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Bicyclic semigroup
About: Bicyclic semigroup is a research topic. Over the lifetime, 1507 publications have been published within this topic receiving 19311 citations.
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TL;DR: In this article, the authors present a new construction of the free inverse monoid on a set X. Contrary to previous constructions of [9, 11], their construction is symmetric and originates from classical ideas of language theory.
20 citations
20 citations
20 citations
TL;DR: In this paper, Iranmanesh and Jafarzadeh showed that the clique number of the symmetric group Sym(X) of permutations on a finite set X is at most 4 or 5.
Abstract: 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, Dolžan 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.
20 citations
TL;DR: In this paper, it was shown that every automorphism of the semigroup of invertible matrices with nonnegative elements over a linearly ordered associative ring on some specially defined subgroup coincides with the composition of an inner automomorphism of a semigroup, an order-preserving automomorphisms of the ring, and a central homothety.
Abstract: In this paper, we prove that every automorphism of the semigroup of invertible matrices with nonnegative elements over a linearly ordered associative ring on some specially defined subgroup coincides with the composition of an inner automorphism of the semigroup, an order-preserving automorphism of the ring, and a central homothety
20 citations