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Fuzzy Prime Filters of Lattice Implication Algebras

Fuzzy Prime Filters of Lattice Implication Algebras Fuzzy Inf. Eng. (2011) 3: 235-246 DOI 10.1007/s12543-011-0080-y ORIGINAL ARTICLE Yi Liu · Ke-yun Qin · Yang Xu Received: 16 May 2010/ Revised: 18 July 2011/ Accepted: 16 August 2011/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract In this paper, some properties of fuzzy filters are given. Besides, the structure of fuzzy filters are further studied. And finally, the concept of fuzzy prime filter is proposed with some equivalent conditions of fuzzy prime filters obtained. Keywords Lattice implication algebras · Fuzzy filters · Fuzzy prime filters 1. Introduction As is known to all, one significant function of artificial intelligence is to make com- puter simulate human being in dealing with uncertain information. And logic estab- lishes the foundations for it. However, certain information process is based on the classic logic. Non-classical logics consist of these logics handling a wide variety of uncertainties (such as fuzziness, randomness, and so on ) and fuzzy reasoning. Therefore, non-classical logic has been proved to be a formal and useful technique for computer science to deal with fuzzy and uncertain information. Many-valued logic, as the extension and development of classical logic, has always been a cru- cial direction in non-classical logic. Lattice-valued logic, an important many-valued logic, has two prominent roles: One is to extend the chain-type truth-valued field of the current logics to some relatively general lattices. The other is that the incom- pletely comparable property of truth value characterized by the general lattice can more effectively reflect the uncertainty of human being’s thinking, judging and de- cision. Hence, lattice-valued logic has been becoming a research field and strongly Yi Liu () Intelligent Control Development Center, Southwest Jiaotong University, Chengdu, Sichuan 610031, P.R.China College of Mathematics and Information Science, Neijiang Normal University, Neijiang , Sichuan 641000, P.R.China email: liuyiyl@126.com Ke-yun Qin, Yang Xu Intelligent Control Development Center, Southwest Jiaotong University, Chengdu, Sichuan 610031, P.R.China 236 Yi Liu · Ke-yun Qin · Yang Xu (2011) influencing the development of algebraic logic, computer science and artificial in- telligent technology. In order to investigate a many-valued logical system whose propositional value is given in a lattice, in 1993, Xu first established the lattice im- plication algebra by combining lattice and implication algebra, and explored many useful structures [1-4]. The filter theory serves a vital function for the development of lattice implication algebras. It is also important to the automated reasoning and approximated reasoning based on lattice implication algebra, for instance, filter-based resolution principle proposed by J. Ma et al [8]. Xu and Qin [6] introduced the concepts of filter and implicative filter in lattice implication algebras, and examined their properties [21, 22]. Jun [14] and other scholars studied several filters in lattice implication algebras [14-16, 19-22]. In particular, Jun [14] gave an equivalent condition of a filter, and provided some equivalent conditions for a filter to be an implicative filter in lattice implication algebras. The concept of fuzzy set was introduced by Zadeh (1965). Since then this idea has been applied to other algebraic structures such as groups, semigroups, rings, modules, vector spaces and topologies. Xu and Qin [2] applied the concept of fuzzy set to lat- tice implication algebras, proposed the notions of fuzzy filters and fuzzy implicative filters [3]. Later on, other scholars put forward relative fuzzy filters like fuzzy (pos- itive) implication filter, fuzzy fantastic filter and discovered some properties [9-12, 17, 18, 24, 25]. This logical algebra has captured close attention from a good many researchers. And elegant results are obtained, which are collected in the monograph [4]. Let (L,∨,∧, O, I) be a bounded lattice with an order-reversing involution , the greatest element I and the smallest element O, and →: L× L −→ L be a mapping. L = (L,∨,∧, ,→, O, I) is called a lattice implication algebra if the following conditions hold for any x, y, z ∈ L: (I ) x → (y → z) = y → (x → z); (I ) x → x = I; (I ) x → y = y → x ; (I ) x → y = y → x = I implies x = y; (I )(x → y) → y = (y → x) → x; (l )(x∨ y) → z = (x → z)∧ (y → z); (l )(x∧ y) → z = (x → z)∨ (y → z). In this paper, we denote L as a lattice implication algebra (L,∨,∧, ,→, O, I). We list some basic properties of lattice implication algebras. It is useful to develop these topics in other sections. Let L be a lattice implication algebra. Then for any x, y, z ∈ L, the following conclusions hold: (1) If I → x = I, then x = I. (2) I → x = x and x → O = x . (3) O → x = I and x → I = I. (4) (x → y) → ((y → z) → (x → z)) = I. Fuzzy Inf. Eng. (2011) 3: 235-246 237 (5) (x → y)∨ (y → x) = I. (6) If x ≤ y, then x → z ≥ y → z and z → x ≤ z → y. (7) x ≤ y if and only if x → y = I. (8) (z → x) → (z → y) = (x∧ z) → y = (x → z) → (x → y). (9) x → (y∨ z) = (y → z) → (x → z). (10) x → (y → z) = (x∨ y) → z if and only if x → (y → z) = x → z = y → z; (11) z ≤ y → x if and only if y ≤ z → x. Let f be a mapping from lattice implication algebra L to L .If 1 2 f (x → y) = f (x) → f (y); f (x∨ y) = f (x)∨ f (y); f (x∧ y) = f (x)∧ f (y); f (x ) = ( f (x)) . Then f is called a lattice implication homomorphism from L to L . 1 2 From the Definition of lattice implication homomorphism, we can easily obtain the following result: If f is a lattice implication homomorphism from L to L , then 1 2 f (I) = I. In lattice implication algebras, we define the binary operation⊗ as follows, for any x, y ∈ L, x⊗ y = (x → y ) . Some properties of operation⊗ can be found in reference [4]. In the paper, the writers carry out in-depth research on the fuzzy filters of the lattice implication algebra. In Section 2, diverse equivalence characterizations of fuzzy filters are carefully displayed. The demonstration of structure of fuzzy filters can be found in section three. In the final part, the notion of fuzzy prime filter and the properties with some equivalence conditions are strongly stated. 