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Fuzzy Inf. Eng. (2009) 4: 401-419 DOI 10.1007/s12543-009-0031-z ORIGINAL ARTICLE Fuzzy Prime Boolean Filters and Their Operations in IMT L-algebras Jia-lu Zhang Received: 6 April 2009/ Revised: 9 November 2009/ Accepted: 26 November 2009/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China 2009 Abstract Some characterizations of fuzzy prime Boolean filters of IMT L-algebras are given. The lattice operations and the order-reversing involution on the setPB(M) of all fuzzy prime Boolean filters of IMT L-algebras are defined. It is showed that the setPB(M) endowed with these operations is a complete quasi-Boolean algebra (a distributive complete lattice with an order-reversing involution). It is derived that the algebra M/F, which is the set of all cosets of F, is isomorphic to the Boolean algebra {0, 1} if F is a fuzzy prime Boolean filter. By introducing an adjoint pair on PB(M), it is proved that the setPB(M) is also a residuated lattice. Keywords IMT L-algebra · Fuzzy filter · Fuzzy prime Boolean filter · Residuated lattice· Quasi-Boolean algebra 1. Introduction Study of multiple-valued logic or fuzzy logic is one of important issues in computer science and artificial intelligence [1, 2]. The monoidal t-norm based logic (briefly, MT L) proposed by Esteva and Godo [3] is the common formal deductive system of all fuzzy logic systems based on left-continuous t-norm. MT L has two impor- tant schematic extensions: weak nilpotent minimum logic (briefly, WNM), involutive monoidal t-norm based logic (briefly, IMT L). IMT L is the schematic extension of MT L by adding an axiom forcing the negation to be an involution, it can also be viewed as the common formal deductive system of all fuzzy logic systems based on left-continuous t-norm with an order-reversing involution, the Lukasiewicz logic Ł proposed by Wang [5] are two standard ex- [4] and the formal deductive system L amples of IMT L. Further, WNM and IMT L have a common schematic extension: nilpotent minimum logic (briefly, NM). In [6], Pei has already proved that L and NM are equivalent logic systems. Jia-lu Zhang () Department of Mathematics, Xiangnan University, Chenzhou, Hunan, 423000, P.R.China email: zjl0735@163.com 402 Jia-lu Zhang (2009) Algebraic method plays an important role in the research of fuzzy logic. Many log- ical algebra systems corresponding to various logical systems are established, such as monoidal t-norm based logic algebras (MT L-algebras), basic logic algebras (BL- algebras), many valued logic algebras (MV -algebras), lattice implication algebras, R -algebras, etc [3, 4, 7-9]. IMT L-algebras are the algebra structures for Esteva and Godo’s IMT L. The standard examples of an IMT L-algebra is the interval [0, 1] endowed with the structure induced by a left-continuous t-norm and an involutive negation. MV -algebras [7], introduced by Chang in 1958, and R -algebras [8], in- troduced by Wang in 2000, are two important classes of IMT L-algebras. For more