Ronnason Chinram and Chutiporn Jirojkul

pp. 231 - 234

Abstract

In 1952, R. A. Good and D. R. Hughes introduced the notion of bi-ideals of semigroups and in 1981, the concept of Γ-semigroups was introduced by M. K. Sen. We have known that Γ-semigroups are a generalization of semigroups. In this research, the notion of bi- Γ-ideals in Γ-semigroups is introduced. We show that bi- Γ-ideals in Γ-semigroups are a generalization of bi-ideals in semigroups and we give some properties for bi- Γ-ideals in Γ-semigroups. We give the two definitions as follows : A Γ-semigroup M is called a bi-simple Γ-semigroup if M is the unique bi-Γ-ideal of M and a bi- Γ-ideal B of M is called a minimal bi- Γ-ideal of M if B does not properly contain any bi-Γ-ideal of M. We show that a bi-Γ-ideal B of a Γ-semigroup M is a minimal bi- Γ-ideal of M if and only if B is a bi-simple Γ-semigroup.