2. Some Properties of Fuzzy Filters Definition 1 [3] Let F be said to be a filter of L . If it satisfies: (F ) I ∈ F. (F ) For any x, y ∈ L, if x ∈ F and x → y ∈ F, then y ∈ F. Lemma 1 [4] Let F be a subset of L. Then F is a filter of L , if and only if : (1) I ∈ F. (2) If x ∈ F and x ≤ y, then y ∈ F. (3) For any x, y ∈ F, then x⊗ y ∈ F. Definition 2 [2] Let A be a fuzzy subset of L. A is said to be a fuzzy filter of L ,ifit satisfies, for any x, y ∈ L (1) A(x) ≤ A(I); (2) A(y) ≥ min{A(x → y), A(x)}. Theorem 1 [25] Let A be a fuzzy set in L. Then A is a fuzzy filter of L if and only if, for any x, y ∈ L (1) A(x⊗ y) ≥ min{A(x), A(y)}; (2) x ≤ y implies A(x) ≤ A(y). 238 Yi Liu · Ke-yun Qin · Yang Xu (2011) Corollary 1 Let A be a fuzzy filter of L . The the following hold for any x, y ∈ L: (1) A(x∨ y) ≥ min{A(x), A(y)}, (2) A(x∧ y) = min{A(x), A(y)}, (3) A(x⊗ y) = min{A(x), A(y)}. Corollary 2 Let A be a fuzzy subset of L. Then A is a fuzzy filter of L if and only if A(x⊗ y) ≥ min{A(x), A(y)} and A(x∨ y) ≥ A(x). Theorem 2 [25] Let A be a fuzzy subset of L. Then A is a fuzzy filter if and only if , for any x, y, z ∈ L, x → (y → z) = I implies min{A(x), A(y)}≤ A(z). Since x ≤ y → z if and only if x ⊗ y ≤ z. So we can easily obtain the following corollaries. Corollary 3 A fuzzy set A in L is a fuzzy filter of L if and only if A satisfies the following condition: A(x) ≥ min{A(a ),··· , A(a )}, 1 n where a → (···→ (a → x)··· ) = I for a ,··· , a ∈ L. n 1 1 n The meet of two fuzzy filters A , A of L is define as 1 2 A ∧ A = A ∩ A . 1 2 1 2 Corollary 4 Let A be fuzzy filters, i = 1, 2. Then A ∧ A is also a fuzzy filter of L . i 1 2 Proof Let x, y, z ∈ L be such that x → (y → z) = I. Since A are fuzzy filters, i = 1, 2, we have A (z) ≥ min{A (x), A (y)} for i = 1, 2. Hence i i i (A ∧ A )(z) = A (z)∧ A (z) 1 2 1 2 ≥ A (x)∧ A (y)∧ A (x)∧ A (y) 1 1 2 2 = min{(A ∧ A )(x), (A ∧ A )(y)}. 1 1 2 2 Therefore, we have A ∧ A is a fuzzy filter of L from Theorem 2. 1 2 Corollary 5 Let A be a family fuzzy filters of L for i ∈ I, where I is an index set. The ∧ A is a fuzzy filter of L . i∈I i Theorem 3 Let A be a fuzzy filter of L . Put A[a] = {x ∈ L|A(a) ≤ A(x)}, then A[a] is a filter of L . Proof It is easy to obtain that I ∈ A[a]. Suppose x, y ∈ L and x → y ∈ A[a], x ∈ A[a]. Then A(a) ≤ A(x → y) and A(a) ≤ A(x). Hence, min{A(x → y), A(x)}≥ A(a). Since A is a fuzzy filter, so A(y) ≥ min{A(x → y), A(x)}≥ A(a), that is, y ∈ A[a]. Therefore A[a] is a filter of L . Fuzzy Inf. Eng. (2011) 3: 235-246 239 Theorem 4 Let A be a fuzzy set of L . Then (1) If A[a] is a filter of L for any a ∈ L, then A(z) ≤ min{A(x → y), A(x))} implies A(z) ≤ A(y). (1) (2) If A satisfies A(x) ≤ A(I) for any x ∈ L and (1), then A[a] is a filter of L . Proof (1) Assume that A[a] is a filter of L for any a ∈ L. Let x, y, z ∈ L such that A(z) ≤ min{A(x → y), A(x))}, then x → y ∈ A[z] and x ∈ A[z]. As A[z] is a filter, hence y ∈ A[z], that is A(z) ≤ A(y). (2) Suppose A to satisfy A(x) ≤ A(I) for any x ∈ L and (2), for any a ∈ L, and x, y ∈ L such that x → y ∈ A[a] and x ∈ A[a], we have A[a] ≤ min{A(x → y), A(x)}. Hence A(a) ≤ A(y) by (1). That is y ∈ A[a]. As A satisfies A(x) ≤ A(I) for any x ∈ L,so A(a) ≤ A(I), that is, I ∈ A[a]. Therefore A[a] is a filter of L . 3. Structure of Fuzzy Filters In this section, we investigate the structure of fuzzy filters of a lattice implication algebra. Theorem 5 Let A be a fuzzy filter of L and L satisfies the filter’s ascending chain condition of L . If a series of elements in Im(A) is strictly increasing, then |Im(A)| < ∞. Proof Assume that |Im(A)| = ∞ and {a } is a strictly increasing series in Im(A), then 1 > ··· > a > a > 0. Put A = {x ∈ L|A(x) ≥ a }, k = 2, 3,··· . Since A is a 2 1 k k fuzzy filter of L ,so A is a filter of L . Let x ∈ A . Then A(x) ≥ a > a . Thus we k k k k−1 have x ∈ A , it follows that A ⊆ A . k−1 k k−1 As a ∈ Im(A), thus there exists an x ∈ L such that A(x ) = a . Hence k−1 k−1 k−1 k−1 x ∈ A . But x  A , then A ⊂ A . Since Im(A) = ∞, so we can obtain k−1 k−1 k−1 k k k−1 an unterminate strictly increasing filter chain: A ⊂ A ⊂ ··· ⊂ A ⊂ A ⊂··· . 1 2 k−1 k Contradict with the ascending condition. Therefore |Im(A)| < ∞. Theorem 6 Let A be a fuzzy filter ofL and A contains finite image, that is,|Im(A)| < ∞. Then L satisfies the filter’s ascending chain condition. Proof Suppose that L dissatisfies a filter ascending condition, so there exists a strictly increasing chain: ∅  F ⊂ F ⊂···⊂··· . 0 1 We define a fuzzy set on L as follows: , if x ∈ F \ F , n = 0, 1, 2,···, ⎨ n+1 n A(x) = n+ 1 1, if x ∈ F . Now, we need to prove A is a fuzzy filter of L . Obviously, we obtain that A(I) ≥ A(x) for any x ∈ L. 240 Yi Liu · Ke-yun Qin · Yang Xu (2011) Let x, y ∈ L,if x → y ∈ F \ F and x ∈ F \ F , where n = 0, 1, 2,··· , k = n+1 n k+1 k 0, 1, 2,··· . Without loss of generality, assume that n ≥ k. Then x ∈ F by F is a n+1 n filter of L , thus y ∈ F . Therefore n+1 min{A(x → y), A(x)} = ≤ A(y). n+ 1 If x → y, x ∈ F , then y ∈ F for F is a filter of L . Therefore A(y) = 1 = 0 0 0 min{A(x → y), A(x)}. If x → y  F , x ∈ F , thus, there exists a positive integer k such that x → y ∈ 0 0 F \ F , hence y ∈ F for F is a filter of L . Therefore k+1 k k+1 k+1 min{A(x → y, A(x))} = ≤ A(y). k+ 1 If x → y ∈ F and x  F .As F = ∩ F , so there exits a positive integer k such 0 0 0 n n=1 that x ∈ F \F , hence y ∈ F . We get r+1 r r+1 min{A(x → y), A(x)} = ≤ A(y). r + 1 In a word, we hold that A is a fuzzy filter of L and A has infinite different val- ues. This contradict with Im(A) < ∞. Therefore L satisfies filter’s ascending chain condition. 