Characterizations of hemirings by the properties of

Annals of Fuzzy Mathematics and Informatics Volume 3, No. 2, (April 2012), pp. 229- 242 ISSN 2093–9310 http://www.afmi.or.kr

@FMI c Kyung Moon Sa Co.

http://www.kyungmoon.com

Characterizations of hemirings by the properties of their interval valued fuzzy ideals Muhammad Shabir, Nosheen Malik, Tahir Mahmood Received 23 March 2011; Revised 14 May 2011; Accepted 22 June 2011

Abstract. In this paper we give some basic results for interval valued fuzzy ideals and characterize regular and weakly regular hemirings by the properties of their interval valued fuzzy ideals.

2010 AMS Classification: 16Y60, 08A72, 03G25, 03E72 Keywords: ideal.

Regular hemiring, Weakly regular hemiring, Interval valued fuzzy

Corresponding Author: Tahir Mahmood ([email protected])

1. Introduction

A

set R 6= ∅ together with two associative binary operations, addition “+” and multiplication “·” such that “·” distributes over “+”, is called semiring and was first introduced by Vandiver [18] in 1934. Additively commutative semirings with zero element are called hemirings. In more recent times semirings have been deeply studied, especially in relation with applications [8]. Semirings have also been used for studying optimization, graph theory, theory of discrete event dynamical systems, matrices, determinants, generalized fuzzy computation, theory of automata, formal language theory, coding theory, analysis of computer programmes [5, 6, 8, 9, 19]. Hemirings, appears in a natural manner, in some applications to the theory of automata, the theory of formal languages and in computer sciences [10, 11, 12, 16]. Ideals of hemirings and semirings play an important role in the structure theory and are very useful for many purposes. Some important types of ideals are discussed in [15, 21]. In [3] J. Ahsan characterize weakly regular hemirings by the properties of their ideals. In 1965 Zadeh [20] introduced the concept of fuzzy sets. Since then fuzzy sets have been applied to many branches in Mathematics. The fuzzification of algebraic structures was initiated by Rosenfeld [17] and he introduced the notion of fuzzy subgroups. In [2] J. Ahsan initiated the study of fuzzy semirings (See also [4]). The fuzzy algebraic structures play an important role in Mathematics with

M. Shabir et al./Ann. Fuzzy Math. Inform. 3 (2012), No. 2, 229–242

wide applications in many other branches such as theoretical physics, computer sciences, control engineering, information sciences, coding theory and topological spaces [1, 10, 19]. The concept of interval valued fuzzy sets in algebra was initiated in [7] by Biswas and further this concept was investigated in [13]. In [14] Xueling defined and discussed interval valued fuzzy ideals in hemirings. In this paper we give some basic results for interval valued fuzzy ideals and characterize regular and weakly regular hemirings by the properties of their interval valued fuzzy ideals. 2. Preliminaries A set R 6= φ together with two binary operations addition “+” and multiplication “·” is called semiring if (R, +) and (R, ·) are semigroups and multiplication is distributive over addition, that is for all a, b, c ∈ R, a (b + c) = ab + ac and (a + b) c = ac + bc An element 0 ∈ R satisfying the conditions, 0x = x0 = 0 and 0 + x = x + 0 = x, for all x ∈ R, is called a zero of the semiring (R, +, ·). And an element 1 ∈ R satisfying the condition, 1x = x1 = x for all x ∈ R, is called identity of the semiring (R, +, ·). A semiring with commutative multiplication is called a commutative semiring. A semiring with commutative addition and zero element is called a hemiring. A non-empty A ⊆ R is called a sub hemiring of R if it contains zero and is closed with respect to the addition and multiplication of R. An element a ∈ R is called multiplicatively idempotent if a2 = a. A hemiring R is called multiplicatively idempotent if each element of R is multiplicatively idempotent. A non-empty A ⊆ R is called a left (right) ideal of R if A is closed under addition and RA ⊆ A (AR ⊆ A). If A and B are left (respectively right) ideals of a hemiring R then A ∩ B is a left (respectively right) ideal of R. If A and B are left (respectively right) ideals of a hemiring R then A + B is the smallest left (respectively right) ideal of R containing both A and B. If A and B are ideals of a hemiring R then AB is an ideal of R contained in A ∩ B. If A ⊆ R, then characteristic function CA of A is a function from X into {0, 1}, defined by 1 if x ∈ A, CA (x) := 0 otherwise, A fuzzy subset λ of a universe X is a function λ : X → [0, 1]. For any two fuzzy subsets λ and µ of X, λ ≤ µ means that, for all x ∈ X, λ(x) ≤ µ(x). The symbols λ ∧ µ and λ ∨ µ will mean the following fuzzy subsets of X (λ ∧ µ)(x) = λ(x) ∧ µ(x) (λ ∨ µ)(x) = λ(x) ∨ µ(x) for all x ∈ X. More generally, if {λi : i ∈ Λ} is a family of fuzzy subsets of X, then ∧i∈Λ λi and ∨i∈Λ λi are defined by (∧i∈Λ λi )(x) = ∧i∈Λ (λi (x)) (∨i∈Λ λi )(x) = ∨i∈Λ (λi (x)) and are called the intersection and the union of the family {λi : i ∈ Λ} of fuzzy subsets of X, respectively. 230


