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Minimal Paths in the Commuting Graphs of Semigroups by João Araújo; Michael Kinyon; Janusz Konieczny is a Mathematics article available to read on EtoBox.

What is Minimal Paths in the Commuting Graphs of Semigroups about?

Let S be a finite non-commutative semigroup. The commuting graph of S, denoted G(S), is the graph whose vertices are the noncentral 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 a 1 -a 2 -• • • -a m such that a 1 ̸ = a m and a 1 a i = a m a i 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.

Who reads Minimal Paths in the Commuting Graphs of Semigroups?

Author

João Araújo; Michael Kinyon; Janusz Konieczny

Publisher

Elsevier Science; Elsevier ; Elsevier Inc.; Elsevier BV (ISSN 0195-6698)

Published

2011

Language

EN

Field

Mathematics (Physical Sciences)