**InternationaleJournals**

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

**VB
Subrahmanyeswara Rao Seetamraju ^{1}, A. Anjaneyulu^{2}, D.
Madhusudana Rao^{3}.**

** ^{1}**Dept. of
Mathematics, V K R, V N B & A G K College of Engg, Gudivada, A.P. India.

** ^{2}**Dept. of
Mathematics, V S R & N V R College, Tenali, A.P. India.

** ^{3}**Dept. of
Mathematics, V S R & N V R College, Tenali, A.P. India.

**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 ρ_{1} and ρ_{2} are two left Г-congruences(resp. right Г-congruences, Γ-congruences) of S, then ρ_{1} o ρ_{2}is 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, *x*Γ*a*
= *x*Г*b* if and only if *y*Г *a* = *y*Г*b* 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