Definition 2.1. Let λ and µ be any two fuzzy subsets of a hemiring R. Then the sum of λ and µ is defined as (λ + µ)(x) = ∨x=y+z [λ(y) ∧ µ(z)] for all x ∈ R. Definition 2.2. Let λ and µ be any two fuzzy subsets of a hemiring R. Then the product of λ and µ is defined as (λµ)(x) = ∨x=Σni=1 yi zi [∧i {λ(yi ) ∧ µ(zi )}] for all x ∈ R. Definition 2.3. A hemiring R is called von Neumann regular if for any a ∈ R there exists x ∈ R such that a = axa or a ∈ aRa for all a ∈ R. Theorem 2.4. A hemiring R is von Neumann regular if and only if for any right ideal A and any left ideal B of R, A ∩ B = AB. Definition 2.5 ([3]). A hemiring R is called left (respectively right) weakly regular if we have a ∈ RaRa (respectively a ∈ aRaR) for all a ∈ R. Remark 2.6. Clearly if R is commutative then R is (right or left) weakly regular if and only if R is von-Neumann regular. Theorem 2.7 ([4]). The following assertions for a hemiring R are equivalent: (i) R is right weakly regular. (ii) All right ideals of R are idempotent. (iii) IJ = I ∩ J for all right ideals I and two-sided ideas J of R. Theorem 2.8. If R is commutative hemiring, then R is fully idempotent if and only if R is von Neumann regular. Theorem 2.9. If R is commutative hemiring then R is fully idempotent iff R is weakly regular. Definition 2.10. Let λ be a fuzzy subset of a hemiring R. Then λ is said to be a fuzzy subhemiring of R if for all x, y ∈ R, (i) λ (x + y) ≥ λ (x) ∧ λ (y) (ii) λ (xy) ≥ λ (x) ∧ λ (y). Definition 2.11. A fuzzy subset λ of a hemiring R is said to be a fuzzy left (respectively right) ideal of the hemiring R if for all x, y ∈ R (i) λ (x + y) ≥ λ (x) ∧ λ (y) , (ii) λ (xy) ≥ λ (y) (respectively λ (xy) ≥ λ (x)). A fuzzy subset λ of a hemiring R is called a fuzzy ideal of hemiring R if it is both fuzzy left and right ideal of R Theorem 2.12. If λ and µ are fuzzy left(respectively right) ideals of a hemiring R then λ ∩ µ is also a fuzzy left (respectively right) ideal of R. Theorem 2.13. If λ and µ are fuzzy left(respectively right) ideals of a hemiring R then their sum λ + µ is also a fuzzy left (respectively right) ideal of R. Theorem 2.14. If λ and µ are fuzzy left (respectively right) ideals of a hemiring R then their product λµ is also a fuzzy left (respectively right) ideal of R. 231


3. Major Section ˜ = Let L be the family of all closed subintervals of [0, 1] with minimal element O 0 ˜ [0, 0] and maximal element I = [1, 1] according to the partial order [α, α ] ≤ [β, β 0 ] if and only if α ≤ β, α0 ≤ β 0 defined on L for all [α, α0 ] , [β, β 0 ] ∈ L. ˜ of a hemiring R is a function Definition 3.1. An interval valued fuzzy subset λ ˜ λ : R → L. ˜ We write λ(x) = [λ− (x), λ+ (x)] ⊆ [0, 1] for all x ∈ R, where λ− , λ+ : R → [0, 1] are lower and upper fuzzy sets of R, giving lower and upper limit of the image interval for each x ∈ R. Note that we have 0 ≤ λ− (x) ≤ λ+ (x) ≤ 1 for all x ∈ R. ˜ = [λ− , λ+ ]. For simplicity we write λ ˜ and µ Definition 3.2. For any two interval valued fuzzy subsets λ ˜ of a hemiring R, union and intersection are defined as ˜∪µ λ ˜ (x) = λ− (x) ∨ µ− (x) , λ+ (x) ∨ µ+ (x) ˜∩µ λ ˜ (x) = λ− (x) ∧ µ− (x) , λ+ (x) ∧ µ+ (x) , for all x ∈ R. More generally if then for all x ∈ R,

n o ˜ i : i ∈ I is a family of interval valued fuzzy subsets of R λ ˜ i )(x) = ∨i λ− (x), ∨i λ+ (x) (∪i λ i i + (∩i λi )(x) = ∧i λ− i (x), ∧i λi (x) .

