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:
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:
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 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

Abstract:
We show that the existence of a nontrivial bounded uniformly continuous (BUC) complete trajectory for a $C_0$-semigroup $T_A(t)$ generated by an operator $A$ in a Banach space $X$ is equivalent to the existence of a solution $Pi = delta_0$ to the homogenous operator equation $Pi S|_mathcal{M} = APi$. Here $S|_mathcal{M}$ generates the shift $C_0$-group $T_S(t)|_mathcal{M}$ in a closed translation-invariant subspace $mathcal{M}$ of $BUC(mathbb{R},X)$, and $delta_0$ is the point evaluation at the origin. If, in addition, $mathcal{M}$ is operator-invariant and $0 eq Pi in mathcal{L}(mathcal{M},X)$ is any solution of $Pi S|_mathcal{M} = APi$, then all functions $t o Pi T_S(t)|_mathcal{M}f$, $f in mathcal{M}$, are complete trajectories for $T_A(t)$ in $mathcal{M}$. We connect these results to the study of regular admissibility of Banach function spaces for $T_A(t)$; among the new results are perturbation theorems for regular admissibility and complete trajectories. Finally, we show how strong stability of a $C_0$-semigroup can be characterized by the nonexistence of nontrivial bounded complete trajectories for the sun-dual semigroup, and by the surjective solvability of an operator equation $Pi S|_mathcal{M} = APi$.

Abstract:
Our aim is to show the way we pass from the results of ordered semigroups (or semigroups) to ordered $\Gamma$-semigroups (or $\Gamma$-semigroups). The results of this note have been transferred from ordered semigroups. The concept of strongly regular $po$-$\Gamma$-semigroups has been first introduced here and a characterization of strongly regular $po$-$\Gamma$-semigroups is given.

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 characterize the intra-regular $\Gamma$-semigroups and the left regular $\Gamma$-semigroups $M$ in which $x\Gamma M\subseteq M\Gamma x$ for every $x\in M$ in terms of filters and we prove, among others, that every intra-regular $\Gamma$-semigroup is decomposable into simple components, and every $\Gamma$-semigroup $M$ for which $x\Gamma M\subseteq M\Gamma x$ is left regular, is decomposable into left simple components.