Original Article
Int. J. Chem. Sci, Volume: 15( 1)

Left  -Filters on  -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 image -Filters on image -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 image -semigroups in 1981. Saha [11] introduced image -semigroups different from the first definition of image -semigroups in the sense of sen.

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

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

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

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

A subset T of S is called a prime if image or image 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 image -filter, right image -filter and image -filter.

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

Theorem (1)

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

1. F is a left image -filter of S.

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

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

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

Let xαy∈S \ F for x, y ∈ S and αimage. Suppose that ximageS \ F and yimageS \ 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 S \ F is a prime right image - ideal.

(2)⇒(1) : If S \ F = image then F = S. Thus F is a left  -filter of S. Next assume that S \ F is a prime right image - ideal of S. Then F is a image -sub semigroup of S. Indeed: Suppose that xαyimageF for x, y∈F and αimage. Then xαy∈S \ F for x, y∈F and αimage. 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 xαyF for x, y∈S and αimage. Then xF. Indeed: If ximageF, then x∈S \ F. Since S \ F is a prime right image -ideal of S, xαy∈(S \ F)imageS ⊆ S \ F. It is impossible. Thus xF. Therefore F is a left filter of S.

Theorem (2)

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

(1) F is a right image filter of S.

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

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

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

Let xαyS \ F for x, y∈S and αimage. Suppose that ximageS \ F and yimageS \ F. Then x∈F and y∈F. Since F is a sub semigroup of S, xαyF. 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 image -ideal.

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

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

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

(1) F is a image filter of S.

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

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

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

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

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

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

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

Conclusion

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

References