InternationaleJournals

Г-CONGRUENCE and Ћ-CLASSES in PO-Г-SEMIGROUPS

VB Subrahmanyeswara Rao Seetamraju1, A. Anjaneyulu2, D. Madhusudana Rao3.

1Dept. of Mathematics, V K R, V N B & A G K College of Engg, Gudivada, A.P. India. Email:

2Dept. of Mathematics, V S R & N V R College, Tenali, A.P. India. Email:

3Dept. of Mathematics, V S R & N V R College, Tenali, A.P. India. Email:

ABSTRACT

The terms Г-congruence,semilattice, semilattice Г-congruence and complete are introduced. It is provedthat an equivalence relation ρ on a po-Г-semigroup S is a Г-congruence if andonly if for all a, b, c, d S, α Γ, a ρ b and c ρ d implies a ρ c ρ b ρ d.  It is proved that if S be a po-Γ- semigroup and ρ1 and ρ2 are two left Г-congruences(resp. right Г-congruences, Γ-congruences) of S, then ρ1 o ρ2is a left Г-congruence (resp.right Г-congruence, Г-congruence) of S.  Further it is also proved that if S is a po-Г-semigroup and are left Γ-congruences (resp. right congruences, congruences) of S, then  is a left Г-congruence(resp. right Г-congruence, Г-congruence) of S.  The term, Ћ-class, Ћ-simple and Ћ-subset areintroduced.  It is proved that if S is a po-Г-semigroup,z S and A is a po-Г-ideal of an Ћ-class (z)Ћ ,then A has no proper completely prime po-Г-ideals. It is proved that if S is a po-Г-semigroup, A is a completely prime po-Г-ideal ofS,then A = {(a)Ћ / a A}.  It is proved that every po-Г-semigroup is a semilattice of Ћ-simple po-Г-semigroups. It is proved that every Ћ-simple po-Г-subsemigroupT of a po-Г-semigroup S is contained in an Ћ-class of S.  It is proved that if A is a po-Г-ideal of apo-Г-semigroup S, then the conditions (i) A is the intersection of all completelyprime po-Г-ideals of S containing A  (ii) A is the intersection of all minimal completely prime po-Г-ideals of Scontaining A  (iii) A is the union of Ћ-classes.  (iv) A is a completely semiprime po-Г-ideal of S are equivalent.  It is proved that if a nonempty subset Ћ of a po-Г-semigroupS is an Ћ-subset,then Ћ is a class of semilattice Г-congruence.  It is proved that a po-Г-semigroup S is separative if and only if S is a semilattice ofstrongly Г-cancellative po-Г-semigroup.  If so, the relation σ defined on S by x σ y if for any a, b S, xΓa = xГb if and only if yГ a = yГb is the greatest band Г-congruence on S all whose classes are strongly Г-cancellative.

MATHEMATICS SUBJECT CLASIFICATION (2010): 06F05, 06F99, 20M10, 20M99

KEY WORDS: Г-congruence, semilattice Г-congruence, Ћ-class.

International eJournal of Mathematics and Engineering

Volume 4, Issue 1, Pages:  1969 - 1974