Some T - Semigroups

Abstract

Let S and gamma be two nonempty sets. Then S is called a gamma-semigroup if there exists a mapping from S X gamma X S into S, denote the image of (a, gamma, b) by a gamma b, satisfying the identity (a alpha b)beta c = a alpha (b beta c) for all a, b, c E gamma. A nonempty subset B of S is called a gamma-subsemigroup of S if B gamma B ... B where B gamma B = {a alpha b}| alpha, b ... B and alpha ... gamma}. It is obvious that, R is a gamma-semigroup under usual addition and multiplication for any nonempyty subset gamma of R and M[subscript mn] (R), the set of all m x n matrices over R, is a gamma-semigroup under multiplication for any nonempty subset gamma of M[subscript nm(R). In this research, we, first, determine real intervals I and gamma such that I is a gamma -subsemigroup of R. Besides, for a fixed nonempty subset T of M[subscript mn] (R), we characterize a nonempty subset gamma of M[subscript nm (R) so that gamma is a gamma -subsemigroup of M[subscript mn (R). Furthermore, we khow that L(V), the set of all linear transformations on a vector space V over a division ring, is a semigroup and a gamma -semigroup under composition for any nonempty subset gamma of L(V). For some particular subsemigroups S of L(V), necessary and sufficient conditions of nonempty subsets gamma of L(V) so that S is a gamma -subsemigroup of L(V) are given.