˜ and µ Definition 3.3. Let λ ˜ be interval valued fuzzy subsets of a hemiring R. Then their sum is defined as ˜+µ (λ ˜)(x) = ∨x=y+z λ− (y) ∧ µ− (z) , λ+ (y) ∧ µ+ (z) for all x ∈ R. ˜ and µ Definition 3.4. Let λ ˜ be interval valued fuzzy subsets of a hemiring R. Then their product is defined as ˜ µ)(x) = ∨x=Σn y z ∧i λ− (yi ) ∧ µ− (zi ), λ+ (yi ) ∧ µ+ (zi ) (λ˜ i=1 i i Pn ˜ µ)(x) = O. ˜ if x can be expressed as x = i=1 yi zi , otherwise (λ˜ Definition 3.5. Let A be a subset of a hemiring R. Then the interval valued characteristic function CÃ of A is defined to be a function CÃ : R → L such that for all x ∈ R I˜ = [1, 1] if x ∈ A CÃ (x) = ˜ = [0, 0] O if x ∈ / A. Clearly the interval valued characteristic function of any subset of R is also an interval valued fuzzy subset of R. The interval valued characteristic function can be used to indicate either membership or non-membership of any member of R in a subset A of R. Note that C˜R (x) = I˜ for all x ∈ R. 232


˜ µ Lemma 3.6. Let λ, ˜ and ν˜ be the interval valued fuzzy subsets of a hemiring R. ˜ ˜ ˜ ⊆ ν˜µ If λ ⊆ µ ˜ then λ˜ ν⊆µ ˜ν˜ and ν˜λ ˜. Pn Proof. Let x ∈ R. If x is not expressed as x = i=1 yi zi , then ˜ ν (x) = O ˜ = (˜ λ˜ µν˜) (x) Otherwise, ˜ ν )(x) (λ˜

= ∨x=Σni=1 yi zi ∧i λ− (yi ) ∧ ν − (zi ), λ+ (yi ) ∧ ν + (zi ) ≤ ∨x=Σni=1 yi zi ∧i µ− (yi ) ∧ ν − (zi ), µ+ (yi ) ∧ ν + (zi ) since λ ⊆ µ =

(˜ µν˜) (x) .

˜ν ⊆ µ ˜ ⊆ ν˜µ Hence λ˜ ˜ν˜. Similarly we can prove that ν˜λ ˜.

˜ be an interval valued fuzzy subset of a hemiring R. Then λ ˜ Definition 3.7. Let λ is said to be an interval valued fuzzy subhemiring of R if for all x, y ∈ R, ˜ (x + y) ≥ λ ˜ (x) ∧ λ ˜ (y) and λ ˜ (xy) ≥ λ ˜ (x) ∧ λ ˜ (y) . λ ˜ be an interval valued fuzzy subset of a hemiring R. Definition 3.8 ([14]). Let λ ˜ Then λ is said to be an interval valued fuzzy left (resp. right) ideal of R if and only if for all x, y ∈ R ˜ (x + y) ≥ λ ˜ (x) ∧ λ ˜ (y) (i) λ ˜ (xy) ≥ λ ˜ (y) respectively λ ˜ (xy) ≥ λ ˜ (x) . (ii) λ An interval valued fuzzy subset of R is called an interval valued fuzzy ideal of hemiring R if it is both interval valued fuzzy left and right ideal of R. ˜ of a hemiring R is an interval valued Remark 3.9. An interval valued fuzzy subset λ fuzzy two sided ideal of R if for all x, y ∈ R ˜ (x + y) ≥ λ ˜ (x) ∧ λ ˜ (y) (i) λ ˜ ˜ ˜ (ii) λ (xy) ≥ λ (x) ∨ λ (y). Remark 3.10. Every interval valued fuzzy ideal of a hemiring R is also an interval valued fuzzy subhemiring of R but the converse is not true. ˜ = [λ− , λ+ ] is an interval valued fuzzy left ideal of R Remark 3.11. Note that if λ then for all x, y ∈ R ˜ (x + y) ≥ λ ˜ (x) ∧ λ ˜ (y) λ − ⇒ λ (x + y), λ+ (x + y) ≥ λ− (x) , λ+ (x) ∧ λ− (y) , λ+ (y) ⇒ λ− (x + y), λ+ (x + y) ≥ [λ− (x) ∧ λ− (y) , λ+ (y) ∧ λ+ (y)] ⇒ λ− (x + y) ≥ λ− (x) ∧ λ− (y) and λ+ (x + y) ≥ λ+ (x) ∧ λ+ (y) and ˜ (xy) ≥ λ ˜ (y) ⇒ λ− (xy), λ+ (xy) ≥ λ− (y), λ+ (y) λ ⇒ λ− (xy) ≥ λ− (y) and λ+ (xy) ≥ λ+ (y). 233