4. Fuzzy Prime Filters Definition 3 A fuzzy filter A of L is said to be fuzzy prime if it is non-constant and A(x∨ y) = A(x)∨ A(y) for any x, y ∈ L. Example 1 Let L = {O, a, b, c, d, I}, the Hasse diagram of L and its operators can be found in [4, Example 2.1.4]. Then L = (L,∨,∧, ,→, O, I) is a lattice implication algebras. We define a fuzzy subset B of L as: A(I) = A(b) = A(c) = 0.6, A(0) = A(d) = A(a) = 0.2. It is routine to verify that A is a fuzzy prime filter of L . Lemma 2 Let A be a fuzzy filter of L . Then A is a constant fuzzy set if and only if A(I) = A(O). Proof Necessariness is obvious and we need to prove the sufficiency: Assume that A satisfies A(I) = A(O). Since A is a fuzzy filter, so A is order- preserving. For any x ∈ L, O ≤ x ≤ I, it follows that A(O) ≤ A(x) ≤ A(I). Hence A(I) = A(O) = A(x) for any x ∈ L. That is A is constant. Theorem 7 Let A be a non-constant fuzzy filter of L . Then the following are equiv- alent: (1) A is a fuzzy prime filter of L . (2) For all x, y ∈ L, if A(x∨ y) = A(I), then A(x) = A(I) or A(y) = A(I). (3) For all x, y ∈ L, A(x → y) = A(I) or A(y → x) = A(I). Fuzzy Inf. Eng. (2011) 3: 235-246 241 Proof (1) ⇒ (2): Assume that A is a fuzzy prime filter of L . Let x, y ∈ L and A(x ∨ y) = A(I). Then A(x) ∨ A(y) = A(x ∨ y) = A(I) and hence A(x) = A(I)or A(y) = A(I). (2) ⇒ (3): As (x → y) ∨ (y → x) = I for any x, y ∈ L. Then A((x → y) ∨ (y → x)) = A(I). From (2), we have that A(x → y) = A(I)or A(y → x) = A(I). So (3) holds. (3) ⇒ (1): Let x, y ∈ L. Suppose that, for instance, A(x → y) = A(I), we have y ≥ (x∧ y)∨ ((x → y)⊗ y) = ((x → y)⊗ x)∨ ((x → y)⊗ y) = (x → y)⊗ (x∨ y). Hence, A(y) ≥ A((x → y)⊗ (x∨ y)) ≥ A(x → y)∧ A(x∨ y) = min{A(1), A(x∨ y)} = A(x∨ y). Since x ∨ y ≥ y,so A(x ∨ y) ≥ A(y). Therefore, A(x ∨ y) = A(y). Similarly, A(x∨ y) = A(x). Thus, A(x∨ y) = A(x)∨ A(y). Analogously, A(y → x) = A(I) implies A(x∨ y) = A(x)∨ A(y). Therefore, A is a fuzzy prime filter of L . Let A be a fuzzy filter of L . We denote L as set {x ∈ L|A(x) = A(I)}. Theorem 8 Let A be a fuzzy filter of L . Then A is a fuzzy prime filter if and only if L is a prime filter of L . Proof Suppose that A is a fuzzy prime filter of L . Since A is non-constant, L is proper. Let x∨ y ∈ L for any x, y ∈ L. Then A(I) = A(x∨ y) = A(x)∨ A(y). Hence A(x) = A(I)or A(y) = A(I). This means that x ∈ L or y ∈ L . Therefore, L is A A A prime. Conversely, assume that L is a prime filter of L . Since L is proper, A is non- A A constant. As (x → y)∨ (y → x) = I for any x, y ∈ L. Then x → y ∈ L or y → x ∈ L . A A That is, A(x → y) = A(I)or A(y → x) = A(I). So A is a fuzzy prime filter of L . Let F be a subset of L andα,β ∈ [0, 1] such thatα>β. Now we define fuzzy set A by α, if x ∈ F, A (x) = β, otherwise. Particularly, A isχ on F atα = 1,β = 0. F F Lemma 3 Let F be a non-empty subset of L. Then A is a fuzzy filter of L if and only if F is a filter of L . Proof Assume that A is a fuzzy filter of L . For any x, y ∈ L,if x, x → y ∈ F, then A (x) = A (x → y) = α.So F F A (y) ≥ min{A (x), A (x → y)} = α. F F F 242 Yi Liu · Ke-yun Qin · Yang Xu (2011) Then y ∈ F. Since x ∈ F, then x → I ∈ F, hence, A (I) ≥ A (x) = α.So I ∈ F. F F Therefore F is a filter of L . Conversely, Let F be a filter of L and x, y ∈ L. Case I If x, y ∈ F, then x⊗ y ∈ F. Thus A (x⊗ y) = α = min{A (x), A (y)}. F F F Case II If x  F and y  F. Then A (x) = β = A (y). Thus A (x ⊗ y) ≥ β = F F F min{A (x), A (y)}. F F Case III If x  F and y ∈ F, then A (x) = β and A (y) = α. Hence A (x⊗ y) ≥ F F F min{A (x), A (y)}. F F From Case I to Case III, we arrive at A (x⊗y) ≥ min{A (x), A (y)} for any x, y ∈ L. F F F Let x, y ∈ L and x ≤ y. Case I If y ∈ F, then A (y) = α ≥ A (x). F F Case II If y  F, then x  F for F is a filter of L and x ≤ y. Thus A (x) = A (y) = β. Therefore, for any x, y ∈ L and x ≤ y,we have A (x) ≤ A (y). So A is a fuzzy F F F filter by Theorem 1. Theorem 9 Let F be a filter of L and A be a fuzzy set in L. Then F is a prime filter of L if and only if A is a fuzzy prime filter of L . Proof Assume that F is a prime filter of L . Since F is proper, A is non-constant. As (x → y) ∨ (y → x) = I ∈ F for any x, y ∈ L. Hence, x → y ∈ F or y → x ∈ F. Hence A (x → y) = α = A (I)or A (y → x) = α = A (I). Therefore A is a fuzzy F F F F F prime filter of L . Conversely, assume that A is a fuzzy filter of L. Then, by Lemma 3, F = L is a F A prime filter of L . Corollary 6 Let F be a filter of L . Then F is a prime filter of L if and only ifχ is a fuzzy prime filter of L . Theorem 10 L is a chain if and only if the fuzzy filter χ is a fuzzy prime filter of {I} L . Proof Assume that L is a chain and A be any fuzzy filter of L . Then x ≤ y or y ≤ x for any x, y ∈ L. Hence x → y = I or y → x = I. Thus A(x → y) = A(I) or A(y → x) = A(I). Therefore A is a fuzzy prime filter of L . Furthermore, every non-constant fuzzy filter A, such that A(I) = 1, is fuzzy prime. Therefore the fuzzy filterχ is a fuzzy prime filter of L . {I} Conversely, let any x, y ∈ L. Since χ is a fuzzy prime filter of L ,so χ (x → {I} {I} y) = χ (I) = 1or χ (y → x) = χ (I) = 1 by Theorem 7. Thus x → y ∈{I} or {I} {I} {I} y → x∈{I}, that is, x ≤ y or x ≤ y for any x, y ∈ L.So L is a chain. Theorem 11 Let A be a non-constant fuzzy filter of L such that for any fuzzy filter A , A of L ,A ∧ A ≤ A implies A ≤ Aor A ≤ A. Then A is a fuzzy prime filter of 1 2 1 2 1 2 L . Proof Assume that A is not a fuzzy prime filter of L . Then L is not a prime filter of L by Theorem 8. Let F , F be two filters such that L = F ∩ F . It follows that 1 2 A 1 2 Fuzzy Inf. Eng. (2011) 3: 235-246 243 L ⊂ F and L ⊂ F (otherwise, if L = F or L = F , then L is irreducible, that A 1 A 2 A 1 A 2 A is, L is prime. Contradiction.). Then there exist x , x ∈ L such that x ∈ F but A 1 2 1 1 x  L , x ∈ F but x  L . Therefore A(x ) < A(I) and A(x ) < A(I). We define 1 A 2 2 2 A 1 2 fuzzy set A and A as follows: 1 2 A(I), if x ∈ F , ⎨ 1 A (x) = 1 ⎪ 0, if x  F . A(I), if x ∈ F , ⎨ 2 A (x) = 0, if x  F . Now, we need to prove A (i = 1, 2) are fuzzy filters of L , but we only to prove A . i 1 Obviously, A (I) ≥ A(x) for any x ∈ L. Let x, y ∈ L: (a) If x ∈ F and x → y ∈ F , then y ∈ F for F is a filter. By the definition 1 1 1 1 of A , it follows that A (x) = A(I) = A (x → y) and A (y) = A(I). Therefore 1 1 1 I A ≥ min{A (x), A (x → y)}. y 1 1 (b) If x  F or x → y  F , it is easy to verify A (y) ≥ min{A (x), A (x → y)}. 