details of these algebras, we refer the reader to [3, 4, 7-10]. Up to now, these relevant logical algebra structures have already been widely stud- ied. In particular, emphasis seems to have been put on filters theory. From logical point of view, various filters correspond to various sets of provable formulae. Among various types filters, prime filters and Boolean filters are two important filters. Prime filters are important filters because the quotient algebras induced by prime filters are linearly ordered, and Ha ´ jek [3] proved the completeness of basic logical system BL by means of prime filters of BL-algebras. In [11, 12], Turunnen studied some proper- ties of the filters and prime filters of BL-algebras (he called them deductive systems and prime deductive systems, respectively). Boolean filters are important filters be- cause the quotient algebras induced by Boolean filters are Boolean algebras, and a BL-algebra is bipartite if and only if it has a proper Boolean filter. Turunnen intro- duced the notion of Boolean filters (Boolean deductive systems) in BL-algebras and derived some characterizations of Boolean filters [13]. In [14, 15], Haveshki and Kondo studied the characterizations of some types of filters of BL-algebras and their relations again. Afterwards, Xu, Qin and Jun [16-18] proposed the notions of positive implicative filter and fuzzy positive implicative filter, and derived several character- izations of fuzzy positive implicative filters in lattice implication algebras. Further, Liu and Li [19-21] proposed the notions of fuzzy Boolean filters and fuzzy positive implicative filters of BL-algebras and R -algebras. Some properties of fuzzy Boolean filters and fuzzy positive implicative filters of BL-algebras and R -algebras are given. Recently, we proved that the filters set is a residuated lattice and the prime Boolean filters set is a quasi-Boolean algebra by introducing some operations among filters [23]. This paper focuses on the operations on the setPB(M) of all fuzzy prime Boolean filters of IMT L-algebras. We define the lattice meet and join operations∨, ∧ and the order-reversing involution¬ onPB(M) and prove that it is a complete quasi-Boolean algebra (a distributive complete lattice with an order-reversing involution). We derive some characterizations of fuzzy prime Boolean filters of IMT L-algebras, and prove that the algebra M/F, which is the set of all cosets of F, is isomorphic to the Boolean algebra {0, 1} if F is a fuzzy prime Boolean filter. Finally, we introduce an adjoint pair (⊗, →)on PB(M) and prove that PB(M) is also a residuated lattice. It should be noted that although the concepts of fuzzy filters, fuzzy prime filters and fuzzy Boolean filters in this paper derived from [19-21], the research content and method in this paper are different from [19-21]. Hence the conclusion in this paper is completely new in the filters theory. Fuzzy Inf. Eng. (2009) 4: 401-419 403 Throughout this paper, let M denote an IMT L-algebra. 