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.

Keywords

• left partial congruence,ip - congruence,normal subsemigroup,ip - congruence pair

References

1. [1] F. Pastijn and M. Petrich., Congruences on regular semigroups, Trans. Amer. Math. Soc., 295(1986), 607-633.
2. [2] G.M.S. Gomes., R-unipotent congruences on regular semigroups, emigroup Forum, 31(1985), 265-280.
3. [3] J.M. Howie, An introduction to semigroup Theory, Clarendon Press, Oxford, 1995,
4. [4] K.S.S. Nambooripad, Structure of regular semigroups I, Mem. Amer. Math. Soc. 22 (1979), no.224.
5. [5] M.K. Sen, M.K. and N.K. Saha., On 􀀀-semigroup I , Bull. Cal. Math. Soc., 78(1986), 180-186.
6. [6] N.K. Saha., On 􀀀-semigroup II, Bull. Cal. Math. Soc, 79(1987), 331-335.
7. [7] N.K. Saha., On 􀀀-semigroup III, Bull. Cal. Math. Soc., 80(1988), 1-12.
8. [8] S. Chattopadhyay., Right inverse 􀀀-semigroup, Bull. Cal. Math. Soc., 93(6),(2001), 435-442.

9. [9] S. Chattopadhyay., Right orthodox 􀀀-semigroup, Southeast Asian Bull. of Mathemat- ics,(2005)29, 1-18.
10. [10] S. Chattopadhyay., Sandwich sets on regular 􀀀-semigroup, Communicated.