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Spectrum of prime L-fuzzy h-ideals of a hemiring
Fuzzy Sets and Systems 161 (2010) 1740 – 1749 www.elsevier.com/locate/fss
Spectrum of prime L-fuzzy h-ideals of a hemiring H.V. Kumbhojkar∗,1 Department of Mathematics, Addis Ababa University, Addis Ababa, Ethiopia Received 3 November 2008; received in revised form 16 September 2009; accepted 6 October 2009 Available online 15 October 2009
Abstract We redefine the concept of prime fuzzy h-ideals of a hemiring so that the fuzzy h-ideals are not necessarily 2-valued. We also introduce the concept of semiprime fuzzy h-ideals. A topological space, called the spectrum of prime fuzzy h-ideals of a commutative hemiring with unity, has been obtained. This topological space is compact and preserves isomorphisms between hemirings. The correspondence associating a hemiring with its spectrum of prime fuzzy h-ideals is shown to define a contravariant functor from the category of commutative hemirings with unity into the category of compact topological spaces. The spectrum of (crisp) prime h-ideals of the hemiring is a subspace which is dense in the spectrum of prime fuzzy h-ideals. Valuation lattices for all the fuzzy sets in the paper are assumed to be complete Heyting algebras. © 2009 Elsevier B.V. All rights reserved. Keywords: Prime fuzzy h-ideal; Semiprime fuzzy h-ideal; Prime spectrum; Compact topological space; Contravariant functor
1. Introduction Hemirings, or semirings with commutative addition and absorbing zero, appear in a natural way in theories of automata, formal languages, theoretical computer science, optimization and graph theory [1,3–6,8]. These areas and fuzzy logic have useful or potentially useful applications in Control Engineering. This may be a reason why there have been attempts to fuzzify basic concepts in semiring theory [3]. This paper is the result of a search for appropriate answers to the questions raised in the conclusion of [18]. The authors of [18] conclude their paper with the following comment: “In our opinion the future study of fuzzy sets in hemirings and semirings can be connected with: (1) investigating semiprime fuzzy h-ideals; (2) establishing a fuzzy spectrum of a hemiring; . . .”. The authors of [18] have introduced the concept of a prime fuzzy h-ideal of a hemiring. However, a major hurdle in developing spectral theory based on their definition of prime fuzzy h-ideals is the restrictive nature of the definition. Prime fuzzy h-ideals, defined in this paper, turn out to be 2-valued fuzzy sets and thus, they are not “really fuzzy”. What makes these prime fuzzy h-ideals more restrictive is, as proved by the authors in [18] that one of their values is already fixed to be 1. One more serious draw-back of the definition is noticed when the authors’ definition is extended to L-fuzzy sets, where L is a lattice different from the lattice of the unit interval [0,1] used in [18]. In many important ∗ Corresponding author. Present address: P.O. Box 1176, Mathematics, Science Faculty, AAU, Addis Ababa, Ethiopia.
E-mail address: [email protected] 1 Permanent address: No. 9, Ashok Apartments, Mali Colony Takala, Kolhapur 416008, Maharashtra, India.
0165-0114/$ - see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2009.10.006
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lattices, zero elements are not prime elements. When a fuzzy h-ideal assumes values in such lattices, then even the characteristic function of a prime h-ideal fails to be a prime fuzzy h-ideal. Prime spectrum or the Zariski topology on the set of prime ideals of a commutative ring with unity plays an important role in the field of commutative algebra, algebraic geometry and lattice theory. Semiprime ideals determine the topology. Since we expect that the spectrum of prime fuzzy h-ideals and semiprime fuzzy h-ideals play analogous roles in the theory of commutative hemirings with unity, a study of these concepts becomes important to us. However, for the reasons stated above, the definition of prime fuzzy h-ideals given in [18] does not lead us far in developing a spectral theory of semirings. One, and the most frequently used, way to fuzzify a mathematical concept is to replace a set with structure in the concept by a fuzzy set with analogous fuzzy structure [7,13]. This approach was followed for fuzzifying prime ideals of a ring in [14–17], prime h-ideals of a hemiring in [18] and prime ideals of a semigroup in [2]. In all cases prime fuzzy ideals turned out to be 2-valued. We resolve this problem successfully by redefining prime fuzzy ideals in the cases of commutative rings with unity [10–12] and semigroups [9]. In this paper, we redefine prime fuzzy h-ideals of a hemiring on similar lines. The prime fuzzy h-ideals defined in this paper are no longer 2-valued. In fact, they may even be infinite-valued. This offers an appropriate setting to introduce a topology on the set of prime fuzzy h-ideals of a hemiring to get the socalled spectrum of prime fuzzy h-ideals. We construct the spectrum of prime fuzzy h-ideals of a commutative hemiring with unity and show that the topology of the spectrum is determined by semiprime fuzzy h-ideals. We prove that this spectrum is compact topological space in which the spectrum of (crisp) prime h-ideals is dense. This spectrum defines contravariant functor from the category of commutative hemirings with unity to the category of compact topological spaces. In this paper, we work in a setting more general than [18] by considering L-fuzzy sets, where L is a Heyting algebra. However, since the lattice L is fixed throughout the paper, we call them fuzzy sets instead of L-fuzzy sets without causing any confusion. 2. Preliminaries Throughout this paper, L stands for a complete Heyting algebra. In other words L is a complete lattice such that for all subsets T of L and all b ∈ L, ∨{a ∧ b|a ∈ T } = (∨{a|a ∈ T }) ∧ b and ∧ {a ∨ b|a ∈ T } = (∧{a|a ∈ T }) ∨ b. The triple (S, +, ·) stands for a hemiring, i.e. (S, +) is a commutative monoid having identity element 0 and (S, ·) is a semigroup satisfying the following identities: a(b + c) = ab + ac and (a + b)c = ac + bc, 0.x = 0 = x.0. A commutative hemiring with unity is a hemiring (S, +, ·) such that (S, · ) is a commutative monoid. We denote the identity element of (S, ·) by 1. With abuse of notation we denote (S, +, .) by S. Unless stated otherwise N stands for the hemiring of non-negative integers with usual operations of addition (+) and multiplication (·). A left ideal A of S is a non-empty set A which is closed under the addition of S and is such that, for all x ∈ S and a ∈ A we have xa ∈ A. A left ideal A of S is called a left h-ideal, if the following condition holds for all x, z ∈ S: (x + a + z = b + z and a, b ∈ A) ⇒ x ∈ A. An L-fuzzy subset (or simply an L-fuzzy set) A of a set X is a function A: X → L. With usual abuse of notation we call A a fuzzy set of X. If ∈ L then the set {x ∈ X |A(x) ≥ } is called an -level cut or in short -cut of A and is denoted by A . A fuzzy set J:S → L is a fuzzy left ideal if for all a, b ∈ S the following conditions are satisfied: (i) J (a + b) ≥ J (a) ∧ J (b) and (ii) J (ab) ≥ J (b). A fuzzy left ideal J:S → L is called a fuzzy left h-ideal, if the following condition is satisfied: x + a + z = b + z ⇒ J (x) ≥ J (a) ∧ J (b)∀x, a, b, z ∈ S. A (fuzzy) right h-ideal is similarly defined. Whenever a statement is made about (fuzzy) left h-ideals, it is to be understood that the analogous statement is made about (fuzzy) right h-ideals. A (fuzzy) h-ideal is one, which is both (fuzzy) right and (fuzzy) left h-ideal. 3. Prime fuzzy left h-ideals Let A be a left ideal of S and h(A) = {x ∈ S|x + a + z = b + z for some a, b, ∈ A and z ∈ S}.
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The following properties of h( A) are well known: 1. 2. 3. 4.
h(A) is the smallest left h-ideal of S containing A. A is an h-ideal iff A = h(A). h(h( A)) = h(A). For any left ideals A and B of S, A ⊆ B ⇒ h(A) ⊆ h(B).