This shows that λ− and λ+ are fuzzy left ideals of R. It’s converse is also true and ˜ = [λ− , λ+ ] is an interval can be proved by reversing the above process. Similarly λ − valued fuzzy right (two sided) ideal of R if and only if λ and λ+ are fuzzy right (two ˜ = [λ− , λ+ ] is an interval valued fuzzy subhemiring of sided) ideals of R. Similarly λ − + R if and only if λ and λ are fuzzy subhemirings of R. ˜ of a hemiring R is an interval Lemma 3.12. An interval valued fuzzy subset λ ˜+λ ˜⊆λ ˜ and λ ˜ 2 ⊆ λ. ˜ valued fuzzy subhemiring of R if and only if λ ˜ be an an interval valued fuzzy subhemiring of R. Then for all x ∈ R Proof. Let λ ˜+λ ˜ (x) = ∨x=y+z λ− (y) ∧ λ− (z) , λ+ (y) ∧ λ+ (z) λ ≤ ∨x=y+z λ− (y + z) , λ+ (y + z) = ∨x=y+z λ− (x) , λ+ (x) ˜ (x) . = λ ˜+λ ˜ ⊆ λ. ˜ And λ ˜ 2 (x) = O ˜ (x) if x cannot be expressed as x = Pn yi zi . ˜≤λ Thus λ i=1 Otherwise, ˜ 2 (x) = ˜λ ˜ (x) λ λ = ∨x=Σni=1 yi zi ∧i λ− (yi ) ∧ λ− (zi ), λ+ (yi ) ∧ λ+ (zi ) ≤ ∨x=Σni=1 yi zi ∧i λ− (yi zi ), ∧i λ+ (yi zi ) " # n n X X − + ≤ ∨x=Σni=1 yi zi λ ( yi zi ), λ ( y i zi ) i=1

i=1

= ∨x=Σni=1 yi zi λ− (x), λ+ (x) ˜ (x) . = λ ˜ 2 ⊆ λ. ˜ Thus λ ˜+λ ˜⊆λ ˜ and Conversely, let λ be an interval valued fuzzy subset of R such that λ 2 ˜ ˜ λ ⊆ λ. Then for all x, y ∈ R ˜ (x + y) ≥ ˜+λ ˜ (x + y) λ λ = ∨x+y=a+b λ− (a) ∧ λ− (b) , λ+ (a) ∧ λ+ (b) ≥ λ− (x) ∧ λ− (y) , λ+ (x) ∧ λ+ (y) = λ− (x) , λ+ (x) ∧ λ− (y) , λ+ (y) ˜ (x) ∧ λ ˜ (y) = λ and ˜ (xy) ≥ λ ˜ 2 (xy) = λ ˜λ ˜ (xy) λ − = ∨xy=Σni=1 yi zi ∧i λ (yi ) ∧ λ− (zi ), λ+ (yi ) ∧ λ+ (zi ) ≥ λ− (x) ∧ λ− (y), λ+ (x) ∧ λ+ (y) − = λ (x) , λ+ (x) ∧ λ− (y) , λ+ (y) ˜ (x) ∧ λ ˜ (y) . = λ 234


˜ (x + y) ≥ λ ˜ (x) ∧ λ ˜ (y) and λ ˜ (xy) ≥ λ ˜ (x) ∧ λ ˜ (y) for all x, y ∈ R. Hence λ ˜ is Thus λ an interval valued fuzzy subhemiring of R. ˜ of a hemiring R is an interval Lemma 3.13. An interval valued fuzzy subset λ ˜+λ ˜⊆λ ˜ and CR λ ˜⊆λ ˜ valued fuzzy left (respectively right) ideal of R if and only if λ ˜ ˜ (respectively λCR ⊆ λ ). ˜ be an interval valued fuzzy left ideal of R. Then for all x ∈ R Proof. Let λ ˜+λ ˜ (x) = ∨x=y+z λ− (y) ∧ λ− (z) , λ+ (y) ∧ λ+ (z) λ ≤ ∨x=y+z λ− (y + z) , λ+ (y + z) = ∨x=y+z λ− (x) , λ+ (x) ˜ (x) . = λ ˜ λ ˜ ⊆ λ. ˜ And CR λ ˜ (x) = O ˜ (x) if x cannot be expressed as x = Pn yi zi , ˜≤λ Thus λ+ i=1 otherwise ˜ (x) = ∨x=Σn y z ∧i C − (yi ) ∧ λ− (zi ), C + (yi ) ∧ λ+ (zi ) CR λ R R i=1 i i ≤ ∨x=Σni=1 yi zi ∧i 1 ∧ λ− (zi ), 1 ∧ λ+ (zi ) = ∨x=Σni=1 yi zi ∧i λ− (zi ), λ+ (zi ) ≤ ∨x=Σni=1 yi zi ∧i λ− (yi zi ), λ+ (yi zi ) # " n n X X + − y i zi ) yi zi ), λ ( ≤ ∨x=Σni=1 yi zi λ ( i=1