2. Preliminaries Here we recall some definitions and results about IMT L-algebra which will be needed in the following. Definition 2.1 (see [3]) An MT L-algebra is a bounded residuated lattice (M,∨,∧,⊗, →, 0, 1), where ∨ and ∧ are the lattice meet and join operations and (⊗,→) is a residuated pair, satisfying the pre-linearity equation (x → y)∨ (y → x) = 1. An I MT L-algebra is a MT L-algebra satisfying ¬¬x = x, where¬x = x → 0. Let M be an IMT L-algebra. For any x, y ∈ M, define x⊕ y = ¬x → y. It is easy to verify that ⊕ are commutative, associative and x⊕ y = ¬(¬x⊗¬y). In the following, x will denote x⊗ x⊗···⊗ x and nx will denote x⊕ x⊕···⊕ x for n ≥ 1. Example 2.2 Let [0,1] be the real unit interval. If ⊗ is a left-continuous t-norm with an order-reversing involution and → is its residuum, then ([0, 1],⊗,→)isan IMT L- algebra called the IMT L-unit interval. Lemma 2.3 (see [3, 8]) Let M be an I MT L-algebra. Then for any x, y, z ∈ M, (P1) (M, ⊗, 1) is a commutative semigroup with the unit element 1, (P2) ⊕ is a monotone operator, (P3) x⊗ y ≤ x∧ y ≤ x∨ y ≤ x⊕ y, (P4) x⊗ y ≤ z if and only if x ≤ y → z, (P5) x⊗ y → z = x → (y → z), x → (y → x⊗ y) = 1, (P6) x → y ≤ x∨ z → y∨ z, x → y ≤ x⊗ z → y⊗ z, (P7) x⊗ (x → y) ≤ y, x ≤ y → x⊗ y, (P8) x⊗¬x = 0, x⊕¬x = 1, (P9) x⊗ (y∨ z) = (x⊗ y)∨ (x⊗ z), (P10) x∨ (y⊗ z) ≥ (x∨ y)⊗ (x∨ z), (P11) (x ↔ y)⊗ (y ↔ z) ≤ x ↔ z, where x ↔ y = (x → y)∧ (y → x), (P12) (x ↔ y)⊗ (z ↔ u) ≤ (x z) ↔ (y u), where ∈{∧, ∨, ⊗, →}. A subset A of M is called a filter of M if (i)1 ∈ A, (ii) x ∈ A and x ≤ y imply y ∈ A, 404 Jia-lu Zhang (2009) (iii) x, y ∈ A implies x⊗ y ∈ A. It is easy to prove that a subset A is a filter if and only if A satisfies (i) and (iv) x, x → y ∈ A imply y ∈ A. A filter A is called prime if (v)∀ a, b ∈ M, a∨ b ∈ A implies a ∈ A or b ∈ A. A filter A is called Boolean if (vi)∀ a ∈ M, a∨¬a ∈ A [3,8,13]. A fuzzy set F of M is defined by a membership function F : M → [0, 1], F(x)is the value of fuzzy set F on the element x. A fuzzy set G is called a fuzzy subset of fuzzy set F, denoted by G ⊆ F,if G(x) ≤ F(x) for all x ∈ M. The intersection F ∩ G and the union F ∪ G of two fuzzy sets F and G are interpreted in the usual sense, i.e., (F ∩ G)(x) = F(x) ∧ G(x) and (F ∪ G)(x) = F(x)∨ G(x) for all x ∈ M. Definition 2.4 (see [19]) A fuzzy set F of M is called a fuzzy filter of M if it satisfies: (F1) F(1) ≥ F(x) for any x ∈ M, (F2) x ≤ y implies F(x) ≤ F(y), (F3) F(x⊗ y) ≥ F(x)∧ F(y) for any x, y ∈ M. The set of all fuzzy filters of M is denoted byF (M). Proposition 2.5 (see [19]) A fuzzy set F of M is a fuzzy filter if and only if F satisfies (F1) and (F4) F(y) ≥ F(x)∧ F(x → y) for any x, y ∈ M. Lemma 2.6 (see [19]) A fuzzy set F of M is a fuzzy filter if and only if F is a filter of M for every t ∈ [0, 1] if F is nonempty, where F = {x| F(x) ≥ t}. t t Let G be a fuzzy set of M. The fuzzy filter generated by G is defined as G = F. F∈F (M) G⊆F Since M can be viewed as a fuzzy set, i.e. M ∈F (M), then the F is nonempty. F∈F (M) G⊆F It shows that G is well-defined. Lemma 2.7 Let G be