The left h-ideal h(A) is called the h-closure of A. Proposition 3.1 (Zhan and Dudek [18]). If S is a hemiring and A and B are left ideals of S, then h(AB) = h(h( A)h(B)). A left h-ideal P of a hemiring S is called prime, if PS and for any left h-ideals A and B of S, AB ⊆ P implies either A ⊆ P or B ⊆ P [18]. An interesting consequence of Proposition 3.1 is the following: Proposition 3.2. A proper left h-ideal P of a hemiring S is prime iff for all a, b ∈ S, aSb ⊆ P implies a ∈ P or b ∈ P. Proof. Let P be a prime h-ideal and aSb ⊆ P for a,b ∈ S. Clearly, we have SaSb ⊆ P. Let A = h(Sa) and B = h(Sb). Our first claim is AB ⊆ P. Observe that every element of AB is a finite sum of the products xy, where x is in A and y is in B. Let for some s, t, u, v, z and w in S, we have x + sa + z = ta + z and y + ub + w = vb + w. We have then, equalities: ay + aub + aw = avb + aw, and x y + say + zy = tay + zy. As aub, avb are elements of P and P is a left h-ideal, ay is in P. Therefore, say, tay and consequently xy are in P. This justifies the claim. Since P is prime we have, either A ⊆ P or B ⊆ P. Suppose A ⊆ P. Let a be the left ideal generated by a. Then clearly a a ⊆ Sa. Therefore, we have, h( a )h( a ) ⊆ h(h( a )h( a )) = h( a a ) ⊆ h(Sa) = A ⊆ P. Therefore, we have h( a ) ⊆ P and hence, a ∈ P. The converse is obvious. If S is a commutative hemiring with unity, the condition in Proposition 3.2 is equivalent to the following condition: ab∈ P ⇒ a ∈ P or b ∈ P. This leads to the following: Proposition 3.3. Let S be a commutative hemiring with unity. Then a proper h-ideal P of S is prime iff for all a, b ∈ S, ab ∈ P ⇒ a ∈ P or b ∈ P. Proposition 3.3 in conjunction with usual applications of Zorn’s Lemma leads us to the following: Theorem 3.4. Every proper h-ideal of a commutative hemiring S with unity is contained in a prime h-ideal of S. It should be noted that Theorem 3.4 depends on Proposition 3.3 and hence on Proposition 3.1. Corollary 3.5. If commutative hemiring S with unity has a proper h-ideal then it has a prime h-ideal. We now, in the light of Propositions 3.2, define prime fuzzy left h-ideal of a hemiring as follows: Definition 3.6. A fuzzy left h-ideal P: S → L is called prime if it is non-constant and, for all a, b ∈ S and ∈ L, the following condition is satisfied: P(asb) ≥ , ∀s ∈ S ⇒ P(a) ≥ or P(b) ≥ . Proposition 3.7. A non-constant fuzzy left h-ideal P : S → L is prime iff its every non-empty level cut is either a prime left h-ideal of S or S itself. Proof. Let P be a prime fuzzy left h-ideal, ∈ L and SP ∅. Let aSb ⊆ P . Then P(asb) ≥ , ∀s ∈ S. Consequently, P(a) ≥ or P(b) ≥ . Therefore, we have a ∈ P or b ∈ P .
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To prove the converse, consider a non-constant fuzzy h-ideal P: S → L. If P is not prime then there exist ∈ L and a, b ∈ S such that P(asb) ≥ , ∀s ∈ S, but ⱕ P(a), and ⱕ P(b). Thus, we have aSb ⊆ P but, a ∈ / P and b ∈ / P and therefore, P is neither a prime h-ideal nor the hemiring S. Corollary 3.8. A left h-ideal P of S is prime iff its characteristics function vP is prime fuzzy left h-ideal. When S is a commutative hemiring with unity, the characterization of a prime fuzzy h-ideal takes the following simpler form: Proposition 3.9. If S is a commutative hemiring with unity, a non-constant fuzzy h-ideal P of S is prime iff P(ab) = P(a) or P(b)∀a, b ∈ S. If, moreover, L is totally ordered, in particular if L = [0, 1], then P is prime iff P(ab) = P(a) ∨ P(b)∀a, b ∈ S. Proof. Let a non-constant fuzzy h-deal P be prime and P(ab) = (say). Then P(asb) = P(sab) ≥ P(s) ∨ P(ab) ≥ , ∀s ∈ S. As P is prime, we have either P(a) ≥ or P(b) ≥ . Suppose P(a) ≥ . Then P(ab) ≥ P(a) ≥ = P(ab). Hence, P(ab) = P(a). Conversely, suppose P(ab) = P(a) or P(b) and P(asb) ≥ , ∀s ∈ S. Selecting s = 1 we have P(ab) ≥ . Hence, by hypothesis, P(a) ≥ or P(b) ≥ . Thus, P is a prime fuzzy h-ideal. If a fuzzy h-deal is prime according to Definition 4.1 of [18], then in view of Proposition 3.7, it is prime according to Definition 3.6. The following examples show that the converse is not true. Example 3.10. Let p be a prime integer. Define a fuzzy set P : N → [0, 1] by P(x) = 1 if x = 0 = if x ∈ pN ∼ 0 = if x ∈ / pN. By Proposition 3.7 P is a prime fuzzy h-deal, for all 0 ≤ < ≤ 1. We will call the fuzzy ideal P a prime fuzzy h-ideal induced by the prime number p and denote it by ( pN) . Clearly, ( pN) is not a prime fuzzy ideal according to Definition 4.1 of [18] unless = 1. One more drawback of Definition 4.1 of [18] of prime fuzzy ideals is highlighted by the following example. If we do not restrict ourselves to the grade membership lattice [0,1], then even the characteristic function of a prime h-ideal may fail to be prime fuzzy h-ideal according to [18]. However, in view of Proposition 3.7, it will remain prime fuzzy h-ideal according to Definition 3.6. Example 3.11. Consider the Boolean algebra L = {0, , , 1}. Let P be the characteristic function of 2N, and A, B, the fuzzy h-ideals of N defined as follows: A(2n) = 1 = B(2n) and A(2n + 1) = , B(2n + 1) = for all n ∈ N . Then AB ⊆ A ∩ B = P. However, clearly, neither A nor B is contained in the fuzzy h-ideal P. Thus Proposition 4.3 and Theorem 4.4 of [18] are no more valid when the lattice [0,1] is replaced by L = {0, , , 1}. However, they do not conflict with Definition 3.6 of prime. In the light of the fact that a prime fuzzy h-ideal can have any number of values, we are, now, in a position to define meaningfully the prime radical of a fuzzy ideal. Definition 3.12. If J : S → L is a fuzzy ideal, then the intersection of all prime fuzzy h-ideals of S containing J is called the h-prime radical of J. We denote it by rh (J ). If the set of prime fuzzy h-ideals of S containing J is empty, we define rh (J ) to be S . Note that rh (J ) is a fuzzy h-ideal containing J.
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4. Semiprime fuzzy h-ideals In this section, we discuss fuzzy analogue of some results on semiprime h-ideals. We will call a left h-ideal I of a hemiring S semiprime, if I S and for any left h-ideal A of S, A A ⊆ I implies A ⊆ I . Proposition 4.1. A proper left h-ideal I of a hemiring S is semiprime iff for all a ∈ S, aSa ⊆ I ⇒ a ∈ I . The proof runs parallel to that of Proposition 3.2. We now define the fuzzy analogue of semiprime h-ideals: Definition 4.2. A fuzzy left h-ideal J : S → L is called semiprime if J is non-constant and, for all a ∈ S and ∈ L, the following condition is satisfied: J (asa) ≥ , ∀s ∈ S ⇒ J (a) ≥ . Proposition 4.3. A non-constant fuzzy left h-ideal J : S → L is semiprime iff its every non-empty level cut is either a semiprime left h-ideal of S or S itself. The proof runs parallel to that of Proposition 3.7 As immediate consequences of Proposition 4.3, we have the following: Corollary 4.4. Intersection of semiprime fuzzy left h-ideals of S is a semiprime fuzzy left h-ideal of S. In particular, intersection of prime fuzzy left h-ideals of S is a semiprime fuzzy left h-ideal of S. Corollary 4.5. A left h-ideal J of S is semiprime iff its characteristics function vJ is semiprime fuzzy left h-ideal. When S is a commutative hemiring with unity, we have the following characterization of semiprime fuzzy h-ideals: Proposition 4.6. Let S be a commutative hemiring with unity. A non-constant fuzzy h-ideal J of S is semiprime iff J (a 2 ) = J (a). The proof is similar to that of Proposition 3.9. Although every prime fuzzy h-ideal of S is a semiprime, the following example shows that, the converse is not true: Example 4.7. Let p1 , p2 . . . be distinct prime numbers in N, J 0 = N and J l = p1 p2 . . . pl N, l = 1, 2, . . . . Then we have J 0 ⊃ J 1 ⊃ · · · ⊃ J n ⊃ J n+1 . . . . As every non-zero element of N has unique prime factorization, J l is a semiprime h-ideal for l = 2, 3, . . . but not a prime h-ideal. Consider the fuzzy h-ideal given below: J : N → [0, 1], J (x) = 1 if x = 0 = l/(l + 1) ∀x ∈ J l ∼ J l+1 , l = 0, 1, 2. . . . Then, by Proposition 4.3, J is a semiprime fuzzy h-ideal. Proposition 3.7 ensures that if we choose l ≥ 2, J is not a prime fuzzy h-ideal. 5. Spectrum of prime fuzzy ideals In the remaining part of this paper, S stands for a commutative hemiring with unity and X stands for the set of all prime fuzzy h-ideals of S with membership grades in a fixed lattice L. We further assume that P(0) = 1. (The hypothesis J (0) = 1, for a fuzzy ideal J : S → L is not really restrictive. It is in full conformity with the principles of fuzziness. For, every ideal must contain the element 0 of S. Moreover, the hypothesis is needed to prove even a simple result that the intersection of two fuzzy ideals is a fuzzy ideal when L is an arbitrary Heyting algebra. For these reasons J (0) = 1 may, in our opinion, be made a part of the definition of a fuzzy ideal J [13].) If A : S → L is any fuzzy set, let V(A) = {P ∈ X |A ⊆ P}. When A = {a} , where a ∈ S, we denote V(A) by V(a). Thus V(a) = {P ∈ X |P(a) = 1}.
