On $\Gamma$-hypersemigroups

Abstract

: The results on Γ -hypersemigroups are obtained either as corollaries of corresponding results on ∨ e or poe - semigroups or on the line of the corresponding results on le -semigroups. It has come to our attention that Theorem 3.22 in [4] cannot be obtained as corollary to Theorem 2.2 of the same paper as for a Γ -hypersemigroup, ( P ∗ ( M ) , Γ , ⊆ ) is a ∨ e -semigroup and not an le -semigroup. Also on p. 1850, l. 12 in [4], the “ le -semigroup” should be changed to “ ∨ e -semigroup”. In the present paper we prove Theorems 3.26 and 3.28 stated without proof in [4]. On this occasion, some further results are given to emphasize what we say. The results on Γ -hypersemigroups are obtained from the more abstract structure of the poe -semigroups. Further investigation on poe -semigroups and le -semigroups is interesting.