1 1 1 1 1 Therefore, A is a fuzzy filter by (a)(b). Similarly, we can prove A a fuzzy filter 1 2 of L . For any x ∈ L, it is easy to verify that (A ∧ A )(x) ≤ A(x), namely, A ∧ A ≤ A. 1 2 1 2 By the hypothesis, A ≤ A or A ≤ A. But, since A (x ) = A(I) > A(x) and A (x ) = 1 2 1 1 2 2 A(I) > A(x), it follows that A > A and A > A. Contradiction. Thus, A is a fuzzy 1 2 filter of L . Theorem 12 Let L be a lattice implication algebra and A is non-constant fuzzy set of L , f : L → L is an onto lattice implication homomorphism. Define a fuzzy set A (x) = A( f (x)). Then A is a fuzzy prime filter of L if and only if A is a fuzzy prime filter of L . Proof Assume that A is non-constant fuzzy prime filter of L , f is an onto lattice implication homomorphism from L to L . Then for any x, y ∈ L, A(x) ≤ A(I) and f (x) → f (I) = f (x → I) = f (I) = I, that is, f (x) ≤ f (I). Then A (x) = A( f (x)) ≤ A( f (I)) = A (I), and f f min{A (x → y), A (x)}= min{A( f (x → y)), A( f (x))} f f = min{A( f (x) → f (y)), A( f (x))} ≤ A( f (y)) = A (y). Therefore, A is a fuzzy filter of L . Moreover, as A is a fuzzy prime filters of L and f is a lattice implication homomorphism, we get A (x)∨ A (y)= A( f (x))∨ A( f (y)) f f = A( f (x)∨ f (y)) = A( f (x∨ y)) = A (x∨ y). f 244 Yi Liu · Ke-yun Qin · Yang Xu (2011) Hence, A is a fuzzy prime filter of L . Conversely, assume that A is a fuzzy prime filter of L . Since f is an onto lattice implication homomorphism, for any x, y ∈ L, there exist a, b ∈ L such that f (a) = x and f (b) = y. Thus, A(x) = A( f (a)) = A (a) ≤ A (I) = A( f (I)) = A(I) f f and min{A(x), A(x → y)} = min{A( f (a)), A( f (a) → f (b))} = min{A( f (a)), A( f (a → b))} = min{A (a), A (a → b)} f f ≤ A (b) = A( f (b)) = A(y). So A is a fuzzy filter of L . Furthermore, A(x∨ y) = A( f (a)∨ f (b)) = A( f (a∨ b)) = A (a∨ b) = A (a)∨ A (b) f f = A( f (a))∨ A( f (b)) = A(x)∨ A(y). Therefore, A is a fuzzy prime filter of L . Theorem 13 Let L be a lattice implication algebra and A be fuzzy set of L . Define a mapping A : L → R as A (x) = A(x)+ 1− A(I), for any x ∈ L. Then A is fuzzy prime filter of L if and only if A is a fuzzy prime filter of L . Proof Suppose A is a fuzzy prime filter of L , then A(x) ≤ A(I) for any x ∈ L. Then A is a fuzzy set of L . Furthermore, for any x, y ∈ L, ∗ ∗ A (I) = A(I)+ 1− A(I) = 1 ≥ A (x) and ∗ ∗ min{A (x), A (x → y)} = min{A(x)+ 1− A(I), A(x → y)+ 1− A(I)} = min{A(x), A(x → y)}+ 1− A(I) ≤ A(y)+ 1− A(I) = A (y). ∗ ∗ Therefore, A is a fuzzy filter of L . Now, we prove A is prime. Since A is prime, it follows that A(x∨ y) = A(x)∨ A(y) Fuzzy Inf. Eng. (2011) 3: 235-246 245 and A(x∨ y)+ 1− A(I) = (A(x)∨ A(y))+ 1− A(I) which implies A(x∨ y)+ 1− A(I) = (A(x)+ 1− A(I))∨ (A(y)+ 1− A(I)). ∗ ∗ ∗ ∗ Hence A (x∨ y) = A (x)∨ A (y) for any x, y ∈ L, and so A is a fuzzy prime filter of L . ∗ ∗ ∗ Conversely, suppose A is a fuzzy prime filter, then A (x) ≤ A (I), that is, A(x) + 1− A(I) ≤ A(I)+ 1− A(I), it follows that A(x) ≤ A(I). ∗ ∗ ∗ Since min{A (x), A (x → y)}≤ A (y), so min{A(x), A(x → y)}≤ A(y). ∗ ∗ ∗ ∗ As A is prime, it follows that A (x ∨ y) = A (x) ∨ A (y), we have A(x ∨ y) = A(x)∨ A(y). Therefore, A is a fuzzy prime filter of L . 5. Conclusion Aiming to explore the many-valued logical system whose propositional value is given in a lattice, Xu [1] took the initiative in the study of the concept of lattice implication algebras. Hence, in order to promote the development of the many-valued logical system, it is of overwhelming necessity to make clear the structure of lattice impli- cation algebras. When an algebraic system is studied, the research of its structure is found to be the investigation of the structure of filters. In this paper, some equivalent characterizations of fuzzy filters of lattice implication algebras are revealed. Conse- quently, the structure of fuzzy filters are further studied. Finally, the notion of fuzzy prime filters is proposed and some equivalent conditions of fuzzy prime filters are ob- tained. We desperately hope that our work would serve as a foundation for enriching corresponding many-valued logical system. Acknowledgments The work is partly supported by the National Natural Science Foundation of China (No.60875034) and the Scientific Research Project of Department of Education of Sichuan Province (No.08zb082,09zb105). References 1. Xu Y (1993) Lattice implication algebras. Journal of Southwest Jiaotong University 1: 20-27 2. Xu Y, Qin K Y (1995) Fuzzy lattice implication algebras. Journal of Southwest Jiaotong University 2: 121-27 3. Xu Y, Qin K Y (1993) On filters of lattice implication algebras. The Journal of Fuzzy Mathematics 2: 251-260 4. Xu Y, Ruan D, Qin K Y, Liu J (2003) Lattice-valued logic-an alternative approach to treat fuzziness and incomparability. Springer, New York 5. Xu Y, Qin K Y (1992) Lattice H implication algebras and lattice implication algebra classes. Journal of Hebei Mining and Civil Engineering Institute 3: 139-143 6. Xu Y, Qin K Y (1993) On filters of lattice implication algebras. The Journal of Fuzzy Mathematics 1(2): 251-260 7. Xu Y, Qin K Y (1995) Fuzzy lattice implication algebras. Journal of Southwest Jiaotong University 30(2): 121-127 8. Ma J, Li W J, Ruan D, Xu Y (2007) Filter-based resolution principle for lattice-valued propositional logic LP(x). Information Science 177: 1046-1062 246 Yi Liu · Ke-yun Qin · Yang Xu (2011) 9. Jun Y B, Xu Y, Ma J (2007) Redefined fuzzy implication filters. Information Science 177: 1422-1429 10. Zhan J M, Jun Y B (2009) Notes on redefined fuzzy implication filters of lattice implication algebras. Information Science 179: 3182-3186 11. Jun Y B (2001) Fuzzy positive implicative and fuzzy associative filters of lattice implication algebras. Fuzzy Sets and Systems 121: 353-357 12. Rom E H, Kim S Y, Xu Y (2001) Some operations on lattice implication algebras. International Journal of Mathematics and Mathematical Science 1: 45-52 13. Jun Y B (1997) Implicative filters of lattice implication algebras. Bull. Korean Math. Soc. 34(2): 193-198 14. Jun Y B (2000) Fantastic filters of lattice implication algebras. Internat. J. Math. and Math. Sci. 24(4): 277-281 15. Jun Y B (2001) On n-fold implicative filters of lattice implication algebras. Internat. J. Math. and Math. Sci. 26(11): 695-699 16. Jun Y B (2001) Fuzzy positive implicative and fuzzy associative filters of lattice implication algebras. Fuzzy Sets and Systems 121: 353-357 17. Jun Y B, Song S Z (2002) On fuzzy implicative filters of lattice implication algebras. The Journal of Fuzzy Mathematics. 