**InternationaleJournals**

**IDEALS IN PARTIALLY
ORDERED TERNARY SEMIGROUPS**

**V. Siva Rami Reddy ^{1}, V.
Sambasiva Rao^{2}, A. Anjaneyulu^{ 3},A. Gangadhara Rao^{ 4}**

**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 ∪TTA∪ATT∪ 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