= ∨ ˜ (x) . = λ

x=Σn i=1 yi zi

i=1

− λ (x), λ+ (x)

˜ ⊆ λ. ˜ Thus CR λ ˜ be an interval valued fuzzy subset of R such that λ ˜+λ ˜⊆λ ˜ and Conversely, let λ ˜ ˜ CR λ ⊆ λ. Then for all x, y ∈ R ˜ (x + y) ≥ ˜+λ ˜ (x + y) λ λ = ∨x+y=a+b λ− (a) ∧ λ− (b) , λ+ (a) ∧ λ+ (b) ≥ λ− (x) ∧ λ− (y) , λ+ (x) ∧ λ+ (y) = λ− (x) , λ+ (x) ∧ λ− (y) , λ+ (y) ˜ (x) ∧ λ ˜ (y) = λ and ˜ (xy) ≥ CR λ ˜ (xy) λ − + = ∨xy=Σni=1 yi zi ∧i CR (yi ) ∧ λ− (zi ), CR (yi ) ∧ λ+ (zi ) − + ≥ CR (x) ∧ λ− (y), CR (x) ∧ λ+ (y) = 1 ∧ λ− (y), 1 ∧ λ+ (y) − = λ (y) , λ+ (y) ˜ (y) . = λ 235


˜ (x + y) ≥ λ ˜ (x) ∧ λ ˜ (y) and λ ˜ (xy) ≥ λ ˜ (y) for all x, y ∈ R. Hence λ ˜ is an Thus λ interval valued fuzzy left ideal of R. Theorem 3.14. A subset A of a hemiring R is a subhemiring of R if and only if the interval valued characteristic function CÃ is an interval valued fuzzy subhemiring of R. Proof. Suppose that A is a subhemiring of R and x, y ∈ R. Case I: If x, y ∈ A then x + y, xy ∈ A. Thus CÃ (x + y) = I˜ = I˜ ∧ I˜ = CÃ (x) ∧ CÃ (y) and CÃ (xy) = I˜ = I˜ ∧ I˜ = CÃ (x) ∧ CÃ (y). ˜ Then Case II: If at least one, say y ∈ / A then CÃ (y) = O. ˜ = CÃ (x) ∧ O ˜ = CÃ (x) ∧ CÃ (y) CÃ (x + y) ≥ O and ˜ = CÃ (x) ∧ O ˜ = CÃ (x) ∧ CÃ (y) . CÃ (xy) ≥ O Thus in both cases CÃ (x + y) ≥ CÃ (x) ∧ CÃ (y) , andCÃ (xy) ≥ CÃ (x) ∧ CÃ (y). Hence CÃ is an interval valued fuzzy subhemiring of R. Conversely, suppose that CÃ is an interval valued fuzzy subhemiring of R and x, y ∈ A. Then CÃ (x + y) ≥ CÃ (x) ∧ CÃ (y) = I˜ ∧ I˜ = I˜ and ˜ CÃ (xy) ≥ CÃ (x) ∧ CÃ (y) = I˜ ∧ I˜ = I. Thus x + y, xy ∈ A for all x, y ∈ A. This shows that A is a subhemiring of R.

Theorem 3.15. A subset A of a hemiring R is a left (respectively right) ideal of R if and only if the interval valued characteristic function CÃ is an interval valued fuzzy left (respectively right) ideal of R. ˜ and µ Theorem 3.16. If λ ˜ are interval valued fuzzy left (respectively right) ideals ˜+µ of R then their sum λ ˜ is also an interval valued fuzzy left (respectively right) ideal of R. ˜+µ Proof. To show that λ ˜ is an interval valued fuzzy left ideal of R, we will prove that ˜+µ ˜+µ ˜+µ (i) λ ˜ (x + y) ≥ λ ˜ (x) ∧ λ ˜ (y) ˜+µ ˜+µ (ii) λ ˜ (yx) ≥ λ ˜ (x) ˜ = [λ− , λ+ ] and µ for all x, y ∈ R. Recall that if λ ˜ = [µ− , µ+ ] are interval valued − + − + fuzzy left ideals of R then λ , λ , µ and µ are fuzzy left ideals of R. Let x, y ∈ R. 236


Then

˜+µ ˜+µ λ ˜ (x) ∧ λ ˜ (y) = ∨x=a+b λ− (a) ∧ µ− (b), λ+ (a) ∧ µ+ (b) ∧ ∨y=u+v λ− (u) ∧ µ− (v), λ+ (u) ∧ µ+ (v) = ∨x=a+b λ− (a) ∧ µ− (b), λ+ (a) ∧ µ+ (b) ∧ ∨y=u+v λ− (u) ∧ µ− (v), λ+ (u) ∧ µ+ (v) = ∨x=a+b, y=u+v λ− (a) ∧ µ− (b), λ+ (a) ∧ µ+ (b) ∧ − λ (u) ∧ µ− (v), λ+ (u) ∧ µ+ (v) = ∨x=a+b, y=u+v [λ− (a) ∧ µ− (b) ∧ λ− (u) ∧ µ− (v), λ+ (a) ∧ µ+ (b) ∧ λ+ (u) ∧ µ+ (v)] − ≤ ∨x=a+b, y=u+v λ (a + u) ∧ µ− (b + v), λ+ (a + u) ∧ µ+ (b + v) ≤ ∨x+y=u0 +v0 λ− (u0 ) ∧ µ− (v 0 ), λ+ (u0 ) ∧ µ+ (v 0 ) ˜+µ = (λ ˜)(x + y)

and ˜+µ (λ ˜)(x)