10(4): 893-900 18. Jun Y B, Xu Y, Qin K Y (1998) Positive implicative and associative filters of lattice implication algebras. Bull. Korean Math. Soc. 35(1): 53-61 19. Kim S Y, Roh E H, Jun Y B (1999) On ultra filters of lattice implication algebras. Scientiae Mathe- maticae 2(2): 201-204 20. Liu J, Xu Y (1996) On certain filters in lattice implication algebras. Chinese Quarterly J. Math. 11(4): 106-111 21. Liu J, Xu Y (1997) Filters and structure of lattice implication algebras. Chinese Science Bulletin 42(18): 1517-1520 22. Jun Y B (2008) Fuzzy prime ideals of pseudo-MV algebras. Soft Comput, 12: 365-372 23. Kondo M, Dudek W A (2008) Filter theory of BL-algebras. Soft Comput. 12: 419-423 24. Xu W T, Xu Y (2008) Structure of fuzzy filters in lattice implication algebras. Chinese Quar. J. Math. 6: 512-518 25. Xu W T, Zou L et al (2007) Some properties of fuzzy filters in lattice implication algebras. Proceeding of 2007 International Conference on Intelligent System and Knowledge Engineering: 400-403 26. Jun Y B, Xu Y, Zhang X H (2005) Fuzzy filters of MTL-algebras. Information Science 175: 120-138 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Fuzzy Information and Engineering Taylor & Francis

Fuzzy Prime Filters of Lattice Implication Algebras

Fuzzy Information and Engineering , Volume 3 (3): 12 – Sep 1, 2011

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Fuzzy Inf. Eng. (2011) 3: 235-246 DOI 10.1007/s12543-011-0080-y ORIGINAL ARTICLE Yi Liu · Ke-yun Qin · Yang Xu Received: 16 May 2010/ Revised: 18 July 2011/ Accepted: 16 August 2011/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract In this paper, some properties of fuzzy filters are given. Besides, the structure of fuzzy filters are further studied. And finally, the concept of fuzzy prime filter is proposed with some equivalent conditions of fuzzy prime filters obtained. Keywords Lattice implication algebras · Fuzzy filters · Fuzzy prime filters 1. Introduction As is known to all, one significant function of artificial intelligence is to make com- puter simulate human being in dealing with uncertain information. And logic estab- lishes the foundations for it. However, certain information process is based on the classic logic. Non-classical logics consist of these logics handling a wide variety of uncertainties (such as fuzziness, randomness, and so on ) and fuzzy reasoning. Therefore, non-classical logic has been proved to be a formal and useful technique for computer science to deal with fuzzy and uncertain information. Many-valued logic, as the extension and development of classical logic, has always been a cru- cial direction in non-classical logic. Lattice-valued logic, an important many-valued logic, has two prominent roles: One is to extend the chain-type truth-valued field of the current logics to some relatively general lattices. The other is that the incom- pletely comparable property of truth value characterized by the general lattice can more effectively reflect the uncertainty of human being’s thinking, judging and de- cision. Hence, lattice-valued logic has been becoming a research field and strongly Yi Liu () Intelligent Control Development Center, Southwest Jiaotong University, Chengdu, Sichuan 610031, P.R.China College of Mathematics and Information Science, Neijiang Normal University, Neijiang , Sichuan 641000, P.R.China email: liuyiyl@126.com Ke-yun Qin, Yang Xu Intelligent Control Development Center, Southwest Jiaotong University, Chengdu, Sichuan 610031, P.R.China 236 Yi Liu · Ke-yun Qin · Yang Xu (2011) influencing the development of algebraic logic, computer science and artificial in- telligent technology. In order to investigate a many-valued logical system whose propositional value is given in a lattice, in 1993, Xu first established the lattice im- plication algebra by combining lattice and implication algebra, and explored many useful structures [1-4]. The filter theory serves a vital function for the development of lattice implication algebras. It is also important to the automated reasoning and approximated reasoning based on lattice implication algebra, for instance, filter-based resolution principle proposed by J. Ma et al [8]. Xu and Qin [6] introduced the concepts of filter and implicative filter in lattice implication algebras, and examined their properties [21, 22]. Jun [14] and other scholars studied several filters in lattice implication algebras [14-16, 19-22]. In particular, Jun [14] gave an equivalent condition of a filter, and provided some equivalent conditions for a filter to be an implicative filter in lattice implication algebras. The concept of fuzzy set was introduced by Zadeh (1965). Since then this idea has been applied to other algebraic structures such as groups, semigroups, rings, modules, vector spaces and topologies. Xu and Qin [2] applied the concept of fuzzy set to lat- tice implication algebras, proposed the notions of fuzzy filters and fuzzy implicative filters [3]. Later on, other scholars put forward relative fuzzy filters like fuzzy (pos- itive) implication filter, fuzzy fantastic filter and discovered some properties [9-12, 17, 18, 24, 25]. This logical algebra has captured close attention from a good many researchers. And elegant results are obtained, which are collected in the monograph [4]. Let (L,∨,∧, O, I) be a bounded lattice with an order-reversing involution , the greatest element I and the smallest element O, and →: L× L −→ L be a mapping. L = (L,∨,∧, ,→, O, I) is called a lattice implication algebra if the following conditions hold for any x, y, z ∈ L: (I ) x → (y → z) = y → (x → z); (I ) x → x = I; (I ) x → y = y → x ; (I ) x → y = y → x = I implies x = y; (I )(x → y) → y = (y → x) → x; (l )(x∨ y) → z = (x → z)∧ (y → z); (l )(x∧ y) → z = (x → z)∨ (y → z). In