Monogenic Semigroups Research Articles

Commuting Regular Graphs for Non-commutative Semigroups

To study the commuting regularity of a semigroup, we use a graph. Indeed, we define a multi-graph for a semigroup and identify this graph for the semidirect product of two monogenic semigroups. For a non-group semigroup S, the ordered pair \((x, y)\) of the elements of \(S\) is called a commuting regular pair if for some \(z \in S\), \(xy = yxzyx\) holds, and \(S\) is called a commuting regular semigroup if every ordered pair of S is commuting regular. As a result of Abueida in 2013 concerning the heterogenous decomposition of uniform complete multi-graphs into the spanning edge-disjoint trees, we show that for a semigroup of order \(n\), the commuting regular graph of \(S\), \(\Gamma(S)\) has at most n spanning edge-disjoint trees.

On Some Monogenic Semigroups with an Associate Subgroup

Let S be any semigroup and a, s ∈ S. If a = asa, then s is an associate of a. A subgroup G of S is an associate subgroup of S if every a ∈ S has a unique associate a* in G. It turns out that G = H z for some idempotent z, the zenith of S. The mapping a → a* is a unary operation on S. We say that S is monogenic if S is generated, as a unary semigroup, by a single element. We embark upon the problem of the structure of monogenic semigroups in this sense by characterizing monogenic ones belonging to completely simple semigroups, normal cryptogroups, orthogroups, combinatorial semigroups, cryptic medial semigroups, cryptic orthodox semigroups, and orthodox monoids. In each of these cases, except one, we construct a free object. The general problem remains open.

A classification of disjoint unions of two or three copies of the free monogenic semigroup

We prove that, up to isomorphism and anti-isomorphism, there are only two types of semigroups which are the union of two copies of the free monogenic semigroup. Similarly, there are only nine types of semigroups which are the union of three copies of the free monogenic semigroup. We provide finite presentations for semigroups of each of these types.

Some properties on the tensor product of graphs obtained by monogenic semigroups

In Das et al. (2013) [8], a new graph Γ(SM) on monogenic semigroups SM (with zero) having elements {0,x,x2,x3,…,xn} has been recently defined. The vertices are the non-zero elements x,x2,x3,…,xn and, for 1⩽i,j⩽n, any two distinct vertices xi and xj are adjacent if xixj=0 in SM. As a continuing study, in Akgunes et al. (2014) [3], it has been investigated some well known indices (first Zagreb index, second Zagreb index, Randić index, geometric–arithmetic index, atom-bond connectivity index, Wiener index, Harary index, first and second Zagreb eccentricity indices, eccentric connectivity index, the degree distance) over Γ(SM).In the light of above references, our main aim in this paper is to extend these studies over Γ(SM) to the tensor product. In detail, we will investigate the diameter, radius, girth, maximum and minimum degree, chromatic number, clique number and domination number for the tensor product of any two (not necessarily different) graphs Γ(SM1) and Γ(SM2).

Some properties on the lexicographic product of graphs obtained by monogenic semigroups

In (Das et al. in J. Inequal. Appl. 2013:44, 2013), a new graph on monogenic semigroups (with zero) having elements was recently defined. The vertices are the non-zero elements and, for , any two distinct vertices and are adjacent if in . As a continuing study, in an unpublished work, some well-known indices (first Zagreb index, second Zagreb index, Randic index, geometric-arithmetic index, atom-bond connectivity index, Wiener index, Harary index, first and second Zagreb eccentricity indices, eccentric connectivity index, the degree distance) over were investigated by the same authors of this paper. In the light of the above references, our main aim in this paper is to extend these studies to the lexicographic product over . In detail, we investigate the diameter, radius, girth, maximum and minimum degree, chromatic number, clique number and domination number for the lexicographic product of any two (not necessarily different) graphs and . MSC:05C10, 05C12, 06A07, 15A18, 15A36.

On disjoint unions of finitely many copies of the free monogenic semigroup

Every semigroup which is a finite disjoint union of copies of the free monogenic semigroup (natural numbers under addition) is finitely presented and residually finite.

On a graph of monogenic semigroups

Abstract 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 Γ ( 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 Γ ( 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 Γ ( S M 1 ) and Γ ( S M 2 ) , we present the spectral properties to the Cartesian product Γ ( S M 1 ) □ Γ ( S M 2 ) . MSC:05C10, 05C12, 06A07, 15A18, 15A36.