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Proposition 5.1. Let A : S → L and B : S → L be two fuzzy sets. Then A ⊆ B ⇒ V(B) ⊆ V( A), V(A) ∪ V(B) ⊆ V( A ∩ B), If I and J are h-ideals of S, then V( I ) ∪ V( J ) = V( I ∩J ). Let J be a fuzzy h-ideal of S generated by A and rh (J ) be the h-prime radical of J . Then we have V( A) = V(J ) = V(rh (J )). (e) If {Ai |i ∈ } is a family of fuzzy subsets of S, then V(∪{Ai |i ∈ }) = ∩{V (Ai )|i ∈ }. (f) If T ⊆ S, then V(T ) = ∩{V(a)|a ∈ T }. (g) V(a) ∪ V(b) = V(ab), for all ab ∈ S.
(a) (b) (c) (d)
Proof. (a) The statement (a) follows readily from the definitions. (b) Observe that A ∩ B ⊆ A and A ∩ B ⊆ B and apply (a) (c) Since, I ∩ J = I ∩J the result (b) assures: V( I ) ∪ V( J ) ⊆ V( I ∩J ). On the other hand let P ∈ V( I ∩J ). Since I ∩J ⊆ P we have P(x) = 1 for all x ∈ I ∩ J . Suppose I P and J P hold. Then there exist x ∈ I and y ∈ J , such that P(x) 1 P(y). But as x y ∈ I ∩ J , we have P(x y) = 1. This contradicts that P is a prime fuzzy h-ideal. Hence we have either I ⊆ P or J ⊆ P and therefore, V( I ∩J ) ⊆ V( I ) ∪ V( J ). (d) To prove the equality V(A) = V(J ) note that if T is the set of all fuzzy h-ideals containing A, then J = ∩{I |I ∈ T }. The result then follows from the fact that the inclusion A ⊆ P holds iff the inclusion J ⊆ P holds. To prove V(J ) = V(rh (J )), we choose T = {P ∈ X |J ⊆ P}. Then rh (J ) = ∩{P|P ∈ T }. The result then follows from the fact that the inclusion J ⊆ P holds iff the inclusion rh (J ) ⊆ P holds. (e) To prove (e) observe that P ∈ V(∪{Ai |i ∈ }) ⇔ ⇔ ⇔ ⇔
∪ {Ai |i ∈ } ⊆ P Ai ⊆ P for all i ∈ P ∈ V(Ai ) for all i ∈ P ∈ ∩{V(Ai )|i ∈ }.
(f) If T ⊆ S, then applying (e) to the identity T = ∪{{a} |a ∈ T } we get the result. (g) The statement (g) follows readily from the definitions. The similarity between the set of (crisp) prime h-ideals of S and that of prime fuzzy h-ideals of S ends here. In general, equality does not hold in Proposition 5.1 (b). In crisp situation, when A and B are h-ideals, the equality holds. The following examples show that the equality does not hold even for fuzzy h-ideals. Example 5.2. Let S = N [x, y] be the hemiring of polynomials in two indeterminates over N. If a, b ∈ S, let a and a, b denote h-ideals of S generated by {a} and {a, b}, respectively. Consider the fuzzy ideal P : S → [0, 1] defined as follows: P(a) = 1 if a = 0 = 0.50 if a ∈ x ∼ 0 = 0.30 if a ∈ x, y ∼ x = 0 everywhere else. Then, by Proposition 3.7, P is a prime fuzzy h-ideal. Consider the fuzzy h-ideals J : S → [0, 1] and K : S → [0, 1] defined as follows: J (a) = 1 if a = 0 = 0.45 if a ∈ x 2 ∼ {0} = 0.40 if a ∈ x 2 , y 2 ∼ x 2 = 0 everywhere else.
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K (a) = 1 if a = 0 = 0.65 if a ∈ x 2 ∼ {0} = 0.20 if a ∈ x 2 , y 2 ∼ x 2 = 0 everywhere else. Then (J ∩ K )(a) = 1 = P(a) if a = 0, = 0.45 < 0.50 = P(a) if a ∈ x 2 ∼ {0} = 0.20 < 0.30 = P(a) if a ∈ x 2 , y 2 ∼ x 2 = 0 ≤ P(a) everywhere else. Therefore, we have J ∩ K ⊆ P. But as J (y 2 ) > P(y 2 ) we have J P and as K (x 2 ) > P(x 2 ) we have K P. Thus, P ∈ V (J ∩ K ), but P ∈ / V(J ) ∪ V(K ). Example 5.3. Let L = {0, , , 1} be the Boolean algebra of four elements. Consider the fuzzy h-ideal J : N → L defined as follows: J (x) = 1 if x ∈ 6N = if x ∈ 2N ∼ 6N = if x ∈ 3N ∼ 6N = 0 everywhere else. Define K : N → L as follows: K (x) = when J (x) = and K (x) = when J (x) = ; K (x) = J (x) everywhere else. Clearly, J ∩ K is the characteristic function of the h-ideal 6N. If P is the characteristic function of 2N, then P is a prime fuzzy h-ideal containing J ∩ K . But J (3) = > 0 = P(3) and K (3) = > 0 = P(3). Thus, the strict inequality may hold in Proposition 5.1(b) even if we restrict ourselves to 2-valued fuzzy h-ideals. However, as will be proved in the remaining part of the paper, the set {V( A )|A ⊆ S} forms a system of closed sets for a topology on the set X of prime fuzzy h-ideals. Let X(a) = X ∼ V(a). Then X(a) = {P ∈ X |P(a) 1}. Let C = {X (a)|a ∈ S}. Theorem 5.4. The set C is a base for a topology on X . The open subsets of X are precisely X(T ), where T is any subset of S. The topology is completely determined by semiprime fuzzy h-ideals of S. Proof. Clearly, we have X(1) = X , and hence ∪{X(a)|a ∈ S} = X . Since X(ab) = X(a) ∩ X(b) for all a, b ∈ S, C is closed under intersection and hence forms a base for a topology on X . A typical open set in this topology will be ∪{X(a)|a ∈ T } for some T ⊆ S. Therefore, we get ∪ {X(a)|a ∈ T } = ∪ {X ∼ V(a)|a ∈ T } = X ∼ ∩{V(a)|a ∈ T } = X ∼ V(T ) = X(T ). The last part follows from the fact that V(T ) = V( T ) = V(rh ( T )), where rh ( T ) is the h-prime fuzzy radical of T . Definition 5.5. The topological space X of prime fuzzy h-ideals of S will be called the spectrum of prime fuzzy h-ideals (or in short prime fuzzy h-spectrum) of S and will be denoted by Fh-spec(S). By Corollaries 3.5 and 3.8 Fh-spec(S) is non-empty when S has a proper h-ideal.
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Example 5.6. (a) Let R+ be the hemiring of non-negative real numbers. Let, for 0 ≤ < 1, O : R+ → [0, 1] be the fuzzy ideal defined by O (x) = 1 if x = 0 = if otherwise. R+ has only one proper ideal, namely 0 which is the prime h-ideal. Therefore, the set of prime fuzzy h-ideals is X = {O |0 ≤ < 1}. Since V(0) = X and V(a) = ∅ for all a 0, Fh-spec(R+ ) is an indiscrete topological space. (b) Let O denote the prime fuzzy h-ideal defined as above except that its domain of definition, now, is N. Then the set of prime fuzzy h-ideals is X = {O |0 ≤ < 1} ∪ {( pN) | p is prime, 0 ≤ < ≤ 1}, where ( pN) is defined in Example 3.10. Since every h-ideal of N is a principal ideal, closed sets of Fh-spec(N) are precisely V(n) and the open m be the prime factorization of n, then sets are precisely X(n) for n ∈ N . If n ≥ 2 let p11 p22 . . . pm m V(n) = V( p11 ) ∪ · · · ∪ V( pm ) = V( p1 ) ∪ · · · ∪ V( pm ) = V( p1 . . . pm ).
