**InternationaleJournals**

**PO-**r**-FILTERSIN PO-**r**-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 Engineering, Gudivada, A.P. India. Email Id: manyam4463@gmail.com

^{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 left po-r-filter, right po-r-filter, po-r-filter, are introduced. It is
proved that a nonempty subset F of a po-r-semigroup S is a left po-r-filter if and only if S/F is a completely prime
right po-r-ideal of S or empty. Further
it is proved that S is a po-r-semigroup and F is a left po-r-filter, then S/F is a prime right po-r-ideal of S or empty and A nonempty subset F of a
commutative po-r-semigroup S is a left po-r-filter if and only if S/F is a prime right po-r-ideal of S or empty.
It is proved that a nonempty subset F of a po-r-semigroup S is a right po-r-filterif and only if S/F is a completely prime
left po-r-ideal of S or empty. It is proved that every po-r-filter F of a po-r-semigroupS is a po-*c-*system.** **Further
it is also proved that a nonempty subset F of a po-r-semigroup S is a po-r-filter if and only if S/F is a completely prime po-r-ideal of S or empty. It is provedthat every po-r-filter F of a po-r-semigroup S is a po-*m*-system. It is proved that,
if a nonempty subset F of a po-r-semigroup S is a po-r-filter, then F is a po-*d*-system of S or empty. Further it is proved that, every po-r-filter F of a po-r-semigroup S is a po-*n*-system of S. It is proved that the po-r-filter of a po-r-semigroup S generated by a nonempty subset A of S is the intersection of all po-r-filters of S containing A. It is proved that if
N(*b*) ⊆ N(*a*),
then N(*a*)\N(*b*), if it is nonempty, is a completely prime po-r-ideal of N(*a*).

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

**KEY
WORDS: **po-r-semigroup, po-r-ideal, prime po-r-ideal, po-r-filter.

**International
eJournal of Mathematical Sciences, Technology and Humanities**

**Volume** **2**,
**Issue** **4**, **Pages**: 669 - 683