Utilize este identificador para referenciar este registo: http://hdl.handle.net/10400.2/3813
Título: The commuting graph of the symmetric inverse semigroup
Autor: Araújo, João
Bentz, Wolfram
Konieczny, Janusz
Palavras-chave: Commuting graphs of semigroups
Symmetric inverse semigroup
Commutative semigroups
Inverse semigroups
Nilpotent semigroups
Clique number
Data: 2014
Citação: Araújo, João; Bentz, Wolfram; Konieczny, Janusz - The commuting graph of the symmetric inverse semigroup. "Israel Journal of Mathematics" [Em linha]. ISSN 0021-2172 (Print) 1565-8511 (Online). (2014), p. 1-29
Resumo: The commuting graph of a finite non-commutative semigroup S, denoted G(S), is a simple graph whose vertices are the non-central elements of S and two distinct vertices x, y are adjacent if xy = yx. Let I(X) be the symmetric inverse semigroup of partial injective transformations on a finite set X. The semigroup I(X) has the symmetric group Sym(X) of permutations on X as its group of units. In 1989, Burns and Goldsmith determined the clique number of the commuting graph of Sym(X). In 2008, Iranmanesh and Jafarzadeh found an upper bound of the diameter of G(Sym(X)), and in 2011, Dol˘zan and Oblak claimed that this upper bound is in fact the exact value.The goal of this paper is to begin the study of the commuting graph of the symmetric inverse semigroup I(X). We calculate the clique number of G(I(X)), the diameters of the commuting graphs of the proper ideals of I(X), and the diameter of G(I(X)) when |X| is even or a power of an odd prime. We show that when |X| is odd and divisible by at least two primes, then the diameter of G(I(X)) is either 4 or 5. In the process, we obtain several results about semigroups, such as a description of all commutative subsemigroups of I(X) of maximum order, and analogous results for commutative inverse and commutative nilpotent subsemigroups of I(X). The paper closes with a number of problems for experts in combinatorics and in group or semigroup theory.
Peer review: yes
URI: http://hdl.handle.net/10400.2/3813
ISSN: 0021-2172
Aparece nas colecções:Matemática e Estatística - Artigos em revistas internacionais / Papers in international journals

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