• PO IDEALS IN PARTIALLY ORDERED SEMIGROUPS

Abstract

In this paper, the terms, partially ordered semigroup, posubsemigroup, posubsemigroup generated by a subset, two sided identity of a posemigroup, left zero, right zero, zero of a posemigroup, poleft ideal,poright ideal, po ideal, po ideal generated bya subset and po ideal generated by an element a in a posemigroup are introduced.It is proved that, if S is a posemigroup and A ⊆S, B ⊆S, then (i) A ⊆ (A],  (ii) ((A]] = (A],  (iii) (A](B] ⊆ (AB] and (iv) A ⊆ B ⇒ A ⊆ (B], (v) A ⊆ B ⇒ (A] ⊆ (B].It is proved that the nonempty intersection of any family of posubsemigroups of a posemigroup S is a posubsemigroup of S.  It is proved that (1) the nonempty intersection of any family of poleft ideals (orporight ideals orpoideals )  of a posemigroupS is a poleft ideal (or po right ideals or po ideals ) of S, (2) the union of any family of poleft ideals (or po right ideals or po ideals )  of a posemigroup S is a poleft ideal (or po right ideals or po ideals ) of S.  Let S be a posemigroup and A is a nonempty subset of S, then it is proved that

(1) L(A) = (A ∪SA]

(2) R(A) = (A ∪ AS] and (3) J(A) = (A ∪SA∪SAS∪AS].