, it follows that a, e 1 , e 2 , . . . , e k , b are pairwise distinct and the path a − e 1 − e 2 − · · · − e k − b is minimal.

Commuting graphs of boundedly generated semigroups

“…In Section 4 we provide a counterexample to another problem from [3] concerning the notion of knit degree (defined in the sequel).…”

“…Another question that was posed in [3] is related to the notion of knit degree: For n = 2 and every n ≥ 4 a band with knit degree n was constructed in [3]. In [3, Section 6(1)] it was guessed that a semigroup with knit degree 3 does not exist.…”

“…Araújo, Kinyon and Konieczny [3] proved that for every natural number n ≥ 2, there exists a semigroup whose commuting graph has diameter n. They also pose the following question:…”

“…In [3], the authors introduce constructions of semigroups with connected commuting graphs having arbitrary diameter. The constructions there have unbounded rank that grows linearly with the diameter.…”

“…For example, if S is a commutative semigroup, then Γ(S) is the empty graph. A study of these graphs in relation with certain algebraic structures can be found, for instance in [1,2,3,4,5].…”