Therefore, the spectral topology is determined by the prime fuzzy h-ideals containing the characteristic functions of the semiprime h-ideals p1 . . . pm N, m = 1, 2, . . . . Clearly, then V( p) = {( pN)1 | 0 ≤ < 1} for every prime integer p and therefore, X( p) = {O |0 ≤ < 1} ∪ {( pN) |0 ≤ < < 1}} ∪ {(qN) |0 ≤ < ≤ 1, q is prime q p}. More generally a typical open set in Fh-spec(N) is X( p1 . . . pm ) = {O |0 ≤ < 1} ∪ {( pN) |0 ≤ < < 1, p is prime, p = p1 , . . . , pm } ∪ {( pN) |0 ≤ < ≤ 1, p is prime, p p1 , . . . , pm }. Proposition 5.7. If F is a subset of Fh-spec(S) then the closure F∗ of F is F∗ = V(v∩{Q1|Q∈F} ), where Q1 = {x ∈ S|Q(x) = 1}. If P, Q ∈ Fh-spec(S), then {P}∗ = {Q}∗ iff P1 = Q1. Thus, Fh-spec(S) is not a T0 -space. Proof. Clearly v∩{Q1|Q∈F} (x) = 1, iff Q(x) = 1 for all Q ∈ F. Therefore, if P ∈ F, then ∩{Q1|Q∈F} ⊆ P and as a result P ∈ V(v∩{Q1|Q∈F} ). Thus, V(v∩{Q1|Q∈F} ) is a closed set containing S, and hence contains F∗ . On the other hand, let v∩{Q1|Q∈F} ⊆ P and P ∈ / F. If X(a) is a basic open set containing P, then P(a) 1 and we have v∩{Q1|Q∈F} (a) 1. Clearly, there exists Q ∈ F such that Q(a)1. In other words, every open set containing P contains at least one element of F. Therefore, if P ∈ V(∩{Q1|Q∈F} ), then P is either in F or is a limit point of F. Hence V(v∩{Q1|Q∈F} ) = F∗ . Theorem 5.8. Fh-spec(S) is a compact topological space and the topological space h-Spec(S) of prime h-ideals of S is dense in Fh-spec(S). Proof. To prove the compactness, it is sufficient to consider a cover consisting of basic open sets. Let {X(a)|a ∈ T }, where T ⊆ S, be one such open cover of X . Then, X = ∪{X(a)|a ∈ T } = X ∼ V(T ). Hence V(T ) = ∅. Let T generate the h-ideal K . We claim that K = S. Suppose K S. Then, by Theorem 3.4, there exists a prime h-ideal P of S containing K . Clearly, then, we have v K ⊆ vP . Since vP is a prime fuzzy h-ideal, we have an absurd result that vP ∈ V(v K ) = ∅. 2 This justifies the claim. n Therefore, there exist elements a1 , a2 , . . . , an ∈ T and s1 , s2 , . . . , sn , z ∈ S such that 1+ m 1 si ai +z = m+1 si ai + z. Consider A = {a1 , a2 , . . . , an } and let A be the h-ideal generated by A. Since A = S, we have A = S and V( A ) = V( A ) = V( S ) = ∅. Therefore, ∪ {X(ai )|i = 1, 2, . . . , n} = ∪ {X ∼ V(ai )|i = 1, 2, . . . , n} = X ∼ ∩{V(ai )|i = 1, 2, . . . , n} = X ∼ V( A ) = X. Thus, {X(ai )|i = 1, 2, . . . , n} is a finite subcover of X. 2 Note that v may not be a prime fuzzy h-ideal if we apply Definition 4.1 of [18] in conjunction with the assumption that L [0, 1]. P
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In the remaining part of the paper let U denote h-Spec(S) and W denote Fh-spec(S). Consider the subset Y of W given by Y = {vP |P ∈ U }. A typical closed subset F of W is F = V(vT ), where T ⊆ S. Hence a typical closed subset F ∗ of the induced subspace Y of W will be of the form F ∗ = F ∩ Y = {vP |T ⊆ P}. With the usual identification of an ideal P of S with the fuzzy ideal vP , Y can be identified with U with the topology defined by the closed sets of the type {P|T ⊆ P}. Let P ∈ W be any prime fuzzy h-ideal not in Y and let X(a) be any basic open set in W containing P. Let P1 = {x ∈ S|P(x) = 1}. Then P1 ∈ Y . As P(a) 1 we have a ∈ / P1 and thus P1 ∈ X (a). Therefore, every P ∈ W is either in Y or is a limit point of Y . Hence, W is a closure of Y and, under the above stated identification, of U . This completes the proof of the theorem. Theorem 5.9. Let S be the category of commutative hemirings with 1 and T be the category of compact topological spaces. Then the correspondence which associates hemiring S in S with the topological space W in T, where W stands for Fh-spec(S) and a morphism f : S → S in S with the morphism f ∗ : W → W in T, given by f ∗ (P ) = f −1 (P’), defines a contravariant functor from S to T. Proof. Observe that if f : S → S and g : S → S are two morphism in S , then f ∗ and g ∗ are well defined and (g ◦ f )∗ = f ∗ ◦ g ∗ . If I : S → S is the identity on S in S , then I ∗ is the identity on W . As for the continuity of f ∗ , observe that if V(a) is a basic closed set in W , then f ∗−1 (V(a)) = {P ∈ W | f ∗ (P ) ∈ V(a)} = {P ∈ W | f ∗ (P )(a) = 1} = {P ∈ W |P f (a) = 1} = V ( f (a)). Therefore, f ∗−1 (V(a)) is a basic closed set in W . Corollary 5.10. If f : S → S is an isomorphism in the category S of commutative hemirings with 1, then f ∗ : W → W is an isomorphism in the category T of compact topological spaces. Thus, Fh-spec(S) is unique for a given hemiring S. Proof. If g : S → S is the inverse of the isomorphism f , then g ∗ is the inverse of f ∗ . 6. Conclusion In the conclusion of their paper [18], Zhan and Dudek have raised two important issues: (1) investigating semiprime fuzzy ideals and (2) establishing fuzzy spectrum of hemirings. Our paper settles both the issues satisfactorily. The main obstacle in tackling these issues was a lack of appropriate definition of a prime (and semiprime) fuzzy h-ideal. Definition 3.6, introduced in this paper, is not only a proper fuzzification of the notion of “primeness” but, in view of Proposition 3.7 and Corollary 3.8, is the best that one can achieve. Moreover, it generalizes the fuzzification in [18], its prime h-ideals can take any number of grade-memberships, the grade-membership is not restricted to the unit interval [0,1] in order to make the concept meaningful and it takes nice form if restricted to [0.1]. Based on the new definition we construct the spectrum of prime fuzzy h-ideals. We have introduced the concept of semiprime fuzzy h-ideals and have shown that they determine the topology of the spectrum. There are some peripheral issues to be settled. We may need a proper definition of the nil-radical of a fuzzy ideal to study irreducibility of the fuzzy spectrum. We need to study other characterizations of semiprime fuzzy h-ideals to refine our results. Fuzzification of primary h-ideals will be helpful in our deeper study of prime h-ideals. Acknowledgements The author is highly grateful to the Editors-in-Chief Professors Bernard De Baets and Didier Dubois, the Area Editor Professor John Mordeson and the referees for all the pains undertaken for the improvement of the paper. Thanks are due to Professor Jianming Zhan for furnishing some recent references promptly.
