• PO IDEALS IN PARTIALLY ORDERED SEMIGROUPS

Padmalatha A., Gangadhara Rao*, A. Anjaneyulu, India.

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 SASASAS].


Keywords


partially ordered semigroup, posubsemigroup, posubsemigroup generated by a subset, cyclic posubsemi -group of a posemigroup, two sided identity of a posemigroup, zero of a posemigroup,poleft ideal, poright ideal, poideal, po ideal generated by a subset an

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