this paper, we denote L as a lattice implication algebra (L,∨,∧, ,→, O, I). We list some basic properties of lattice implication algebras. It is useful to develop these topics in other sections. Let L be a lattice implication algebra. Then for any x, y, z ∈ L, the following conclusions hold: (1) If I → x = I, then x = I. (2) I → x = x and x → O = x . (3) O → x = I and x → I = I. (4) (x → y) → ((y → z) → (x → z)) = I. Fuzzy Inf. Eng. (2011) 3: 235-246 237 (5) (x → y)∨ (y → x) = I. (6) If x ≤ y, then x → z ≥ y → z and z → x ≤ z → y. (7) x ≤ y if and only if x → y = I. (8) (z → x) → (z → y) = (x∧ z) → y = (x → z) → (x → y). (9) x → (y∨ z) = (y → z) → (x → z). (10) x → (y → z) = (x∨ y) → z if and only if x → (y → z) = x → z = y → z; (11) z ≤ y → x if and only if y ≤ z → x. Let f be a mapping from lattice implication algebra L to L .If 1 2 f (x → y) = f (x) → f (y); f (x∨ y) = f (x)∨ f (y); f (x∧ y) = f (x)∧ f (y); f (x ) = ( f (x)) . Then f is called a lattice implication homomorphism from L to L . 1 2 From the Definition of lattice implication homomorphism, we can easily obtain the following result: If f is a lattice implication homomorphism from L to L , then 1 2 f (I) = I. In lattice implication algebras, we define the binary operation⊗ as follows, for any x, y ∈ L, x⊗ y = (x → y ) . Some properties of operation⊗ can be found in reference [4]. In the paper, the writers carry out in-depth research on the fuzzy filters of the lattice implication algebra. In Section 2, diverse equivalence characterizations of fuzzy filters are carefully displayed. The demonstration of structure of fuzzy filters can be found in section three. In the final part, the notion of fuzzy prime filter and the properties with some equivalence conditions are strongly stated. 2. Some Properties of Fuzzy Filters Definition 1 [3] Let F be said to be a filter of L . If it satisfies: (F ) I ∈ F. (F ) For any x, y ∈ L, if x ∈ F and x → y ∈ F, then y ∈ F. Lemma 1 [4] Let F be a subset of L. Then F is a filter of L , if and only if : (1) I ∈ F. (2) If x ∈ F and x ≤ y, then y ∈ F. (3) For any x, y ∈ F, then x⊗ y ∈ F. Definition 2 [2] Let A be a fuzzy subset of L. A is said to be a fuzzy filter of L ,ifit satisfies, for any x, y ∈ L (1) A(x) ≤ A(I); (2) A(y) ≥ min{A(x → y), A(x)}. Theorem 1 [25] Let A be a fuzzy set in L. Then A is a fuzzy filter of L if and only if, for any x, y ∈ L (1) A(x⊗ y) ≥ min{A(x), A(y)}; (2) x ≤ y implies A(x) ≤ A(y). 238 Yi Liu · Ke-yun Qin · Yang Xu (2011) Corollary 1 Let A be a fuzzy filter of L . The the following hold for any x, y ∈ L: (1) A(x∨ y) ≥ min{A(x), A(y)}, (2) A(x∧ y) = min{A(x), A(y)}, (3) A(x⊗ y) = min{A(x), A(y)}. Corollary 2 Let A be a fuzzy subset of L. Then A is a fuzzy filter of L if and only if A(x⊗ y) ≥ min{A(x), A(y)} and A(x∨ y) ≥ A(x). Theorem 2 [25] Let A be a fuzzy subset of L. Then A is a fuzzy filter if and only if , for any x, y, z ∈ L, x → (y → z) = I implies min{A(x), A(y)}≤ A(z). Since x ≤ y → z if and only if x ⊗ y ≤ z. So we can easily obtain the following corollaries. Corollary 3 A fuzzy set A in L is a fuzzy filter of L if and only if A satisfies the following condition: A(x) ≥ min{A(a ),··· , A(a )}, 1 n where a → (···→ (a → x)··· ) = I for a ,··· , a ∈ L. n 1 1 n The meet of two fuzzy filters A , A of L is define as 1 2 A ∧ A = A ∩ A . 1 2 1 2 Corollary 4 Let A be fuzzy filters, i = 1, 2. Then A ∧ A is also a fuzzy filter of L . i 1 2 Proof Let x, y, z ∈ L be such that x → (y → z) = I. Since A are fuzzy filters, i = 1, 2, we have A (z) ≥ min{A (x), A (y)} for i = 1, 2. Hence i i i (A ∧ A )(z) = A (z)∧ A (z) 1 2 1 2 ≥ A (x)∧ A (y)∧ A (x)∧ A (y) 1 1 2 2 = min{(A ∧ A )(x), (A ∧ A )(y)}. 1 1 2 2 Therefore, we have A ∧ A is a fuzzy filter of L from Theorem 2. 1 2 Corollary 5 Let A be a family fuzzy filters of L for i ∈ I, where I is an index set. The ∧ A is a fuzzy filter of L . i∈I i Theorem 3 Let A be a fuzzy filter of L . Put A[a] = {x ∈ L|A(a) ≤ A(x)}, then A[a] is a filter of L . Proof It is easy to obtain that I ∈ A[a]. Suppose x, y ∈ L and x → y ∈ A[a], x ∈ A[a]. Then A(a) ≤ A(x → y) and A(a) ≤ A(x). Hence, min{A(x → y), A(x)}≥ A(a). Since A is a fuzzy filter, so A(y) ≥ min{A(x → y), A(x)}≥ A(a), that is, y ∈ A[a]. Therefore A[a] is a filter of L . Fuzzy Inf. Eng. (2011) 3: 235-246 239 Theorem 4 Let A be a fuzzy set of L . Then (1) If A[a] is a filter of L for any a ∈ L, then A(z) ≤ min{A(x → y), A(x))} implies A(z) ≤ A(y). (1) (2) If A satisfies A(x) ≤ A(I) for any x ∈ L and (1), then A[a] is a filter of L . Proof (1) Assume that A[a] is a filter of L for any a ∈ L. Let x, y, z ∈ L such that A(z) ≤ min{A(x → y), A(x))}, then x → y ∈ A[z] and x ∈ A[z]. As A[z] is a filter, hence y ∈ A[z], that is A(z) ≤ A(y). (2) Suppose A to satisfy A(x) ≤ A(I) for any x ∈ L and (2), for any a ∈ L, and x, y ∈ L such that x → y ∈ A[a] and x ∈ A[a], we have A[a] ≤ min{A(x → y), A(x)}. Hence A(a) ≤ A(y) by (1). That is y ∈ A[a]. As A satisfies A(x) ≤ A(I) for any x ∈ L,so A(a) ≤ A(I), that is, I ∈ A[a]. Therefore A[a] is a filter of L . 3. Structure of Fuzzy Filters In this section, we investigate the structure of fuzzy filters of a lattice implication algebra. Theorem 5 Let A be a fuzzy filter of L and L satisfies the filter’s ascending chain condition of L . If a series of elements in Im(A) is strictly increasing, then |Im(A)| < ∞. Proof Assume that |Im(A)| = ∞ and {a } is a strictly increasing series in Im(A), then 1 > ··· > a > a > 0. Put A = {x ∈ L|A(x) ≥ a }, k = 2, 3,··· . Since A is a 2 1 k k fuzzy filter of L ,so A is a filter of L . Let x ∈ A . Then A(x) ≥ a > a . Thus we k k k k−1 have x ∈ A , it follows that A ⊆ A . k−1 k k−1 As a ∈ Im(A), thus there exists an x ∈ L such that A(x ) = a . Hence k−1 k−1 k−1 k−1 x ∈ A . But x  A , then A ⊂ A . Since Im(A) = ∞, so we can obtain k−1 k−1 k−1 k k k−1 an unterminate strictly increasing filter chain: A ⊂ A ⊂ ··· ⊂ A ⊂ A ⊂··· . 1 2 k−1 k Contradict with the ascending condition. Therefore |Im(A)| < ∞. Theorem 6 Let A be a fuzzy filter ofL and A contains finite image, that is,|Im(A)| < ∞. Then L satisfies the filter’s ascending chain condition. Proof Suppose that L dissatisfies a filter ascending condition, so there exists a strictly increasing chain: ∅  F ⊂ F ⊂···⊂··· . 