= ∨x=u+v [λ− (u) ∧ µ− (v), λ+ (u) ∧ µ+ (v)] ≤ ∨x=u+v λ− (yu) ∧ µ− (yv), λ+ (yu) ∧ µ+ (yv) ≤ ∨yx=u0 +v0 λ− (u0 ) ∧ µ− (v 0 ), λ+ (u0 ) ∧ µ+ (v 0 ) ˜+µ = (λ ˜)(yx).

˜+µ ˜+µ ˜+µ ˜+µ ˜+µ Thus λ ˜ (x + y) ≥ λ ˜ (x) ∧ λ ˜ (y) and λ ˜ (yx) ≥ λ ˜ (x) ˜+µ for all x, y ∈ R. Hence λ ˜ is an interval valued fuzzy left ideal of R. ˜ and µ Theorem 3.17. If λ ˜ are interval valued fuzzy left (respectively right) ideals ˜ µ is also an interval valued fuzzy left (respectively right) ideal of a hemiring R then λ˜ of R. Proof. Let x, y ∈ R. Then

˜ µ (x) = ∨x=Σn y z {∧i λ− (yi ) ∧ µ− (zi ) , λ+ (yi ) ∧ µ+ (zi ) } λ˜ i=1 i i

and

˜ µ (y) = ∨y=Σn y0 z0 {∧i λ− (y 0 ) ∧ µ− (z 0 ) , λ+ (y 0 ) ∧ µ+ (z 0 ) }. λ˜ i i i i i=1 i i 237


Now ˜ µ)(x) ∧ (λ˜ ˜ µ)(y) = ∨x=Σn y z {∧i λ− (yi ) ∧ µ− (zi ) , λ+ (yi ) ∧ µ+ (zi ) } (λ˜ i=1 i i n o ∧ ∨y=Σnj=1 y0j z0j ∧j λ− yj0 ∧ µ− zj0 , λ+ yj0 ∧ µ+ zj0 = ∨x=Σni=1 yi zi {∧i λ− (yi ) ∧ µ− (zi ) , λ+ (yi ) ∧ µ+ (zi ) } n o ∧ ∨y=Σnj=1 y0j z0j ∧j λ− yj0 ∧ µ− zj0 , λ+ yj0 ∧ µ+ zj0 = ∨x=Σni=1 yi zi , y=Σnj=1 y0j z0j ∧i λ− (yi ) ∧ µ− (zi ) , λ+ (yi ) ∧ µ+ (zi ) ∧ ∧j λ− yj0 ∧ µ− zj0 , λ+ yj0 ∧ µ+ zj0 = ∨x+y=Σni=1 yi zi +Σnj=1 y0j z0j [{∧i [λ− (yi ) ∧ µ− (zi )]} ∧ {∧j [λ− yj0 ∧ µ− zj0 ]}, {∧i [λ+ (yi ) ∧ µ+ (zi )]} ∧ {∧j [λ+ yj0 ∧ µ+ zj0 ]}] ≤ ∨x+y=Prk=1 uk vk {∧k [λ− (uk ) ∧ µ− (vk ) , λ+ (uk ) ∧ µ+ (vk )]} ˜ µ)(x + y). = (λ˜ For a, x ∈ R

˜ µ (x) λ˜

= ∨x=Σni=1 yi zi {∧i λ− (yi ) ∧ µ− (zi ) , λ+ (yi ) ∧ µ+ (zi ) } ≤ ∨x=Σni=1 yi zi {∧i λ− (ayi ) ∧ µ− (zi ) , λ+ (ayi ) ∧ µ+ (zi ) } = ∨ax=Σni=1 ayi zi {∧i λ− (ayi ) ∧ µ− (zi ) , λ+ (ayi ) ∧ µ+ (zi ) } ≤ ∨ax=Σni=1 y0i z0i {∧i λ− (yi0 ) ∧ µ− (zi0 ) , λ+ (yi0 ) ∧ µ+ (zi0 ) } ˜ µ)(ax). = (λ˜

