ON RIGHT INVERSE $\Gamma$-SEMIGROUP

Abstract

Let S = fa; b; c; : : : g and 􀀀 = f ; ; ; : : : g be two nonempty sets. S is called a 􀀀-semigroup if a b 2 S, for all 2 􀀀 and a; b 2 S and (a b) c = a (b c), for all a; b; c 2 S and for all ; 2 􀀀. An element e 2 S is said to be -idempotent for some 2 􀀀 if e e = e. A 􀀀- semigroup S is called regular 􀀀-semigroup if each element of S is regular i.e, for each a 2 S there exists an element x 2 S and there exist ; 2 􀀀 such that a = a x a. A regular 􀀀-semigroup S is called a right inverse 􀀀-semigroup if for any - idempotent e and -idempotent f of S, e f e = f e. In this paper we introduce ip - congruence on regular 􀀀-semigroup and ip - congruence pair on right inverse 􀀀-semigroup and investigate some results relating this pair.