Presentation of semidirect product of monogenic semigroups

By studying the semigroup presented by π = � A, B | A n+1 = A, B m+1 = B, BA = A n−1 B, B m = A n � , for every positive integers m, n ≥ 3 we show that, for even values of m this is an appropriate presentation for the semidirect product of monogenic semigroup S = � a | a n+1 = aby the monogenic semigroup T = � b | b m+1 = b� . Moreover, for odd values of m it is a commutative semigroup of order (3+(−1) n )m 2 . The interests of presentability of prod- ucts of semigroups are quite useful in the study of more effective and intrinsic properties of the products (like as the computation of their characters and studying their Green J-classes). Our method of proof based on using the theoretic definitions and hereby we will deduced the presentation of the direct product of monogenic semigroups for every integers n, m ≥ 3. Mathematics Subject Classification: 20M30, 20E45

On the automorphisms of direct product of monogenic semigroups and monoids

This paper investigates the automorphism group of monogenic [4] semigroups (or monoids) to find its relationship with the automorphism group of cyclic groups. Then, by considering a presentation related to the direct product of monogenic semigroups, verify the relationship between the automorphism group of These and the automorphism group of the group presented by the same presentation. This study gives us some [4]explicit formulas for computing the order of automorphism groups of these algebraic structures.

Finite state wreath powers of transformation semigroups

The finite state wreath power of a transformation semigroup is introduced. It is proved that the finite state wreath power of nontrivial semigroup is not finitely generated and in some cases even does not contain irreducible generating systems. The free product of two monogenic semigroups of index 1 and period m is constructed in the finite state wreath power of corresponding monogenic monoid.

INVERSE SEMIGROUPS OF PARTIAL AUTOMATON PERMUTATIONS

The inverse semigroup of partial automaton permutations over a finite alphabet is characterized in terms of wreath products. The permutation conjugacy relation in this semigroup and the Green's relations are described. Criteria of primary conjugacy and conjugacy are given for certain naturally defined families of partial automaton permutations. Sufficient conditions under which an inverse semigroup admits a level transitive action are presented. We give explicit examples (monogenic inverse semigroups and some commutative Clifford semigroups) of inverse semigroups generated by finite automata.

Automatic subsemigroups of free products

We consider the automaticity of subsemigroups of free products of semigroups, proving that subsemigroups of free products, with all generators having length greater than one in the free product, are automatic. As a corollary, we show that if S is a free product of semigroups that are either finite or free, then any finitely generated subsemigroup of S is automatic. In particular, any finitely generated subsemigroup of a free product of finite or monogenic semigroups is automatic.

Splittability over Semigroups

Cleavability (also known as splittability) is well established within a topological framework. The definition was applied to semigroups by D.J. Marron and T.B.M. McMaster. Here the theory is further developed as we examine the properties that are enjoyed by a semigroup that is splittable over a given collection of semigroups. In particular we consider splittability over finite monogenic semigroups and over 0-simple semigroups.

Interassociates of Monogenic Semigroups

Given a monogenic semigroup, (S, ·) , we characterize all interassociates of (S, ·) . We determine necessary and sufficient conditions under which an interassociate of (S, ·) is isomorphic to (S, ·) . Finally, we find necessary and sufficient conditions under which two interassociates of (S, ·) are isomorphic.

Structure of monogenic inverse semigroups

Some properties of monogenic inverse semigroups are considered. In particular, in a free monogenic inverse semigroup we study the disposition of idempotents, describe the structure of ideals, classify the congruences.

Monogenic inverse semigroups and their C* -algebras

SynopsisBy using the Halmos-Wallen description of power partial isometries on Hilbert space, we give a complete description of all monogenic inverse semigroups,ℐ. We also describe the full C*-algebra C*ℐ and the reduced C*-algebra C*(ℐ) with particular emphasis on the case of the free monogenic inverse semigroupℑℐt.

Direct products of finite monogenic inverse semigroups

SynopsisWe characterize direct products of finite monogenic inverse semigroups; we show that a finite monogenic inverse semigroup which is not a group is directly indecomposable and that a finite semigroup which is decomposable into a direct product of monogenic inverse semigroups which are not groups is uniquely so decomposable. We determine when a finite semigroup can be decomposed into a direct product of non-group monogenic inverse semigroups and show how the direct factors, if they exist, can be found.

Direct products of monogenic semigroups

Direct products of monogenic semigroups