Green's relations and congruences for ?-semigroups [i.e. Gamma-semigroups]

Abstract

Let S be a Γ-semigroup and α a fixed element in Γ. Define ab = aαb for all a, b ∈ S. Then S is a semigroup and we denote this semigroup by Sα. Green’s relations L, R, H and D on a Γ-semigroups S were defined by N. K. Saha in the year 1987. The L-class, R-class, H-class and D-class containing the element a of a Γ-semigroup S will be written as La, Ra, Ha and Da, respectively. We study Green’s relations for Γ-semigroups and give some interesting properties. For example, we prove that if a and b are elements in a Γ-semigroup S such that aDb, then |La| = |Lb|, |Ra| = |Rb| and |Ha| = |Hb|. We also observe that if a is an element in a Γ-semigroup S and α ∈ Γ, then HaαHa ∩ Ha = ∅ or HaαHa = Ha. Moreover, if HaαHa = Ha, then Ha is a subsemigroup of Sα. Furthermore, we study congruences for Γ-semigroups and give some connections between congruences and their quotient sets on Green’s relations. We also define two congruences ρr and ρl on a Γ-semigroup S as follows: ρr = {(a, b) ∈ S × S | aγt = bγt for all t ∈ S and γ ∈ Γ}; ρl = {(a, b) ∈ S × S | tγa = tγb for all t ∈ S and γ ∈ Γ}. If S is a regular Γ-semigroup, we obtain that ρr and ρl are the minimum right and left reductive congruences on S, respectively.