0 1 We define a fuzzy set on L as follows: , if x ∈ F \ F , n = 0, 1, 2,···, ⎨ n+1 n A(x) = n+ 1 1, if x ∈ F . Now, we need to prove A is a fuzzy filter of L . Obviously, we obtain that A(I) ≥ A(x) for any x ∈ L. 240 Yi Liu · Ke-yun Qin · Yang Xu (2011) Let x, y ∈ L,if x → y ∈ F \ F and x ∈ F \ F , where n = 0, 1, 2,··· , k = n+1 n k+1 k 0, 1, 2,··· . Without loss of generality, assume that n ≥ k. Then x ∈ F by F is a n+1 n filter of L , thus y ∈ F . Therefore n+1 min{A(x → y), A(x)} = ≤ A(y). n+ 1 If x → y, x ∈ F , then y ∈ F for F is a filter of L . Therefore A(y) = 1 = 0 0 0 min{A(x → y), A(x)}. If x → y  F , x ∈ F , thus, there exists a positive integer k such that x → y ∈ 0 0 F \ F , hence y ∈ F for F is a filter of L . Therefore k+1 k k+1 k+1 min{A(x → y, A(x))} = ≤ A(y). k+ 1 If x → y ∈ F and x  F .As F = ∩ F , so there exits a positive integer k such 0 0 0 n n=1 that x ∈ F \F , hence y ∈ F . We get r+1 r r+1 min{A(x → y), A(x)} = ≤ A(y). r + 1 In a word, we hold that A is a fuzzy filter of L and A has infinite different val- ues. This contradict with Im(A) < ∞. Therefore L satisfies filter’s ascending chain condition. 4. Fuzzy Prime Filters Definition 3 A fuzzy filter A of L is said to be fuzzy prime if it is non-constant and A(x∨ y) = A(x)∨ A(y) for any x, y ∈ L. Example 1 Let L = {O, a, b, c, d, I}, the Hasse diagram of L and its operators can be found in [4, Example 2.1.4]. Then L = (L,∨,∧, ,→, O, I) is a lattice implication algebras. We define a fuzzy subset B of L as: A(I) = A(b) = A(c) = 0.6, A(0) = A(d) = A(a) = 0.2. It is routine to verify that A is a fuzzy prime filter of L . Lemma 2 Let A be a fuzzy filter of L . Then A is a constant fuzzy set if and only if A(I) = A(O). Proof Necessariness is obvious and we need to prove the sufficiency: Assume that A satisfies A(I) = A(O). Since A is a fuzzy filter, so A is order- preserving. For any x ∈ L, O ≤ x ≤ I, it follows that A(O) ≤ A(x) ≤ A(I). Hence A(I) = A(O) = A(x) for any x ∈ L. That is A is constant. Theorem 7 Let A be a non-constant fuzzy filter of L . Then the following are equiv- alent: (1) A is a fuzzy prime filter of L . (2) For all x, y ∈ L, if A(x∨ y) = A(I), then A(x) = A(I) or A(y) = A(I). (3) For all x, y ∈ L, A(x → y) = A(I) or A(y → x) = A(I). Fuzzy Inf. Eng. (2011) 3: 235-246 241 Proof (1) ⇒ (2): Assume that A is a fuzzy prime filter of L . Let x, y ∈ L and A(x ∨ y) = A(I). Then A(x) ∨ A(y) = A(x ∨ y) = A(I) and hence A(x) = A(I)or A(y) = A(I). (2) ⇒ (3): As (x → y) ∨ (y → x) = I for any x, y ∈ L. Then A((x → y) ∨ (y → x)) = A(I). From (2), we have that A(x → y) = A(I)or A(y → x) = A(I). So (3) holds. (3) ⇒ (1): Let x, y ∈ L. Suppose that, for instance, A(x → y) = A(I), we have y ≥ (x∧ y)∨ ((x → y)⊗ y) = ((x → y)⊗ x)∨ ((x → y)⊗ y) = (x → y)⊗ (x∨ y). Hence, A(y) ≥ A((x → y)⊗ (x∨ y)) ≥ A(x → y)∧ A(x∨ y) = min{A(1), A(x∨ y)} = A(x∨ y). Since x ∨ y ≥ y,so A(x ∨ y) ≥ A(y). Therefore, A(x ∨ y) = A(y). Similarly, A(x∨ y) = A(x). Thus, A(x∨ y) = A(x)∨ A(y). Analogously, A(y → x) = A(I) implies A(x∨ y) = A(x)∨ A(y). Therefore, A is a fuzzy prime filter of L . Let A be a fuzzy filter of L . We denote L as set {x ∈ L|A(x) = A(I)}. Theorem 8 Let A be a fuzzy filter of L . Then A is a fuzzy prime filter if and only if L is a prime filter of L . Proof Suppose that A is a fuzzy prime filter of L . Since A is non-constant, L is proper. Let x∨ y ∈ L for any x, y ∈ L. Then A(I) = A(x∨ y) = A(x)∨ A(y). Hence A(x) = A(I)or A(y) = A(I). This means that x ∈ L or y ∈ L . Therefore, L is A A A prime. Conversely, assume that L is a prime filter of L . Since L is proper, A is non- A A constant. As (x → y)∨ (y → x) = I for any x, y ∈ L. Then x → y ∈ L or y → x ∈ L . A A That is, A(x → y) = A(I)or A(y → x) = A(I). So A is a fuzzy prime filter of L . Let F be a subset of L andα,β ∈ [0, 1] such thatα>β. Now we define fuzzy set A by α, if x ∈ F, A (x) = β, otherwise. Particularly, A isχ on F atα = 1,β = 0. F F Lemma 3 Let F be a non-empty subset of L. Then A is a fuzzy filter of L if and only if F is a filter of L . Proof Assume that A is a fuzzy filter of L . For any x, y ∈ L,if x, x → y ∈ F, then A (x) = A (x → y) = α.So F F A (y) ≥ min{A (x), A (x → y)} = α. F F F 242 Yi Liu · Ke-yun Qin · Yang Xu (2011) Then y ∈ F. Since x ∈ F, then x → I ∈ F, hence, A (I) ≥ A (x) = α.So I ∈ F. F F Therefore F is a filter of L . Conversely, Let F be a filter of L and x, y ∈ L. Case I If x, y ∈ F, then x⊗ y ∈ F. Thus A (x⊗ y) = α = min{A (x), A (y)}. F F F Case II If x  F and y  F. Then A (x) = β = A (y). Thus A (x ⊗ y) ≥ β = F F F min{A (x), A (y)}. F F Case III If x  F and y ∈ F, then A (x) = β and A (y) = α. Hence A (x⊗ y) ≥ F F F min{A (x), A (y)}. F F From Case I to Case III, we arrive at A (x⊗y) ≥ min{A (x), A (y)} for any x, y ∈ L. F F F Let x, y ∈ L and x ≤ y. Case I If y ∈ F, then A (y) = α ≥ A (x). F F Case II If y  F, then x  F for F is a filter of L and x ≤ y. Thus A (x) = A (y) = β. Therefore, for any x, y ∈ L and x ≤ y,we have A (x) ≤ A (y). So A is a fuzzy F F F filter by Theorem 1. Theorem 9 Let F be a filter of L and A be a fuzzy set in L. Then F is a prime filter of L if and only if A is a fuzzy prime filter of L . Proof Assume that F is a prime filter of L . Since F is proper, A is non-constant. As (x → y) ∨ (y → x) = I ∈ F for any x, y ∈ L. Hence, x → y ∈ F or y → x ∈ F. Hence A (x → y) = α = A (I)or A (y → x) = α = A (I). Therefore A is a fuzzy F F F F F prime filter of L . Conversely, assume that A is a fuzzy filter of L. Then, by Lemma 3, F = L is a F A prime filter of L . Corollary 6 Let F be a filter of L . Then F is a prime filter of L if and only ifχ is a fuzzy prime filter of L . Theorem 10 L is a chain if and only if the fuzzy filter χ is a fuzzy prime filter of {I} L . Proof Assume that L is a chain and A be any fuzzy filter of L . Then x ≤ y or y ≤ x for any x, y ∈ L. Hence x → y = I or y → x = I. Thus A(x → y) = A(I) or A(y → x) = A(I). Therefore A is a fuzzy prime filter of L . Furthermore, every non-constant fuzzy filter A, such that A(I) = 1, is fuzzy prime. Therefore