Abstract:
In this paper, we take Q_{16} subsemilattice of D and we will calculate the number of right unit, idempotent and regular elements α of B_{X} (Q_{16}) satisfied that V (D, α) = Q_{16} for a finite set X. Also we will give a formula for calculate idempotent and regular elements of B_{X} (Q) defined by an X-semilattice of unions D.

Abstract:
For a regular biordered set E, the notion of E-diagram and the associated regular semigroup was introduced in our previous paper (1995). Given a regular biordered set E, an E-diagram in a category C is a collection of objects, indexed by the elements of E and morphisms of C satisfying certain compatibility conditions. With such an E-diagram A we associate a regular semigroup RegE(A) having E as its biordered set of idempotents. This regular semigroup is analogous to automorphism group of a group. This paper provides an application of RegE(A) to the idempotent-separating extensions of regular semigroups. We introduced the concept of crossed pair and used it to describe all extensions of a regular semigroup S by a group E-diagram A. In this paper, the necessary and sufficient condition for the existence of an extension of S by A is provided. Also we study cohomology and obstruction theories and find a relationship with extension theory for regular semigroups.

Abstract:
As we know if D is a complete X-semilattice of unions then semigroup Bx(D)possesses a right unit iff D is an XI-semilattice of unions. The investigation of those a-idempotent and regular elements of semigroups B_{x}(D) requires an investigation of XI-subsemilattices of semilattice D for which V(D,a)=Q∈∑_{2}(X,8) . Because the semilattice Q of the class ∑_{2}(X,8) are not always XI -semilattices, there is a need of full description for those idempotent and regular elements when V(D,a)=Q . For the case where X is a finite set we derive formulas by calculating the numbers of such regular elements and right units for which V(D,a)=Q .

Abstract:
Let $a$ be a non-invertible transformation of a finite set and let $G$ be a group of permutations on that same set. Then $\genset{G, a}\setminus G$ is a subsemigroup, consisting of all non-invertible transformations, in the semigroup generated by $G$ and $a$. Likewise, the conjugates $a^g=g^{-1}ag$ of $a$ by elements $g\in G$ generate a semigroup denoted $\genset{a^g | g\in G}$. We classify the finite permutation groups $G$ on a finite set $X$ such that the semigroups $\genset{G,a}$, $\genset{G, a}\setminus G$, and $\genset{a^g | g\in G}$ are regular for all transformations of $X$. We also classify the permutation groups $G$ on a finite set $X$ such that the semigroups $\genset{G, a}\setminus G$ and $\genset{a^g | g\in G}$ are generated by their idempotents for all non-invertible transformations of $X$.

Abstract:
Difficulties encountered in studying generators of semigroup of binary relations defined by a complete X -semilattice of unions D arise because of the fact that they are not regular as a rule, which makes their investigation problematic. In this work, for special D, it has been seen that the semigroup , which are defined by semilattice D, can be generated by the set .

Abstract:
It is proved that, given a (von Neumann) regular semigroup with finitely many left and right ideals, if every maximal subgroup is presentable by a finite complete rewriting system, then so is the semigroup. To achieve this, the following two results are proved: the property of being defined by a finite complete rewriting system is preserved when taking an ideal extension by a semigroup defined by a finite complete rewriting system; a completely 0-simple semigroup with finitely many left and right ideals admits a presentation by a finite complete rewriting system provided all of its maximal subgroups do.

Abstract:
In the paper, we define the equivalence relations on $Pi^*$-regulur semigroups, to show $L^*$, $R^*$, $H^*$, $J^*$-class contains an idempotent with some characterizations.

Abstract:
We provide short and direct proofs for some classical theorems proved by Howie, Levi and McFadden concerning idempotent generated semigroups of transformations on a finite set.

Abstract:
One-parameter semigroups of antitriangle idempotent supermatrices and corresponding superoperator semigroups are introduced and investigated. It is shown that $t$-linear idempotent superoperators and exponential superoperators are mutually dual in some sense, and the first gives additional to exponential solution to the initial Cauchy problem. The corresponding functional equation and analog of resolvent are found for them. Differential and functional equations for idempotent (super)operators are derived for their general $t$ power-type dependence.

Abstract:
In this paper we discuss the structure of -semihypergroups. We prove some basic results and present several examples of -semihypergroups. Also, we obtain some properties of regular and strongly regular relations on a -semihypergroup and construct a -semigroup from a -semihypergroup by using the notion of fundamental relation. 1. Introduction The algebraic hyperstructure notion was introduced in 1934 by the French mathematician Marty [1], at the 8th Congress of Scandinavian Mathematicians. He published some notes on hypergroups, using them in different contexts: algebraic functions, rational fractions, and noncommutative groups. Algebraic hyperstructures are a suitable generalization of classical algebraic structures. In a classical algebraic structure, the composition of two elements is an element, while in an algebraic hyperstructure, the composition of two elements is a set. Around the 1940s, the general aspects of the theory, the connections with groups, and various applications in geometry were studied. The theory knew an important progress starting with the 1970s, when its research area enlarged. A recent book on hyperstructures [2] points out to their applications in cryptography, codes, automata, probability, geometry, lattices, binary relations, graphs, and hypergraphs. Many authors studied different aspects of semihypergroups, for instance, Bonansinga and Corsini [3], Corsini [4], Davvaz [5], Davvaz and Poursalavati [6], Fasino and Freni [7], Gu？an [8], Hasankhani [9], Leoreanu [10], and Onipchuk [11]. In 1986, Sen and Saha [12] defined the notion of a -semigroup as a generalization of a semigroup. One can see that -semigroups are generalizations of semigroups. Many classical notions of semigroups have been extended to -semigroups and a lot of results on -semigroups are published by a lot of mathematicians, for instance, Chattopadhyay [13, 14], Hila [15], Saha [16], Sen et al. [12, 17–20], Seth [21], and Sardar et al. [22]. In [23–25], Anvariyeh et al. introduced the notion of a -semihypergroup as a generalization of a semihypergroup. Many classical notions of semigroups and semihypergroups have been extended to -semihypergroups and a lot of results on -semihypergroups are obtained. 2. Basic Definitions In this section, we recall certain definitions and results needed for our purpose. Let and be two nonempty sets, the set of all mapping from to , and a set of some mappings from to . The usual composition of two elements of cannot be defined. But if we take from and from , then the usual mapping composition can be defined. Also, we see that and