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References [1] A.V. Aho, J.D. Ullman, Introduction to Automata Theory, Languages and Computation, Addison-Wesley, Reading, MA, 1979. [2] J. Ahsan, K. Saifullah, Semigroups characterized by their fuzzy ideals, Technical Report no. 147, King Fahd University of Petroleum and Minerals, Dhahran, 1993. [3] K. Glazek, A Guide to Literature on Semirings and their Applications in Mathematics and Information Sciences: with Complete Bibliography, Kluwer Academic Publishers, Dordrecht, 2002. [4] J.S. Golan, Semirings and their Applications, Kluwer Academic Publishers, Dordrecht, 1999. [5] J.S. Golan, Power Algebra over Semirings with Applications in Mathematics and Computer Science, Kluwer Academic Publishers, Dordrecht, 1999. [6] U. Hebisch, H.J. Weinert, Semirings, Algebraic Theory, and Applications in Computer Science, World Scientific, Singapore, 1998. [7] M. Kondo, W.A. Dudek, On transfer principle in fuzzy theory, Mathware Soft. Comput. 12 (2005) 41–55. [8] W. Kuich, A. Salomaa, Semirings, Automata, Languages, Springer, Berlin, 1986. [9] H.V. Kumbhojkar, On prime fuzzy ideals of a semigroup, in: Proc. of 26th Annual Iranian Mathematical Conference, 1995, pp. 193–199. [10] H.V. Kumbhojkar, Spectrum of prime fuzzy ideals, Fuzzy Sets and Systems 62 (1994) 101–109. [11] H.V. Kumbhojkar, Some comments on spectrum of prime fuzzy ideals, Fuzzy Sets and Systems 85 (1997) 109–114. [12] H.V. Kumbhojkar, M.S. Bapat, On prime and primary fuzzy ideals and their radicals, Fuzzy Sets and Systems 53 (1993) 203–216. [13] H.V. Kumbhojkar, N. Kuroki, Y. Suzuki, Some comments on fuzzifying a concept, J. Fuzzy Math. 3 (1995) 83–88. [14] D.S. Malik, J.N. Mordeson, Fuzzy prime ideals of a ring, Fuzzy Sets and Systems 37 (1990) 93–98. [15] J.N. Mordeson, P.S. Nair, Fuzzy Mathematics, Physica-Verlag, Warsaw, Poland, 1998. [16] T.K. Mukherjee, M.K. Sen, On fuzzy ideals of ring I, Fuzzy Sets and Systems 21 (1987) 90–104. [17] U.M. Swami, K.L.N. Swami, Fuzzy prime ideals of a ring, J. Math. Anal. Appl. 134 (1988) 94–103. [18] J. Zhan, W.A. Dudek, Fuzzy h-ideals of hemirings, Inform. Sci. 177 (2007) 876–886.
Spectrum of prime L-fuzzy h-ideals of a hemiring H.V. Kumbhojkar∗,1 Department of Mathematics, Addis Ababa University, Addis Ababa, Ethiopia Received 3 November 2008; received in revised form 16 September 2009; accepted 6 October 2009 Available online 15 October 2009
Abstract We redefine the concept of prime fuzzy h-ideals of a hemiring so that the fuzzy h-ideals are not necessarily 2-valued. We also introduce the concept of semiprime fuzzy h-ideals. A topological space, called the spectrum of prime fuzzy h-ideals of a commutative hemiring with unity, has been obtained. This topological space is compact and preserves isomorphisms between hemirings. The correspondence associating a hemiring with its spectrum of prime fuzzy h-ideals is shown to define a contravariant functor from the category of commutative hemirings with unity into the category of compact topological spaces. The spectrum of (crisp) prime h-ideals of the hemiring is a subspace which is dense in the spectrum of prime fuzzy h-ideals. Valuation lattices for all the fuzzy sets in the paper are assumed to be complete Heyting algebras. © 2009 Elsevier B.V. All rights reserved. Keywords: Prime fuzzy h-ideal; Semiprime fuzzy h-ideal; Prime spectrum; Compact topological space; Contravariant functor
1. Introduction Hemirings, or semirings with commutative addition and absorbing zero, appear in a natural way in theories of automata, formal languages, theoretical computer science, optimization and graph theory [1,3–6,8]. These areas and fuzzy logic have useful or potentially useful applications in Control Engineering. This may be a reason why there have been attempts to fuzzify basic concepts in semiring theory [3]. This paper is the result of a search for appropriate answers to the questions raised in the conclusion of [18]. The authors of [18] conclude their paper with the following comment: “In our opinion the future study of fuzzy sets in hemirings and semirings can be connected with: (1) investigating semiprime fuzzy h-ideals; (2) establishing a fuzzy spectrum of a hemiring; . . .”. The authors of [18] have introduced the concept of a prime fuzzy h-ideal of a hemiring. However, a major hurdle in developing spectral theory based on their definition of prime fuzzy h-ideals is the restrictive nature of the definition. Prime fuzzy h-ideals, defined in this paper, turn out to be 2-valued fuzzy sets and thus, they are not “really fuzzy”. What makes these prime fuzzy h-ideals more restrictive is, as proved by the authors in [18] that one of their values is already fixed to be 1. One more serious draw-back of the definition is noticed when the authors’ definition is extended to L-fuzzy sets, where L is a lattice different from the lattice of the unit interval [0,1] used in [18]. In many important ∗ Corresponding author. Present address: P.O. Box 1176, Mathematics, Science Faculty, AAU, Addis Ababa, Ethiopia.
E-mail address: [email protected] 1 Permanent address: No. 9, Ashok Apartments, Mali Colony Takala, Kolhapur 416008, Maharashtra, India.
0165-0114/$ - see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2009.10.006
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lattices, zero elements are not prime elements. When a fuzzy h-ideal assumes values in such lattices, then even the characteristic function of a prime h-ideal fails to be a prime fuzzy h-ideal. Prime spectrum or the Zariski topology on the set of prime ideals of a commutative ring with unity plays an important role in the field of commutative algebra, algebraic geometry and lattice theory. Semiprime ideals determine the topology. Since we expect that the spectrum of prime fuzzy h-ideals and semiprime fuzzy h-ideals play analogous roles in the theory of commutative hemirings with unity, a study of these concepts becomes important to us. However, for the reasons stated above, the definition of prime fuzzy h-ideals given in [18] does not lead us far in developing a spectral theory of semirings. One, and the most frequently used, way to fuzzify a mathematical concept is to replace a set with structure in the concept by a fuzzy set with analogous fuzzy structure [7,13]. This approach was followed for fuzzifying prime ideals of a ring in [14–17], prime h-ideals of a hemiring in [18] and prime ideals of a semigroup in [2]. In all cases prime fuzzy ideals turned out to be 2-valued. We resolve this problem successfully by redefining prime fuzzy ideals in the cases of commutative rings with unity [10–12] and semigroups [9]. In this paper, we redefine prime fuzzy h-ideals of a hemiring on similar lines. The prime fuzzy h-ideals defined in this paper are no longer 2-valued. In fact, they may even be infinite-valued. This offers an appropriate setting to introduce a topology on the set of prime fuzzy h-ideals of a hemiring to get the socalled spectrum of prime fuzzy h-ideals. We construct the spectrum of prime fuzzy h-ideals of a commutative hemiring with unity and show that the topology of the spectrum is determined by semiprime fuzzy h-ideals. We prove that this spectrum is compact topological space in which the spectrum of (crisp) prime h-ideals is dense. This spectrum defines contravariant functor from the category of commutative hemirings with unity to the category of compact topological spaces. In this paper, we work in a setting more general than [18] by considering L-fuzzy sets, where L is a Heyting algebra. However, since the lattice L is fixed throughout the paper, we call them fuzzy sets instead of L-fuzzy sets without causing any confusion. 2. Preliminaries Throughout this paper, L stands for a complete Heyting algebra. In other words L is a complete lattice such that for all subsets T of L and all b ∈ L, ∨{a ∧ b|a ∈ T } = (∨{a|a ∈ T }) ∧ b and ∧ {a ∨ b|a ∈ T } = (∧{a|a ∈ T }) ∨ b. The triple (S, +, ·) stands for a hemiring, i.e. (S, +) is a commutative monoid having identity element 0 and (S, ·) is a semigroup satisfying the following identities: a(b + c) = ab + ac and (a + b)c = ac + bc, 0.x = 0 = x.0. A commutative hemiring with unity is a hemiring (S, +, ·) such that (S, · ) is a commutative monoid. We denote the identity element of (S, ·) by 1. With abuse of notation we denote (S, +, .) by S. Unless stated otherwise N stands for the hemiring of non-negative integers with usual operations of addition (+) and multiplication (·). A left ideal A of S is a non-empty set A which is closed under the addition of S and is such that, for all x ∈ S and a ∈ A we have xa ∈ A. A left ideal A of S is called a left h-ideal, if the following condition holds for all x, z ∈ S: (x + a + z = b + z and a, b ∈ A) ⇒ x ∈ A. An L-fuzzy subset (or simply an L-fuzzy set) A of a set X is a function A: X → L. With usual abuse of notation we call A a fuzzy set of X. If ∈ L then the set {x ∈ X |A(x) ≥ } is called an -level cut or in short -cut of A and is denoted by A . A fuzzy set J:S → L is a fuzzy left ideal if for all a, b ∈ S the following conditions are satisfied: (i) J (a + b) ≥ J (a) ∧ J (b) and (ii) J (ab) ≥ J (b). A fuzzy left ideal J:S → L is called a fuzzy left h-ideal, if the following condition is satisfied: x + a + z = b + z ⇒ J (x) ≥ J (a) ∧ J (b)∀x, a, b, z ∈ S. A (fuzzy) right h-ideal is similarly defined. Whenever a statement is made about (fuzzy) left h-ideals, it is to be understood that the analogous statement is made about (fuzzy) right h-ideals. A (fuzzy) h-ideal is one, which is both (fuzzy) right and (fuzzy) left h-ideal. 3. Prime fuzzy left h-ideals Let A be a left ideal of S and h(A) = {x ∈ S|x + a + z = b + z for some a, b, ∈ A and z ∈ S}.