the fuzzy filterχ is a fuzzy prime filter of L . {I} Conversely, let any x, y ∈ L. Since χ is a fuzzy prime filter of L ,so χ (x → {I} {I} y) = χ (I) = 1or χ (y → x) = χ (I) = 1 by Theorem 7. Thus x → y ∈{I} or {I} {I} {I} y → x∈{I}, that is, x ≤ y or x ≤ y for any x, y ∈ L.So L is a chain. Theorem 11 Let A be a non-constant fuzzy filter of L such that for any fuzzy filter A , A of L ,A ∧ A ≤ A implies A ≤ Aor A ≤ A. Then A is a fuzzy prime filter of 1 2 1 2 1 2 L . Proof Assume that A is not a fuzzy prime filter of L . Then L is not a prime filter of L by Theorem 8. Let F , F be two filters such that L = F ∩ F . It follows that 1 2 A 1 2 Fuzzy Inf. Eng. (2011) 3: 235-246 243 L ⊂ F and L ⊂ F (otherwise, if L = F or L = F , then L is irreducible, that A 1 A 2 A 1 A 2 A is, L is prime. Contradiction.). Then there exist x , x ∈ L such that x ∈ F but A 1 2 1 1 x  L , x ∈ F but x  L . Therefore A(x ) < A(I) and A(x ) < A(I). We define 1 A 2 2 2 A 1 2 fuzzy set A and A as follows: 1 2 A(I), if x ∈ F , ⎨ 1 A (x) = 1 ⎪ 0, if x  F . A(I), if x ∈ F , ⎨ 2 A (x) = 0, if x  F . Now, we need to prove A (i = 1, 2) are fuzzy filters of L , but we only to prove A . i 1 Obviously, A (I) ≥ A(x) for any x ∈ L. Let x, y ∈ L: (a) If x ∈ F and x → y ∈ F , then y ∈ F for F is a filter. By the definition 1 1 1 1 of A , it follows that A (x) = A(I) = A (x → y) and A (y) = A(I). Therefore 1 1 1 I A ≥ min{A (x), A (x → y)}. y 1 1 (b) If x  F or x → y  F , it is easy to verify A (y) ≥ min{A (x), A (x → y)}. 1 1 1 1 1 Therefore, A is a fuzzy filter by (a)(b). Similarly, we can prove A a fuzzy filter 1 2 of L . For any x ∈ L, it is easy to verify that (A ∧ A )(x) ≤ A(x), namely, A ∧ A ≤ A. 1 2 1 2 By the hypothesis, A ≤ A or A ≤ A. But, since A (x ) = A(I) > A(x) and A (x ) = 1 2 1 1 2 2 A(I) > A(x), it follows that A > A and A > A. Contradiction. Thus, A is a fuzzy 1 2 filter of L . Theorem 12 Let L be a lattice implication algebra and A is non-constant fuzzy set of L , f : L → L is an onto lattice implication homomorphism. Define a fuzzy set A (x) = A( f (x)). Then A is a fuzzy prime filter of L if and only if A is a fuzzy prime filter of L . Proof Assume that A is non-constant fuzzy prime filter of L , f is an onto lattice implication homomorphism from L to L . Then for any x, y ∈ L, A(x) ≤ A(I) and f (x) → f (I) = f (x → I) = f (I) = I, that is, f (x) ≤ f (I). Then A (x) = A( f (x)) ≤ A( f (I)) = A (I), and f f min{A (x → y), A (x)}= min{A( f (x → y)), A( f (x))} f f = min{A( f (x) → f (y)), A( f (x))} ≤ A( f (y)) = A (y). Therefore, A is a fuzzy filter of L . Moreover, as A is a fuzzy prime filters of L and f is a lattice implication homomorphism, we get A (x)∨ A (y)= A( f (x))∨ A( f (y)) f f = A( f (x)∨ f (y)) = A( f (x∨ y)) = A (x∨ y). f 244 Yi Liu · Ke-yun Qin · Yang Xu (2011) Hence, A is a fuzzy prime filter of L . Conversely, assume that A is a fuzzy prime filter of L . Since f is an onto lattice implication homomorphism, for any x, y ∈ L, there exist a, b ∈ L such that f (a) = x and f (b) = y. Thus, A(x) = A( f (a)) = A (a) ≤ A (I) = A( f (I)) = A(I) f f and min{A(x), A(x → y)} = min{A( f (a)), A( f (a) → f (b))} = min{A( f (a)), A( f (a → b))} = min{A (a), A (a → b)} f f ≤ A (b) = A( f (b)) = A(y). So A is a fuzzy filter of L . Furthermore, A(x∨ y) = A( f (a)∨ f (b)) = A( f (a∨ b)) = A (a∨ b) = A (a)∨ A (b) f f = A( f (a))∨ A( f (b)) = A(x)∨ A(y). Therefore, A is a fuzzy prime filter of L . Theorem 13 Let L be a lattice implication algebra and A be fuzzy set of L . Define a mapping A : L → R as A (x) = A(x)+ 1− A(I), for any x ∈ L. Then A is fuzzy prime filter of L if and only if A is a fuzzy prime filter of L . Proof Suppose A is a fuzzy prime filter of L , then A(x) ≤ A(I) for any x ∈ L. Then A is a fuzzy set of L . Furthermore, for any x, y ∈ L, ∗ ∗ A (I) = A(I)+ 1− A(I) = 1 ≥ A (x) and ∗ ∗ min{A (x), A (x → y)} = min{A(x)+ 1− A(I), A(x → y)+ 1− A(I)} = min{A(x), A(x → y)}+ 1− A(I) ≤ A(y)+ 1− A(I) = A (y). ∗ ∗ Therefore, A is a fuzzy filter of L . Now, we prove A is prime. Since A is prime, it follows that A(x∨ y) = A(x)∨ A(y) Fuzzy Inf. Eng. (2011) 3: 235-246 245 and A(x∨ y)+ 1− A(I) = (A(x)∨ A(y))+ 1− A(I) which implies A(x∨ y)+ 1− A(I) = (A(x)+ 1− A(I))∨ (A(y)+ 1− A(I)). ∗ ∗ ∗ ∗ Hence A (x∨ y) = A (x)∨ A (y) for any x, y ∈ L, and so A is a fuzzy prime filter of L . ∗ ∗ ∗ Conversely, suppose A is a fuzzy prime filter, then A (x) ≤ A (I), that is, A(x) + 1− A(I) ≤ A(I)+ 1− A(I), it follows that A(x) ≤ A(I). ∗ ∗ ∗ Since min{A (x), A (x → y)}≤ A (y), so min{A(x), A(x → y)}≤ A(y). ∗ ∗ ∗ ∗ As A is prime, it follows that A (x ∨ y) = A (x) ∨ A (y), we have A(x ∨ y) = A(x)∨ A(y). Therefore, A is a fuzzy prime filter of L . 5. Conclusion Aiming to explore the many-valued logical system whose propositional value is given in a lattice, Xu [1] took the initiative in the study of the concept of lattice implication algebras. Hence, in order to promote the development of the many-valued logical system, it is of overwhelming necessity to make clear the structure of lattice impli- cation algebras. When an algebraic system is studied, the research of its structure is found to be the investigation of the structure of filters. In this paper, some equivalent characterizations of fuzzy filters of lattice implication algebras are revealed. Conse- quently, the structure of fuzzy filters are further studied. Finally, the notion of fuzzy prime filters is proposed and some equivalent conditions of fuzzy prime filters are ob- tained. We desperately hope that our work would serve as a foundation for enriching corresponding many-valued logical system. Acknowledgments The work is partly supported by the National Natural Science Foundation of China (No.60875034) and the Scientific Research Project of Department of Education of Sichuan Province (No.08zb082,09zb105). References 1. Xu Y (1993) Lattice implication algebras. 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Journal

Fuzzy Information and EngineeringTaylor & Francis

Published: Sep 1, 2011

Keywords: Lattice implication algebras; Fuzzy filters; Fuzzy prime filters

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