a nonempty fuzzy set of M. Then for x ∈ M, G(x) = { G(a )| a , a ,··· , a ∈ M, x ≥ a ⊗ a ⊗···⊗ a }. k 1 2 n 1 2 n k=1 The proof is an easy translation of the proof of [20,Theorem 3.11]. Example 2.8 Let L = {0, a, b, c, d, 1} be a set with Fig. 1 as a partial order. Define Fuzzy Inf. Eng. (2009) 4: 401-419 405 two binary operations “⊗ ” and “ → ”on M as Table 1 and Table 2 respectively: a b d c Fig. 1: Partial order set in L = {0, a, b, c, d, 1}. Table 1: Two binary operation “⊗ ”. ⊗ 0 abcd 1 0 000 0 0 0 a 0 ad 0 da b 0 dc c 0 b c 00 cc 0 c d 0 d 00 0 d 1 0 abcd 1 Table 2: Twobinary operation “ → ”. → 0 abcd 1 0 111 1 1 1 a c 1 bc b 1 b da 1 ba 1 c aa 11 a 1 d b 11 b 11 1 0 abcd 1 The operation “¬” is defined as ¬x = x → 0. Then M is an IMT L-algebra. Define a fuzzy set F of M by 0.8, for x∈{a, 1}, F(x) = 0.4, for x∈{0, c, d, b}. One can easily check that F is a fuzzy filter of M. Define a fuzzy set G of M by 0.9, for x = 1, G(x) = 0.7, for x∈{b, c}, 0.3, for x∈{0, a, d}. 406 Jia-lu Zhang (2009) Then the fuzzy filter generated by G is 0.9, for x∈{b, c, 1}, G(x) = 0.3, for x∈{0, a, d}. 3. Fuzzy Prime Boolean Filters of IMT L-algebras In this section, we investigate the properties of fuzzy prime Boolean filters of IMT L- algebras. Definition 3.1 (see [21]) Let F be a fuzzy filter of M. F is called a fuzzy Boolean filter if F(x∨¬x) = F(1) for any x ∈ M. Definition 3.2 (see [21]) Let F be a fuzzy filter of M. F is called a fuzzy prime filter if F(x∨ y) = F(x)∨ F(y) for any x, y ∈ M. A fuzzy filter is called fuzzy prime Boolean filter if it is both a fuzzy Boolean filter and a fuzzy prime filter. The set of all fuzzy prime Boolean filters of M is denoted byPB(M). Example 3.3 One easily check that the fuzzy filter F of M in Example 2.8 is a fuzzy prime filter. Since F(b∨¬b) = F(b∨ d) = F(b) = 0.4 F(1), we know that it is not a fuzzy Boolean filter. If we define a fuzzy set H of M by 0.8, for x∈{1, b, c}, H(x) = 0.5, for x∈{0, a, d}, then H is a fuzzy Boolean filter of M and it is also a fuzzy prime filter of M, i.e. it is a fuzzy prime Boolean filter of M. The following theorem gives some characterizations of fuzzy prime Boolean fil- ters. Theorem 3.4 Let F be a fuzzy filter of M. The following are equivalent: (1) F is a fuzzy prime Boolean filter, (2) F(x⊕ y) ≤ F(x)∨ F(y) for any x, y ∈ M, (3) F(x⊕ y) = F(x)∨ F(y) for any x, y ∈ M, (4) For each t ∈ [0, 1], F = ∅ or x⊕ y ∈ F implies x ∈ F or y ∈ F , t t t t (5) For each t ∈ [0, 1], F = ∅ or F is a prime Boolean filter, t t (6) F is a prime Boolean filter, F(1) (7) F(x) = F(1) or F(¬x) = F(1) for any x ∈ M, (8) F(x) F(1) and F(y) F(1) imply F(x → y) = F(1) and F(y → x) = F(1) for any x, y ∈ M. Proof (1) ⇒ (2) Let F be a fuzzy prime Boolean filter. Since F is a fuzzy Boolean filter, we have F(x∨¬x) = F(1) for any x ∈ M, and F is also a fuzzy prime filter, which implies that F(x ∨¬x) = F(x) ∨ F(¬x) = F(1) for any x ∈ M. Hence Fuzzy Inf. Eng. (2009) 4: 401-419 407 F(x) = F(1) or F(¬x) = F(1) for any x ∈ M. Let x, y ∈ M.If F(x) = F(1) or