Ends for subsemigroups of finite index

Israel Journal of Mathematics, 1999

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Let ,.) (S be a finite semigroup and let T be a non-empty subset of

AKCE International Journal of Graphs and Combinatorics, 2019

The Square element graph over a semigroup S is a simple undirected graph Sq(S) whose vertex set consists precisely of all the non-zero elements of S, and two vertices a, b are adjacent if and only if either ab or ba belongs to the set {t 2 : t ∈ S} \ {1}, where 1 is the identity of the semigroup (if it exists). In this paper, we study the various properties of Sq(S). In particular, we concentrate on square element graphs over three important classes of semigroups. First, we consider the semigroup Ω n formed by the ideals of Z n. Afterwards, we consider the symmetric groups S n and the dihedral groups D n. For each type of semigroups mentioned, we look into the structural and other graph-theoretic properties of the corresponding square element graphs. c

Israel Journal of Mathematics, 2015

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, Dolzan and Oblak claimed that this upper bound is in fact the exact value.

Mathematica Slovaca, 2014

We characterize maximal subsemigroups of the monoid T(X) of all transformations on the set X = ℕ of natural numbers containing a given subsemigroup W of T(X) such that T(X) is finitely generated over W. This paper gives a contribution to the characterization of maximal subsemigroups on the monoid of all transformations on an infinite set.

Journal of Inequalities and Applications, 2013

Let us consider the finite monogenic semigroup S M with zero having elements {x, x 2 , x 3 ,. .. , x n }. There exists an undirected graph (S M) associated with S M whose vertices are the non-zero elements x, x 2 , x 3 ,. .. , x n and, f or 1 ≤ i, j ≤ n, any two distinct vertices x i and x j are adjacent if i + j > n. In this paper, the diameter, girth, maximum and minimum degrees, domination number, chromatic number, clique number, degree sequence, irregularity index and also perfectness of (S M) have been established. In fact, some of the results obtained in this section are sharper and stricter than the results presented in DeMeyer et al. (Semigroup Forum 65:206-214, 2002). Moreover, the number of triangles for this special graph has been calculated. In the final part of the paper, by considering two (not necessarily different) graphs (S 1 M) and (S 2 M), we present the spectral properties to the Cartesian product (S 1 M) (S 2 M).

Any transformation on a set S is called a Cayley function on S if there exists a semigroup operation on S such that β is an inner-translation. In this paper we describe a method to generate a semigroup with k number of idempotents, study some properties of such semigroups like greens relations and bi-ordered sets.

2019

Any transformation on a set S is called a Cayley function on S if there exists a semigroup operation on S such that β is an inner-translation. In this paper we describe a method to generate a semigroup with k number of idempotents, study some properties of such semigroups like greens relations and bi-ordered sets.

Semigroup Forum, 2012

We associate a graph N S with a semigroup S (called the upper non-nilpotent graph of S). The vertices of this graph are the elements of S and two vertices are adjacent if they generate a semigroup that is not nilpotent (in the sense of Malcev). In case S is a group this graph has been introduced by A. Abdollahi and M. Zarrin and some remarkable properties have been proved. The aim of this paper is to study this graph (and some related graphs, such as the non-commuting graph) and to discover the algebraic structure of S determined by the associated graph. It is shown that if a finite semigroup S has empty upper non-nilpotent graph then S is positively Engel. On the other hand, a semigroup has a complete upper non-nilpotent graph if and only if it is a completely simple semigroup that is a band. One of the main results states that if all connected N S -components of a semigroup S are complete (with at least two elements) then S is a band that is a semilattice of its connected components and, moreover, S is an iterated total ideal extension of its connected components. We also show that some graphs, such as a cycle Cn on n vertices (with n ≥ 5), are not the upper nonnilpotent graph of a semigroup. Also, there is precisely one graph on 4 vertices that is not the upper non-nilpotent graph of a semigroup with 4 elements. This work also is a continuation of earlier work by Okniński, Riley and the first named author on (Malcev) nilpotent semigroups.

Communications in Algebra, 2005

A formula for the rank of an arbitrary finite completely 0 -simple semigroup, represented as a Rees matrix semigroup M 0 [G; I, Λ; P ] , is given. The result generalises that of Ruškuc concerning the rank of connected finite completely 0 -simple semigroups. The rank is expressed in terms of |I| , |Λ| , the number of connected components k of P and a number rmin which we define. We go on to show that the number rmin is expressible in terms of a family of subgroups of G the members of which are in one to one correspondence with, and determined by the non-zero entries of, the components of P . A number of applications are given, including a generalisation of a result of Gomes and Howie concerning the rank of an arbitrary Brandt semigroup B(G, {1, . . . , n}) .