˜ µ (x) for all a, x, y ∈ R. ˜ µ (ax) ≥ λ˜ ˜ µ)(x + y) ≥ (λ˜ ˜ µ)(x) ∧ (λ˜ ˜ µ)(y) and λ˜ Thus (λ˜ ˜ µ is an interval valued fuzzy left ideal of R. Hence λ˜ ˜ and µ Remark 3.18. If λ ˜ are interval valued fuzzy left (respectively right) ideals ˜ of R then λ ∩ µ ˜ is an interval valued fuzzy left (respectively right) ideal of R. ˜ µ ˜ µ. Now we characterize regular and weakly regular hemirings In general, λ∩ ˜ 6= λ˜ by the properties of their interval valued fuzzy ideals. Theorem 3.19. The following assertions for a hemiring R are equivalent: (i) R is von Neumann regular. (ii) For any right ideal A and any left ideal B of R, A ∩ B = AB. ˜ and interval valued fuzzy left ideal (iii) For any interval valued fuzzy right ideal λ ˜ ˜ µ, λ ∩ µ ˜ = λ˜ µ. Proof. (i) ⇔ (ii) follows from Theorem 2.4. 238


˜ be an interval valued fuzzy right ideal and µ (i) ⇒ (iii) Let λ ˜ be an interval valued fuzzy left ideal of R. Then for any x ∈ R

˜ µ (x) λ˜

= ∨x=Σni=1 yi zi ∧i λ− (yi ) ∧ µ− (zi ) , λ+ (yi ) ∧ µ+ (zi ) ≤ ∨x=Σni=1 yi zi ∧i λ− (yi zi ) ∧ µ− (yi zi ) , λ+ (yi zi ) ∧ µ+ (yi zi ) hn o i ˜ (yi zi ) ∧ {∧i µ ˜ (yi zi )} = ∨x=Σni=1 yi zi ∧i λ ( ! !) n n X X ˜ y i zi ∧ µ ≤ ∨x=Σn y z λ ˜ y i zi i=1 i i

i=1

= ∨x=Σni=1 yi zi

i=1

n o ˜ (x) ∧ µ λ ˜ (x)

˜ (x) ∧ µ = λ ˜ (x) ˜∩µ = λ ˜ (x) . ˜µ ⊆ λ ˜∩µ Thus λ˜ ˜. Since R is regular so for x ∈ R there exists a ∈ R such that ˜ ˜ (x) ∧ µ ˜ (x) ∧ µ x = xax. Now λ ∩ µ ˜ (x) = λ ˜ (x) ≤ λ ˜ (ax) . Thus

− − + + ˜∩µ ˜ µ (x) . λ ˜ (x) ≤ ∨x=Σm ∧ λ (y ) ∧ µ (z ) , λ (y ) ∧ µ (z ) = λ˜ y z i i i i i j j j=1

˜∩µ ˜ µ, and hence λ˜ ˜µ = λ ˜∩µ This implies λ ˜ ⊆ λ˜ ˜. (iii) ⇒ (i) Let I be a right ideal and J be a left ideal of R. Then the interval valued characteristic functions C˜I and C˜J are interval valued fuzzy right and interval valued fuzzy left ideals of R and by hypothesis C˜IJ = C˜I .C˜J = C˜I ∩ C˜J which implies that C˜IJ = C˜I∩J . Thus IJ = I ∩ J. Hence R is regular.

Theorem 3.20. The following assertions for a hemiring R with 1 are equivalent: (i) R is right weakly regular. (ii) All right ideals of R are idempotent. (iii) IJ = I ∩ J for all right ideals I and two-sided ideas J of R. (iv) All interval valued fuzzy right ideals of R are fully idempotent. ˜µ = λ ˜ ∩µ ˜ and interval valued fuzzy (v) λ˜ ˜ for all interval valued fuzzy right ideals λ two-sided ideals µ ˜ of R. If R is commutative, then the above assertions are equivalent to (vi) R is von-Neumann regular. Proof. (i) ⇔ (ii) ⇔ (iii) and (i) ⇔ (vi) follows from Theorem 2.7. 239


˜ be an interval valued fuzzy right ideal of R and x ∈ R . Then (i)⇒ (iv) Let λ ˜ 2 (x) = ˜λ ˜ (x) λ λ. = ∨x=Σni=1 yi zi ∧i λ− (yi ) ∧ λ− (zi ) , λ+ (yi ) ∧ λ+ (zi ) ≤ ∨x=Σni=1 yi zi ∧i λ− (yi zi ) ∧ λ− (zi ) , λ+ (yi zi ) ∧ λ+ (zi ) = ∨x=Σni=1 yi zi ∧i λ− (yi zi ) ∧ λ− (zi ) , ∧i λ+ (yi zi ) ∧ λ+ (zi ) " # ! ! n n X X yi zi ∧ λ− (zi ) , λ+ yi zi ∧ λ+ (zi ) ≤ ∨x=Σni=1 yi zi λ− i=1

i=1

∨x=Σni=1 yi zi λ− (x) ∧ λ− (zi ) , λ+ (x) ∧ λ+ (zi ) ≤ ∨x=Σni=1 yi zi λ− (x) , λ+ (x) ˜ (x) = λ