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The following properties of h( A) are well known: 1. 2. 3. 4.
h(A) is the smallest left h-ideal of S containing A. A is an h-ideal iff A = h(A). h(h( A)) = h(A). For any left ideals A and B of S, A ⊆ B ⇒ h(A) ⊆ h(B).
The left h-ideal h(A) is called the h-closure of A. Proposition 3.1 (Zhan and Dudek [18]). If S is a hemiring and A and B are left ideals of S, then h(AB) = h(h( A)h(B)). A left h-ideal P of a hemiring S is called prime, if PS and for any left h-ideals A and B of S, AB ⊆ P implies either A ⊆ P or B ⊆ P [18]. An interesting consequence of Proposition 3.1 is the following: Proposition 3.2. A proper left h-ideal P of a hemiring S is prime iff for all a, b ∈ S, aSb ⊆ P implies a ∈ P or b ∈ P. Proof. Let P be a prime h-ideal and aSb ⊆ P for a,b ∈ S. Clearly, we have SaSb ⊆ P. Let A = h(Sa) and B = h(Sb). Our first claim is AB ⊆ P. Observe that every element of AB is a finite sum of the products xy, where x is in A and y is in B. Let for some s, t, u, v, z and w in S, we have x + sa + z = ta + z and y + ub + w = vb + w. We have then, equalities: ay + aub + aw = avb + aw, and x y + say + zy = tay + zy. As aub, avb are elements of P and P is a left h-ideal, ay is in P. Therefore, say, tay and consequently xy are in P. This justifies the claim. Since P is prime we have, either A ⊆ P or B ⊆ P. Suppose A ⊆ P. Let a be the left ideal generated by a. Then clearly a a ⊆ Sa. Therefore, we have, h( a )h( a ) ⊆ h(h( a )h( a )) = h( a a ) ⊆ h(Sa) = A ⊆ P. Therefore, we have h( a ) ⊆ P and hence, a ∈ P. The converse is obvious. If S is a commutative hemiring with unity, the condition in Proposition 3.2 is equivalent to the following condition: ab∈ P ⇒ a ∈ P or b ∈ P. This leads to the following: Proposition 3.3. Let S be a commutative hemiring with unity. Then a proper h-ideal P of S is prime iff for all a, b ∈ S, ab ∈ P ⇒ a ∈ P or b ∈ P. Proposition 3.3 in conjunction with usual applications of Zorn’s Lemma leads us to the following: Theorem 3.4. Every proper h-ideal of a commutative hemiring S with unity is contained in a prime h-ideal of S. It should be noted that Theorem 3.4 depends on Proposition 3.3 and hence on Proposition 3.1. Corollary 3.5. If commutative hemiring S with unity has a proper h-ideal then it has a prime h-ideal. We now, in the light of Propositions 3.2, define prime fuzzy left h-ideal of a hemiring as follows: Definition 3.6. A fuzzy left h-ideal P: S → L is called prime if it is non-constant and, for all a, b ∈ S and ∈ L, the following condition is satisfied: P(asb) ≥ , ∀s ∈ S ⇒ P(a) ≥ or P(b) ≥ . Proposition 3.7. A non-constant fuzzy left h-ideal P : S → L is prime iff its every non-empty level cut is either a prime left h-ideal of S or S itself. Proof. Let P be a prime fuzzy left h-ideal, ∈ L and SP ∅. Let aSb ⊆ P . Then P(asb) ≥ , ∀s ∈ S. Consequently, P(a) ≥ or P(b) ≥ . Therefore, we have a ∈ P or b ∈ P .
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To prove the converse, consider a non-constant fuzzy h-ideal P: S → L. If P is not prime then there exist ∈ L and a, b ∈ S such that P(asb) ≥ , ∀s ∈ S, but ⱕ P(a), and ⱕ P(b). Thus, we have aSb ⊆ P but, a ∈ / P and b ∈ / P and therefore, P is neither a prime h-ideal nor the hemiring S. Corollary 3.8. A left h-ideal P of S is prime iff its characteristics function vP is prime fuzzy left h-ideal. When S is a commutative hemiring with unity, the characterization of a prime fuzzy h-ideal takes the following simpler form: Proposition 3.9. If S is a commutative hemiring with unity, a non-constant fuzzy h-ideal P of S is prime iff P(ab) = P(a) or P(b)∀a, b ∈ S. If, moreover, L is totally ordered, in particular if L = [0, 1], then P is prime iff P(ab) = P(a) ∨ P(b)∀a, b ∈ S. Proof. Let a non-constant fuzzy h-deal P be prime and P(ab) = (say). Then P(asb) = P(sab) ≥ P(s) ∨ P(ab) ≥ , ∀s ∈ S. As P is prime, we have either P(a) ≥ or P(b) ≥ . Suppose P(a) ≥ . Then P(ab) ≥ P(a) ≥ = P(ab). Hence, P(ab) = P(a). Conversely, suppose P(ab) = P(a) or P(b) and P(asb) ≥ , ∀s ∈ S. Selecting s = 1 we have P(ab) ≥ . Hence, by hypothesis, P(a) ≥ or P(b) ≥ . Thus, P is a prime fuzzy h-ideal. If a fuzzy h-deal is prime according to Definition 4.1 of [18], then in view of Proposition 3.7, it is prime according to Definition 3.6. The following examples show that the converse is not true. Example 3.10. Let p be a prime integer. Define a fuzzy set P : N → [0, 1] by P(x) = 1 if x = 0 = if x ∈ pN ∼ 0 = if x ∈ / pN. By Proposition 3.7 P is a prime fuzzy h-deal, for all 0 ≤ < ≤ 1. We will call the fuzzy ideal P a prime fuzzy h-ideal induced by the prime number p and denote it by ( pN) . Clearly, ( pN) is not a prime fuzzy ideal according to Definition 4.1 of [18] unless = 1. One more drawback of Definition 4.1 of [18] of prime fuzzy ideals is highlighted by the following example. If we do not restrict ourselves to the grade membership lattice [0,1], then even the characteristic function of a prime h-ideal may fail to be prime fuzzy h-ideal according to [18]. However, in view of Proposition 3.7, it will remain prime fuzzy h-ideal according to Definition 3.6. Example 3.11. Consider the Boolean algebra L = {0, , , 1}. Let P be the characteristic function of 2N, and A, B, the fuzzy h-ideals of N defined as follows: A(2n) = 1 = B(2n) and A(2n + 1) = , B(2n + 1) = for all n ∈ N . Then AB ⊆ A ∩ B = P. However, clearly, neither A nor B is contained in the fuzzy h-ideal P. Thus Proposition 4.3 and Theorem 4.4 of [18] are no more valid when the lattice [0,1] is replaced by L = {0, , , 1}. However, they do not conflict with Definition 3.6 of prime. In the light of the fact that a prime fuzzy h-ideal can have any number of values, we are, now, in a position to define meaningfully the prime radical of a fuzzy ideal. Definition 3.12. If J : S → L is a fuzzy ideal, then the intersection of all prime fuzzy h-ideals of S containing J is called the h-prime radical of J. We denote it by rh (J ). If the set of prime fuzzy h-ideals of S containing J is empty, we define rh (J ) to be S . Note that rh (J ) is a fuzzy h-ideal containing J.