F(y) = F(1), then F(x⊕ y) ≤ F(1) = F(x)∨ F(y). If F(¬x) = F(¬y) = F(1), then F(¬(x⊕ y)) = F(¬x⊗¬y) ≥ F(¬x)∧ F(¬y) = F(1). Thus F(x)∨ F(y) ≥ F(0) ≥ F(x⊕ y → 0)∧ F(x⊕ y) = F(¬(x⊕ y))∧ F(x⊕ y) ≥ F(x⊕ y). This shows that F(x⊕ y) ≤ F(x)∨ F(y) for all x, y ∈ M. (2) ⇒ (3) Since x⊕ y ≥ x∨ y, we know F(x⊕ y) ≥ F(x)∨ F(y). Hence (3) holds by (2). (3) ⇒ (4) For each t ∈ [0, 1], if F ∅ and x⊕ y ∈ F , then F(x⊕ y) ≥ t. Since t t F(x⊕ y) = F(x)∨ F(y), we know F(x)∨ F(y) ≥ t, then F(x) ≥ t or F(y) ≥ t,so x ∈ F or y ∈ F . Therefore (4) holds. (4) ⇒ (5) For each t ∈ [0, 1], if F ∅ and x∨ y ∈ F . Recall that x∨ y ≤ x⊕ y, t t we have x ⊕ y ∈ F . Hence x ∈ F or y ∈ F by (4). This means that F is a prime t t t t filter. From (4) and x⊕¬x = 1 ∈ F , it follows that x ∈ F or ¬x ∈ F . Notice that t t t x ≤ x∨¬x, ¬x ≤ x∨¬x,wehave x∨¬x ∈ F . This shows that F is also a Boolean t t filter. Thus (5) holds. (5) ⇒ (6) This is trivial. (6) ⇒ (7) For any x ∈ M,we have x∨¬x ∈ F by (6). Since F is also a F(1) F(1) prime filter, we obtain x ∈ F or¬x ∈ F . Thus F(x) = F(1) or F(¬x) = F(1). F(1) F(1) (7) ⇒ (8) Suppose that F(x) F(1) and F(y) F(1). Then F(¬x) = F(1) and F(¬y) = F(1) by (7). Hence F(x → y) ≥ F(x → 0) = F(¬x) = F(1). It follows from this and F(x → y) ≤ F(1) that F(x → y) = F(1). Similarly, it follows from F(y) F(1) that F(y → x) = F(1). (8) ⇒ (1) For any x ∈ M,if F(x) = F(1) then F(x∨¬x) ≥ F(x)∨ F(¬x) = F(1). If F(x) F(1) then F(0) F(1). It follows from (8) that F(¬x) = F(x → 0) = F(1). Hence F(x∨¬x) ≥ F(x)∨ F(¬x) = F(1). This shows that F is a fuzzy Boolean filter of M. For any x, y ∈ M,if F(x) = F(1) or F(y) = F(1) then F(x∨ y) ≤ F(1) = F(x) ≤ F(x)∨ F(y) or F(x∨ y) ≤ F(1) = F(y) ≤ F(x)∨ F(y). 408 Jia-lu Zhang (2009) If F(x) F(1) and F(y) F(1) then F(x → y) = F(1) and F(y → x) = F(1) by (8). Thus F(x) ≥ F(x∨ y)∧ F(x∨ y → x) = F(x∨ y)∧ F((x → x)∧ (y → x)) = F(x∨ y)∧ F(x → x)∧ F(y → x) = F(x∨ y)∧ F(1) = F(x∨ y). It follows that F(x∨ y) ≤ F(x)∨ F(y). This shows that F is also a fuzzy prime filter of M. Theorem 3.5 Let F be a fuzzy prime Boolean filter of M and G be a fuzzy filter of M. If F ≤ G, F(1) = G(1), then G is also a fuzzy prime Boolean filter. Proof Since F is a fuzzy prime Boolean filter, then F(x) = F(1) or F(¬x) = F(1) for all x ∈ M.By F ≤ G, F(1) = G(1), we have G(x) = G(1) or G(¬x) = G(1) for all x ∈ M. Using Theorem 3.4 G is also a fuzzy prime Boolean filter. Theorem 3.6 Let F be a fuzzy prime Boolean filter and a ∈ M. Define F (x) = F(a → x) for all x ∈ M. Then F is the generated fuzzy filter of F ∨χ , where χ is a a a the characteristic function of the singleton set{a}. Proof˜ First, we prove that F is a fuzzy filter. Clearly,∀x ∈ M, F (1) ≥ F (x). a a a F (x)∧ F (x → y) = F(a → x)∧ F(a → (x → y)) a a ≤ F((a → x)⊗ (a → (x → y))) ≤ F(a → (a → y)). Since F is a fuzzy Boolean filter, we have F(a → y) ≥ F(a∨¬a)∧ F(a∨¬a → (a → y)) = F(1)∧ F(a∨¬a → (a → y)) = F(a∨¬a → (a → y)). Recall that a∨¬a → (a → y) = (a → (a → y))∧ (¬a → (a → y)) = a → (a → y), we obtain F(a∨¬a → (a → y)) = F(a → (a → y)). Hence