Minimal paths in the commuting graphs of semigroups

Araújo, João; Kinyon, Michael; Konieczny, Janusz

http://hdl.handle.net/10400.2/2007

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Authors

Araújo, João

Kinyon, Michael

Konieczny, Janusz

Abstract(s)

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.