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4. Semiprime fuzzy h-ideals In this section, we discuss fuzzy analogue of some results on semiprime h-ideals. We will call a left h-ideal I of a hemiring S semiprime, if I S and for any left h-ideal A of S, A A ⊆ I implies A ⊆ I . Proposition 4.1. A proper left h-ideal I of a hemiring S is semiprime iff for all a ∈ S, aSa ⊆ I ⇒ a ∈ I . The proof runs parallel to that of Proposition 3.2. We now define the fuzzy analogue of semiprime h-ideals: Definition 4.2. A fuzzy left h-ideal J : S → L is called semiprime if J is non-constant and, for all a ∈ S and ∈ L, the following condition is satisfied: J (asa) ≥ , ∀s ∈ S ⇒ J (a) ≥ . Proposition 4.3. A non-constant fuzzy left h-ideal J : S → L is semiprime iff its every non-empty level cut is either a semiprime left h-ideal of S or S itself. The proof runs parallel to that of Proposition 3.7 As immediate consequences of Proposition 4.3, we have the following: Corollary 4.4. Intersection of semiprime fuzzy left h-ideals of S is a semiprime fuzzy left h-ideal of S. In particular, intersection of prime fuzzy left h-ideals of S is a semiprime fuzzy left h-ideal of S. Corollary 4.5. A left h-ideal J of S is semiprime iff its characteristics function vJ is semiprime fuzzy left h-ideal. When S is a commutative hemiring with unity, we have the following characterization of semiprime fuzzy h-ideals: Proposition 4.6. Let S be a commutative hemiring with unity. A non-constant fuzzy h-ideal J of S is semiprime iff J (a 2 ) = J (a). The proof is similar to that of Proposition 3.9. Although every prime fuzzy h-ideal of S is a semiprime, the following example shows that, the converse is not true: Example 4.7. Let p1 , p2 . . . be distinct prime numbers in N, J 0 = N and J l = p1 p2 . . . pl N, l = 1, 2, . . . . Then we have J 0 ⊃ J 1 ⊃ · · · ⊃ J n ⊃ J n+1 . . . . As every non-zero element of N has unique prime factorization, J l is a semiprime h-ideal for l = 2, 3, . . . but not a prime h-ideal. Consider the fuzzy h-ideal given below: J : N → [0, 1], J (x) = 1 if x = 0 = l/(l + 1) ∀x ∈ J l ∼ J l+1 , l = 0, 1, 2. . . . Then, by Proposition 4.3, J is a semiprime fuzzy h-ideal. Proposition 3.7 ensures that if we choose l ≥ 2, J is not a prime fuzzy h-ideal. 5. Spectrum of prime fuzzy ideals In the remaining part of this paper, S stands for a commutative hemiring with unity and X stands for the set of all prime fuzzy h-ideals of S with membership grades in a fixed lattice L. We further assume that P(0) = 1. (The hypothesis J (0) = 1, for a fuzzy ideal J : S → L is not really restrictive. It is in full conformity with the principles of fuzziness. For, every ideal must contain the element 0 of S. Moreover, the hypothesis is needed to prove even a simple result that the intersection of two fuzzy ideals is a fuzzy ideal when L is an arbitrary Heyting algebra. For these reasons J (0) = 1 may, in our opinion, be made a part of the definition of a fuzzy ideal J [13].) If A : S → L is any fuzzy set, let V(A) = {P ∈ X |A ⊆ P}. When A = {a} , where a ∈ S, we denote V(A) by V(a). Thus V(a) = {P ∈ X |P(a) = 1}.
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Proposition 5.1. Let A : S → L and B : S → L be two fuzzy sets. Then A ⊆ B ⇒ V(B) ⊆ V( A), V(A) ∪ V(B) ⊆ V( A ∩ B), If I and J are h-ideals of S, then V( I ) ∪ V( J ) = V( I ∩J ). Let J be a fuzzy h-ideal of S generated by A and rh (J ) be the h-prime radical of J . Then we have V( A) = V(J ) = V(rh (J )). (e) If {Ai |i ∈ } is a family of fuzzy subsets of S, then V(∪{Ai |i ∈ }) = ∩{V (Ai )|i ∈ }. (f) If T ⊆ S, then V(T ) = ∩{V(a)|a ∈ T }. (g) V(a) ∪ V(b) = V(ab), for all ab ∈ S.
(a) (b) (c) (d)
Proof. (a) The statement (a) follows readily from the definitions. (b) Observe that A ∩ B ⊆ A and A ∩ B ⊆ B and apply (a) (c) Since, I ∩ J = I ∩J the result (b) assures: V( I ) ∪ V( J ) ⊆ V( I ∩J ). On the other hand let P ∈ V( I ∩J ). Since I ∩J ⊆ P we have P(x) = 1 for all x ∈ I ∩ J . Suppose I P and J P hold. Then there exist x ∈ I and y ∈ J , such that P(x) 1 P(y). But as x y ∈ I ∩ J , we have P(x y) = 1. This contradicts that P is a prime fuzzy h-ideal. Hence we have either I ⊆ P or J ⊆ P and therefore, V( I ∩J ) ⊆ V( I ) ∪ V( J ). (d) To prove the equality V(A) = V(J ) note that if T is the set of all fuzzy h-ideals containing A, then J = ∩{I |I ∈ T }. The result then follows from the fact that the inclusion A ⊆ P holds iff the inclusion J ⊆ P holds. To prove V(J ) = V(rh (J )), we choose T = {P ∈ X |J ⊆ P}. Then rh (J ) = ∩{P|P ∈ T }. The result then follows from the fact that the inclusion J ⊆ P holds iff the inclusion rh (J ) ⊆ P holds. (e) To prove (e) observe that P ∈ V(∪{Ai |i ∈ }) ⇔ ⇔ ⇔ ⇔
∪ {Ai |i ∈ } ⊆ P Ai ⊆ P for all i ∈ P ∈ V(Ai ) for all i ∈ P ∈ ∩{V(Ai )|i ∈ }.
(f) If T ⊆ S, then applying (e) to the identity T = ∪{{a} |a ∈ T } we get the result. (g) The statement (g) follows readily from the definitions. The similarity between the set of (crisp) prime h-ideals of S and that of prime fuzzy h-ideals of S ends here. In general, equality does not hold in Proposition 5.1 (b). In crisp situation, when A and B are h-ideals, the equality holds. The following examples show that the equality does not hold even for fuzzy h-ideals. Example 5.2. Let S = N [x, y] be the hemiring of polynomials in two indeterminates over N. If a, b ∈ S, let a and a, b denote h-ideals of S generated by {a} and {a, b}, respectively. Consider the fuzzy ideal P : S → [0, 1] defined as follows: P(a) = 1 if a = 0 = 0.50 if a ∈ x ∼ 0 = 0.30 if a ∈ x, y ∼ x = 0 everywhere else. Then, by Proposition 3.7, P is a prime fuzzy h-ideal. Consider the fuzzy h-ideals J : S → [0, 1] and K : S → [0, 1] defined as follows: J (a) = 1 if a = 0 = 0.45 if a ∈ x 2 ∼ {0} = 0.40 if a ∈ x 2 , y 2 ∼ x 2 = 0 everywhere else.
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K (a) = 1 if a = 0 = 0.65 if a ∈ x 2 ∼ {0} = 0.20 if a ∈ x 2 , y 2 ∼ x 2 = 0 everywhere else. Then (J ∩ K )(a) = 1 = P(a) if a = 0, = 0.45 < 0.50 = P(a) if a ∈ x 2 ∼ {0} = 0.20 < 0.30 = P(a) if a ∈ x 2 , y 2 ∼ x 2 = 0 ≤ P(a) everywhere else. Therefore, we have J ∩ K ⊆ P. But as J (y 2 ) > P(y 2 ) we have J P and as K (x 2 ) > P(x 2 ) we have K P. Thus, P ∈ V (J ∩ K ), but P ∈ / V(J ) ∪ V(K ). Example 5.3. Let L = {0, , , 1} be the Boolean algebra of four elements. Consider the fuzzy h-ideal J : N → L defined as follows: J (x) = 1 if x ∈ 6N = if x ∈ 2N ∼ 6N = if x ∈ 3N ∼ 6N = 0 everywhere else. Define K : N → L as follows: K (x) = when J (x) = and K (x) = when J (x) = ; K (x) = J (x) everywhere else. Clearly, J ∩ K is the characteristic function of the h-ideal 6N. If P is the characteristic function of 2N, then P is a prime fuzzy h-ideal containing J ∩ K . But J (3) = > 0 = P(3) and K (3) = > 0 = P(3). Thus, the strict inequality may hold in Proposition 5.1(b) even if we restrict ourselves to 2-valued fuzzy h-ideals. However, as will be proved in the remaining part of the paper, the set {V( A )|A ⊆ S} forms a system of closed sets for a topology on the set X of prime fuzzy h-ideals. Let X(a) = X ∼ V(a). Then X(a) = {P ∈ X |P(a) 1}. Let C = {X (a)|a ∈ S}. Theorem 5.4. The set C is a base for a topology on X . The open subsets of X are precisely X(T ), where T is any subset of S. The topology is completely determined by semiprime fuzzy h-ideals of S. Proof. Clearly, we have X(1) = X , and hence ∪{X(a)|a ∈ S} = X . Since X(ab) = X(a) ∩ X(b) for all a, b ∈ S, C is closed under intersection and hence forms a base for a topology on X . A typical open set in this topology will be ∪{X(a)|a ∈ T } for some T ⊆ S. Therefore, we get ∪ {X(a)|a ∈ T } = ∪ {X ∼ V(a)|a ∈ T } = X ∼ ∩{V(a)|a ∈ T } = X ∼ V(T ) = X(T ). The last part follows from the fact that V(T ) = V( T ) = V(rh ( T )), where rh ( T ) is the h-prime fuzzy radical of T . Definition 5.5. The topological space X of prime fuzzy h-ideals of S will be called the spectrum of prime fuzzy h-ideals (or in short prime fuzzy h-spectrum) of S and will be denoted by Fh-spec(S). By Corollaries 3.5 and 3.8 Fh-spec(S) is non-empty when S has a proper h-ideal.
