Original Article

, Volume: 15( 1)

• View PDF
• Download PDF

Left  -Filters on  -Semigroups

*Correspondence:

Jyothi V , Department of Mathematics, K.L. University, Guntur, Andhra Pradesh, India, Tel:9581423642; E-mail: jyothi.mindspace@gmail.com

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 xαyS \ 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 , 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 - 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 xαyF for x, y∈F and α∈. Then xαy∈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 xF, 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 yF. 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

1. AnjeaneyuluA. Structure and ideal theory of semigroups, thesis,ANU. 1980.
2. Petrich M. The introduction to semigroups. Merrill Publishing Company, Columbus, Ohio. 1973.
3. Lee SK, Kwon YI. ?On left regular po-semigroups?, Comm. Korean Math Soc. 1998;13:1-6.
4. KehayopuluN.On filters generalized in po-semigroups.MathJapan.1990;35:789-96.
5. KehayopuluN.Remark on ordered semigroups.MathJapan.1990;35:1061-63.
6. Kehayopulu N.On left regular orderedsemigroups. MathJapan.1990;35:1057-60.
7. LeeSK,Kwon YI. ?On left regular po-semigroups?, Comm. Korean Math Soc.1998;13:1-6.
8. Madhusudhana RaoD, Anjaneyulu A,Gangadhara RaoA. Prime-radicals in semigroups. Inte-j math Eng. 2011; 116:1074-81.
9. Clifford AH, Preston GB. The algebraic theory of semigroups. Am math soc Providence II. 1967.
10. Sen MK.On-semigroups. Proceeding of international Conference of Algebra and its Application, Decker publication, New York-301.
11. Saha NK. On-semigroups-III. BullCulMathSoc.1988;116:1-12.