• 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].
Keywords
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