Original Article

, Volume: 15( 1)

# Left Ã¯ÂÂ -Filters on Ã¯ÂÂ -Semigroups

- *Correspondence:
- Jyothi V , Department of Mathematics, K.L. University, Guntur, Andhra Pradesh, India,
**Tel:**9581423642;**E-mail:**[email protected]

**Received:** February 03, 2017; **Accepted:** February 23, 2017; **Published:** March 06, 2017

**Citation:** Jyothi V, Sarala Y, Madhusudhana Rao D, et al. Left -Filters on -Semigroups. Int J Chem Sci. 2017;15(1):104.

### Abstract

In this article we define left -filters, right -filters and prime left -ideal in -semigroup and characterize -semigroups in terms of these notions. Finally, we give the relation between the left -filters and the prime right -ideals

### Keywords

*Nano powder; Hexagonal wurtzite structure; Chemical precipitation; X-ray diffraction*

### Introduction

Anjaneyulu [1] initiated the study of ideals in semigroups Petrich [2] made a study on filters in general semigroups. Lee and Lee [3] introduced the notion of a left filter in a PO semigroup. Kehayopulu [4-6] gave the characterization of the filters of S in terms of prime ideals in ordered semigroups [7-9]. Sen [10] introduced -semigroups in 1981. Saha [11] introduced -semigroups different from the first definition of -semigroups in the sense of sen.

Let S and be two nonempty sets. Then S is said to be a -semigroup if there exist a mapping from SXXS →S which maps (a,α,b)→aαb satisfying the condition and [8].

Let S be a -semigroup. If A and B are two subsets of S, we shall denote the set by AB.

Let S be a - semigroup. A non-empty subset *A* of S is called a right -ideal of S if . A non empty subset
A of a -semigroup S is a right -ideal of S if ,, implies [8].

Let S be a -semigroup. A non empty *A* of S is called a left -ideal of S if . A nonempty subset *A* of
a -semigroup S is a right -ideal of S if , , implies . *A* is called an -ideal of S
if it is a right and left -ideal of S.

A subset *T *of *S* is called a prime if or for subsets *A,B *of *S*. *T* is called a prime
right ideal if *T* is prime as a right ideal. *T* is called a prime left ideal if *T* is a prime as a left ideal. *T* is called a prime
ideal if *T* is prime as an ideal [11].

We now introduce the left -filter, right -filter and -filter.

A -sub semigroup *F* of a -semigroup *S* is called a left -filter of S if for
. A -semigroup F of a -semigroup S is called a right -filter of S if for
. [13].

**Theorem (1)**

Let *S* be a -semigroup and F a non-empty subset of S. The following are equivalent.

1. F is a left -filter of S.

2. S \ F = or S \ F is a prime right -ideal.

**Proof:** (1)⇒(2) : Suppose that S \ F # . Let x∈S \ F;α∈ and y∈S . Then *xαy*∈S \ F . Indeed: If
*xαy*S \ F ; then xαy∈F. Since F is a left -filter, x∈F. It is impossible. Thus *xαy*∈S \ F, and so
(S \ F)S ⊆ S \ F. Therefore S \ F is a right ideal.

Next, we shall prove that S \ F is a prime.

Let *xαy*∈S \ F for *x, y *∈ S and* α*∈. Suppose that xS \ F and yS \ F. Then x∈F and y∈F. Since
F is a sub semigroup of S , x*α*y∈F. It is impossible. Thus x∈S \ F or y∈S \ F. Hence S \ F is a prime, and so
S \ F is a prime right - ideal.

(2)⇒(1) : If *S* \ *F* = then *F* = *S*. Thus F is a left -filter of S. Next assume that *S* \ *F* is a prime right -
ideal of *S*. Then *F* is a -sub semigroup of S. Indeed: Suppose that x*α*yF for x, y∈F and *α*∈. Then x*α*y∈S \ F for x, y∈F and *α*∈. Since *S* \ *F* is prime, x, y∈S \ F. It is impossible. Thus *xαy*∈F and so *F* is a sub semigroup of *S*.

Let* xαy*∈*F* for x, y∈S and *α*∈. Then *x*∈*F*. Indeed: If *x**F*, then x∈*S* \ *F*. Since *S *\ *F* is a prime
right -ideal of S, *xαy*∈(S \ F)S ⊆ S \ F. It is impossible. Thus *x*∈*F*. Therefore *F* is a left filter of *S*.

**Theorem (2)**

Let *S* be a -semigroup and *F* be a non-empty subset of *S*. The following are equivalent.

(1) F is a right filter of S.

(2) S \ F = or S \ F is a prime left -ideal.

**Proof:** (1)⇒(2) :Suppose that *S* \ *F* =. Let y∈S \ F; *α*∈ and y∈S. Then *xαy*∈S \ *F*. Indeed: If
*xαy*∈S \ F; then *xαy*∈*F*. Since *F* is a right -filter, y∈*F*. It is impossible. Thus *xαy*∈S \ *F*, and so
S(S \ F) ⊆ S \ F. Therefore S \ F is a left -ideal.

Next, we shall prove that S \ F is a prime.

Let* xαy*∈*S* \ *F* for x, y∈S and *α*∈. Suppose that xS \ F and yS \ F. Then x∈F and y∈F.
Since F is a sub semigroup of S, *xαy*∈*F*. It is impossible. Thus x∈S \ F or y∈S \ F. Hence S \ F is a prime
and so that S \ F is a prime left -ideal.

(2)⇒(1) : If S \ F = then *S* = *F* . Thus *F* is a right -filter of S. Next assume that S \ F is a prime left -
ideal of S. Then F is a -sub semigroup of S. Indeed: Suppose that for x, y∈F and *α*∈. Then *xαy*∈S \ F
for x, y∈F and *α*∈. Since S \ F is a prime, x, y∈S \ F. It is impossible. Thus* xαy*∈F; *α*∈ and so F
is a sub semigroup of S.

Let *xαy*∈F for x, y∈S and *α*∈. Then yF. Indeed: If yF, then y∈S \ F. Since S \ F is a prime
right ideal of S, *xαy* ∈ S(S \ F) S \ F. It is impossible. Thus y∈F. Therefore F is a right filter of S.
From theorem 2.6 and 2.7, we get the following.

**Corollary: **Let S be a -semigroup and *F* be a non-empty subset of *S*. The following are equivalent.

(1) F is a filter of S.

(2) S \ F = or S \ F is a prime -ideal of S.

**Proof: (1)⇒(2) : **Assume that S \ F =.

By theorem (1), S \ F is a right ideal.

By theorem (2), S \ F is a left ideal.

By theorem (1) and (2), S \ F is a ideal.

By theorem (2) and (2), S \ F is a prime ideal of S.

(2)⇒(1) : If S \ F = then F = S. Thus F is a -filter of S. Next assume that S \ F is a prime -ideal of
S. By theorem (1) and (2). F is a -subsemigroup of S. Let *xαy* ∈F for x, y∈S and *α*∈. By theorem (1);
F is a left -filter of S. By theorem (2); F is a right -filter of S. Therefore F is a -filter of S.

### Conclusion

This concept is used in filters of chemistry, physical chemistry, electronics.

### References

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