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Example 5.6. (a) Let R+ be the hemiring of non-negative real numbers. Let, for 0 ≤ < 1, O : R+ → [0, 1] be the fuzzy ideal defined by O (x) = 1 if x = 0 = if otherwise. R+ has only one proper ideal, namely 0 which is the prime h-ideal. Therefore, the set of prime fuzzy h-ideals is X = {O |0 ≤ < 1}. Since V(0) = X and V(a) = ∅ for all a 0, Fh-spec(R+ ) is an indiscrete topological space. (b) Let O denote the prime fuzzy h-ideal defined as above except that its domain of definition, now, is N. Then the set of prime fuzzy h-ideals is X = {O |0 ≤ < 1} ∪ {( pN) | p is prime, 0 ≤ < ≤ 1}, where ( pN) is defined in Example 3.10. Since every h-ideal of N is a principal ideal, closed sets of Fh-spec(N) are precisely V(n) and the open m be the prime factorization of n, then sets are precisely X(n) for n ∈ N . If n ≥ 2 let p11 p22 . . . pm m V(n) = V( p11 ) ∪ · · · ∪ V( pm ) = V( p1 ) ∪ · · · ∪ V( pm ) = V( p1 . . . pm ).
Therefore, the spectral topology is determined by the prime fuzzy h-ideals containing the characteristic functions of the semiprime h-ideals p1 . . . pm N, m = 1, 2, . . . . Clearly, then V( p) = {( pN)1 | 0 ≤ < 1} for every prime integer p and therefore, X( p) = {O |0 ≤ < 1} ∪ {( pN) |0 ≤ < < 1}} ∪ {(qN) |0 ≤ < ≤ 1, q is prime q p}. More generally a typical open set in Fh-spec(N) is X( p1 . . . pm ) = {O |0 ≤ < 1} ∪ {( pN) |0 ≤ < < 1, p is prime, p = p1 , . . . , pm } ∪ {( pN) |0 ≤ < ≤ 1, p is prime, p p1 , . . . , pm }. Proposition 5.7. If F is a subset of Fh-spec(S) then the closure F∗ of F is F∗ = V(v∩{Q1|Q∈F} ), where Q1 = {x ∈ S|Q(x) = 1}. If P, Q ∈ Fh-spec(S), then {P}∗ = {Q}∗ iff P1 = Q1. Thus, Fh-spec(S) is not a T0 -space. Proof. Clearly v∩{Q1|Q∈F} (x) = 1, iff Q(x) = 1 for all Q ∈ F. Therefore, if P ∈ F, then ∩{Q1|Q∈F} ⊆ P and as a result P ∈ V(v∩{Q1|Q∈F} ). Thus, V(v∩{Q1|Q∈F} ) is a closed set containing S, and hence contains F∗ . On the other hand, let v∩{Q1|Q∈F} ⊆ P and P ∈ / F. If X(a) is a basic open set containing P, then P(a) 1 and we have v∩{Q1|Q∈F} (a) 1. Clearly, there exists Q ∈ F such that Q(a)1. In other words, every open set containing P contains at least one element of F. Therefore, if P ∈ V(∩{Q1|Q∈F} ), then P is either in F or is a limit point of F. Hence V(v∩{Q1|Q∈F} ) = F∗ . Theorem 5.8. Fh-spec(S) is a compact topological space and the topological space h-Spec(S) of prime h-ideals of S is dense in Fh-spec(S). Proof. To prove the compactness, it is sufficient to consider a cover consisting of basic open sets. Let {X(a)|a ∈ T }, where T ⊆ S, be one such open cover of X . Then, X = ∪{X(a)|a ∈ T } = X ∼ V(T ). Hence V(T ) = ∅. Let T generate the h-ideal K . We claim that K = S. Suppose K S. Then, by Theorem 3.4, there exists a prime h-ideal P of S containing K . Clearly, then, we have v K ⊆ vP . Since vP is a prime fuzzy h-ideal, we have an absurd result that vP ∈ V(v K ) = ∅. 2 This justifies the claim. n Therefore, there exist elements a1 , a2 , . . . , an ∈ T and s1 , s2 , . . . , sn , z ∈ S such that 1+ m 1 si ai +z = m+1 si ai + z. Consider A = {a1 , a2 , . . . , an } and let A be the h-ideal generated by A. Since A = S, we have A = S and V( A ) = V( A ) = V( S ) = ∅. Therefore, ∪ {X(ai )|i = 1, 2, . . . , n} = ∪ {X ∼ V(ai )|i = 1, 2, . . . , n} = X ∼ ∩{V(ai )|i = 1, 2, . . . , n} = X ∼ V( A ) = X. Thus, {X(ai )|i = 1, 2, . . . , n} is a finite subcover of X. 2 Note that v may not be a prime fuzzy h-ideal if we apply Definition 4.1 of [18] in conjunction with the assumption that L [0, 1]. P
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In the remaining part of the paper let U denote h-Spec(S) and W denote Fh-spec(S). Consider the subset Y of W given by Y = {vP |P ∈ U }. A typical closed subset F of W is F = V(vT ), where T ⊆ S. Hence a typical closed subset F ∗ of the induced subspace Y of W will be of the form F ∗ = F ∩ Y = {vP |T ⊆ P}. With the usual identification of an ideal P of S with the fuzzy ideal vP , Y can be identified with U with the topology defined by the closed sets of the type {P|T ⊆ P}. Let P ∈ W be any prime fuzzy h-ideal not in Y and let X(a) be any basic open set in W containing P. Let P1 = {x ∈ S|P(x) = 1}. Then P1 ∈ Y . As P(a) 1 we have a ∈ / P1 and thus P1 ∈ X (a). Therefore, every P ∈ W is either in Y or is a limit point of Y . Hence, W is a closure of Y and, under the above stated identification, of U . This completes the proof of the theorem. Theorem 5.9. Let S be the category of commutative hemirings with 1 and T be the category of compact topological spaces. Then the correspondence which associates hemiring S in S with the topological space W in T, where W stands for Fh-spec(S) and a morphism f : S → S in S with the morphism f ∗ : W → W in T, given by f ∗ (P ) = f −1 (P’), defines a contravariant functor from S to T. Proof. Observe that if f : S → S and g : S → S are two morphism in S , then f ∗ and g ∗ are well defined and (g ◦ f )∗ = f ∗ ◦ g ∗ . If I : S → S is the identity on S in S , then I ∗ is the identity on W . As for the continuity of f ∗ , observe that if V(a) is a basic closed set in W , then f ∗−1 (V(a)) = {P ∈ W | f ∗ (P ) ∈ V(a)} = {P ∈ W | f ∗ (P )(a) = 1} = {P ∈ W |P f (a) = 1} = V ( f (a)). Therefore, f ∗−1 (V(a)) is a basic closed set in W . Corollary 5.10. If f : S → S is an isomorphism in the category S of commutative hemirings with 1, then f ∗ : W → W is an isomorphism in the category T of compact topological spaces. Thus, Fh-spec(S) is unique for a given hemiring S. Proof. If g : S → S is the inverse of the isomorphism f , then g ∗ is the inverse of f ∗ . 6. Conclusion In the conclusion of their paper [18], Zhan and Dudek have raised two important issues: (1) investigating semiprime fuzzy ideals and (2) establishing fuzzy spectrum of hemirings. Our paper settles both the issues satisfactorily. The main obstacle in tackling these issues was a lack of appropriate definition of a prime (and semiprime) fuzzy h-ideal. Definition 3.6, introduced in this paper, is not only a proper fuzzification of the notion of “primeness” but, in view of Proposition 3.7 and Corollary 3.8, is the best that one can achieve. Moreover, it generalizes the fuzzification in [18], its prime h-ideals can take any number of grade-memberships, the grade-membership is not restricted to the unit interval [0,1] in order to make the concept meaningful and it takes nice form if restricted to [0.1]. Based on the new definition we construct the spectrum of prime fuzzy h-ideals. We have introduced the concept of semiprime fuzzy h-ideals and have shown that they determine the topology of the spectrum. There are some peripheral issues to be settled. We may need a proper definition of the nil-radical of a fuzzy ideal to study irreducibility of the fuzzy spectrum. We need to study other characterizations of semiprime fuzzy h-ideals to refine our results. Fuzzification of primary h-ideals will be helpful in our deeper study of prime h-ideals. Acknowledgements The author is highly grateful to the Editors-in-Chief Professors Bernard De Baets and Didier Dubois, the Area Editor Professor John Mordeson and the referees for all the pains undertaken for the improvement of the paper. Thanks are due to Professor Jianming Zhan for furnishing some recent references promptly.
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