≤

˜ 2 ⊆ λ. ˜ Now since R is right weakly regular so x ∈ xRxR. Hence we can write and so Pλ n ˜ (x) ∧ λ ˜ (x) ≤ λ ˜ (xai ) ∧ λ ˜ (xbi ) x = i=1 xai xbi where ai , bi ∈ R and n ∈ N. Now λ for all i, which implies that λ (x) ≤ ∧i {λ (xai ) ∧ λ (xbi )} = ∧i λ− (xai ) ∧ λ− (xbi ) , λ+ (xai ) ∧ λ+ (xbi ) ≤ ∨x=Σni=1 xai xbi ∧i λ− (xai ) ∧ λ− (xbi ) , λ+ (xai ) ∧ λ+ (xbi ) − ≤ ∨x=Σm ∧j λ (yj ) ∧ λ− (zj ) , λ+ (yj ) ∧ λ+ (zj ) j=1 yj zj ˜λ ˜ (x) = λ. 2 ˜ (x) . = λ ˜⊆λ ˜ 2 , and thus λ ˜=λ ˜ 2 . Therefore λ is fully idempotent. Hence λ (iv)⇒ (i) Let x ∈ R and let A = xR be a right ideal of R generated by x. Then x ∈ A and the characteristic function CÃ of A is interval valued fuzzy right ideal of 2 R and by hypothesis CÃ = CÃ · CÃ = CÃ2 . Hence A = A2 and so x ∈ A2 = (xR) because x ∈ A. Hence x ∈ xRxR. Thus R is right weakly regular. (i) ⇒ (v) Let λ be an interval valued fuzzy right ideal and µ ˜ be an interval valued fuzzy two-sided ideal of R. Then for any x ∈ R ˜ µ (x) = ∨x=Σn y z ∧i λ− (yi ) ∧ µ− (zi ) , λ+ (yi ) ∧ µ+ (zi ) λ˜ i=1 i i ≤ ∨x=Σni=1 yi zi ∧i λ− (yi zi ) ∧ µ− (yi zi ) , λ+ (yi zi ) ∧ µ+ (yi zi ) = ∨x=Σni=1 yi zi [{∧i λ (yi zi )} ∧ {∧i µ ˜ (yi zi )}] ( ! !) n n X X ˜ ≤ ∨x=Σn y z λ y i zi ∧ µ ˜ y i zi i=1 i i

i=1

= ∨

x=Σn i=1 yi zi

i=1

n o ˜ (x) ∧ µ λ ˜ (x)

˜ (x) ∧ µ = λ ˜ (x) ˜ = λ∩µ ˜ (x) 240


˜µ ⊆ λ ˜∩µ and P so λ˜ ˜. Since R is right weakly regular so each x ∈ R can be written as n x = i=1 xai xbi , where ai , bi ∈ R, n ∈ N. Now ˜∩µ ˜ (x) ∧ µ ˜ (xai ) ∧ µ λ ˜ (x) = λ ˜ (x) ≤ λ ˜ (xbi ) for all i and thus (λ ∩ µ ˜) (x) ≤ ∧i {(λ (xai ) ∧ µ (xbi ))} = ∧i λ− (xai ) ∧ µ− (xbi ) , λ+ (xai ) ∧ µ+ (xbi ) ≤ ∨x=Σni=1 xai xbi ∧i λ− (xai ) ∧ µ− (xbi ) , λ+ (xai ) ∧ µ+ (xbi ) − ≤ ∨x=Σm ∧i λ (yi ) ∧ µ− (zi ) , λ+ (yi ) ∧ µ+ (zi ) j=1 yj zj =

(λ˜ µ) (x)

which implies λ ∩ µ ˜ ⊆ λ˜ µ. Hence λ˜ µ=λ∩µ ˜. (v) ⇒ (iii) Let I be a right ideal and J be a two-sided ideal of R. Then the characteristic functions C˜I and C˜J are interval valued fuzzy right and interval valued fuzzy two-sided ideals of R and by hypothesis C˜IJ = C˜I .C˜J = C˜I ∩ C˜J which implies C˜IJ = C˜I∩J . Hence IJ = I ∩ J.

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[18] H. S. Vandiver, Note on a simple type of algebra in which cancellation law of addition does not hold, Bull. Amer. Math. Soc. 40 (1934) 914–920 [19] W. Wechler, The concept of fuzziness in automata and language theory, Akademic verlog, Berlin, 1978. [20] L. A. Zadeh, Fuzzy Sets, Inform. Control 8 (1965) 338–353. [21] J. Zhan and W. A. Dudek, Fuzzy h-ideals of hemirings, Inform. Sci. 177 (2007) 876–886.

M. Shabir ([email protected]) Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan T. Mahmood ([email protected]) Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan N. Malik Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan

242

Characterizations of hemirings by the properties of

Characterizations of hemirings by the properties of

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