# 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

, 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. 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

### Journal

Fuzzy Information and EngineeringTaylor & Francis

Published: Sep 1, 2011

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