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IDEALS IN PARTIALLY ORDERED TERNARY SEMIGROUPS

V. Siva Rami Reddy1, V. Sambasiva Rao2, A. Anjaneyulu 3,A. Gangadhara Rao 4
1Dept. of Mathematics, NRI Engineering College, Guntur,
2Acharya Nagarjuna University, Guntur, 3,4V S R & N V R College, Tenali.

ABSTRACT

In this paper, the terms, partially ordered ternary semigroup, po ternary subsemigroup, po ternary subsemigroup generated by a subset, two sided identity of a po ternary semigroup, left zero, right zero, zero of a po ternary semigroup, po left ideal, po lateral ideal, po right ideal, po  two sided ideal and po ideal in a po ternary semigroup are introduced. It is proved that, if T is a po ternary semigroup and A T, B T, then (i) A (A],  (ii) ((A]] = (A],  (iii) (A](B](C] (ABC] and (iv) A B A (B], (v) A B (A] (B]. It is proved that the nonempty intersection of any family of po ternary subsemigroups of a po ternary semigroup T is a po ternary subsemigroup of T.  It is proved that (1) the nonempty intersection of any family of po left ideals (or po lateral ideals or po right ideals or po two sided ideals or po ideals )  of a po ternary semigroup T is a po left ideal ( or po lateral ideals or po right ideals or po two sided ideals or po ideals) of T, (2) the union of any family of po left ideals (or po lateral ideals or po right ideals or po two sided ideals or po ideals)  of a po ternary semigroup T is a po left ideal
(or po lateral ideals or po right ideals or po two sided ideals or po ideals) of T.  Let T be a po-ternary semigroup and A is a nonempty subset of T, then it is proved that (1) L(A) = (A
TTA],
(2)
M(A) = (A
TAT TTATT], (3) T(A) = (A TTAATT TTATT] and (4) J(A) =
(A
TTA∪TAT∪ATT TTATT].

Mathematical subject classification (2010) : 20M07; 20M11; 20M12.

KEY WORDS : partially ordered ternary semigroup, po ternary subsemigroup, po ternary subsemigroup generated by a subset, cyclic po ternary subsemigroup of a po ternary semigroup, two sided identity of a po ternary semigroup, zero of a po ternary semigroup, po left ideal, po lateral ideal, po right ideal, two sided po ideal and po ideal.

International eJournal of Mathematical Sciences, Technology and Humanities

Volume 4, Issue 1, Pages:  1177 - 1194