F (y) = F(a → y) ≥ F(a → (a → y)) ≥ F (x)∧ F (x → y). a a a It follows that F is a fuzzy filter. Next, we prove that F is the least fuzzy filter which contains F ∨χ . Let G be a a a fuzzy filter and F ∨χ ≤ G. Then G(x) ≥ G(a)∧ G(a → x) ≥ (F ∨χ )(a)∧ F(a → x) = F(a → x) = F (x). a a This shows that F = F ∨χ . a a Definition 3.7 Let F be a fuzzy filter of M and x ∈ M. The fuzzy set F : M → [0, 1] defined by F (y) = F(x ↔ y) is called the coset of fuzzy filter F. Fuzzy Inf. Eng. (2009) 4: 401-419 409 x y Lemma 3.8 If F is a fuzzy filter, then F = F if and only if F(x ↔ y) = F(1). x y x y Proof If F = F , then F (x) = F (x), i.e., F(x ↔ x) = F(1) = F(x ↔ y). Con- versely, let F(x ↔ y) = F(1). Since (x ↔ z) ≥ (x ↔ y)⊗ (y ↔ z), we have F(x ↔ z) ≥ F(x ↔ y)∧ F(y ↔ z) = F(y ↔ z). Similarly, F(y ↔ z) ≥ F(x ↔ z). This means that F(x ↔ z) = F(y ↔ z). There- x y fore F = F . 0 1 Theorem 3.9 Let F be a fuzzy prime Boolean filter. Then M/F = {F , F }. Further- more, M/F is isomorphic to the Boolean algebra{0, 1}. Proof Since F is a fuzzy prime Boolean filter, we know that∀x ∈ M, F(x) = F(1) or x 1 F(¬x) = F(1). Using Lemma 3.8, if F(x) = F(1) then F = F .If F(¬x) = F(1), x 0 x 1 x 0 i.e. F(0 ↔ x) = F(1), then F = F . This means that F = F or F = F . Thus 0 1 M/F = {F , F }. Furthermore, M/F is isomorphic to the Boolean algebra{0, 1}. 4. The Lattice Operations on SetPB(M) of All Fuzzy Prime Boolean Filters In this section, we introduce some lattice operations on setPB(M) of all fuzzy prime Boolean filters and investigate their properties. Definition 4.1 Let F (M) be the set of all fuzzy filters of M. Define a partial order “ ≤ ” on F (M):F ≤ G if and only if F ⊆ G, i.e. F(x) ≤ G(x) for all x ∈ M. Define “∧ ” and “∨ ” operations on F (M) as follows: ∀ F ∈F (M)(i ∈ I), F = F , F = F . i i i i i∈I i∈I i∈I i∈I Obviously, F , F are fuzzy filters of M. i∈I i i∈I i Theorem 4.2 (see [20]) (F (M),∅, ≤, ∧, ∨) is a bounded distributive lattice. Theorem 4.3 If∀ i ∈ I, F is a fuzzy prime Boolean filter, then F is also a fuzzy i i i∈I prime Boolean filter. Proof Obviously, F is a fuzzy filter. In the following we prove that it is also a i∈I fuzzy prime Boolean filter. For x, y ∈ M, ( F )(x⊕ y) = { ( F )(a )| x⊕ y ≥ a ⊗ a ⊗···⊗ a }. i i j 1 2 m i∈I i∈I j=1 For any a , a , ··· , a , x⊕ y ≥ a ⊗ a ⊗···⊗ a ,wehave 1 2 m 1 2 m ( F )(a ) ≤ ( F )(a ) i 1 i 1 i∈I i∈I 410 Jia-lu Zhang (2009) and ( F )(a )∧ ( F )(a ) ≤ ( F )(a )∧ ( F )(a ) ≤ ( F )(a ⊗ a ). i 1 i 2 i 1 i 2 i 1 2 i∈I i∈I i∈I i∈I i∈I Following this process, we get m m−1 ( F )(a ) = ( F )(a ) ( F )(a ) i j i j i m j=1 i∈I j=1 i∈I i∈I ≤ ( F )(a ⊗ a ⊗···⊗ a )∧ ( F )(a ). i 1 2 m−1 i m i∈I i∈I Putting b = a ⊗ a ⊗···⊗ a ,wehave b⊗ a = a ⊗ a ⊗···⊗ a ≤ x⊕ y. Hence 1 2 m−1 m 1 2 m ( F )(a ) ≤ ( F )(b)∧ ( F )(a ) i j i i m j=1 i∈I i∈I i∈I ≤ ( F )(b)∧ ( F )(b → b⊗ a ) i i m i∈I i∈I ≤ ( F )(b)∧ ( F )(b → x⊕ y) i i i∈I i∈I = ( F )(b)∧ ( F )(¬b⊕ (x⊕ y)) i i i∈I i∈I = ( F )(b)∧ [ F (x)∨ F (¬b⊕ y)] i i i i∈I i∈I i∈I = ( F )(b)∧ [( F )(x)∨ ( F )(¬b⊕ y)] i i i i∈I i∈I i∈I ≤ ( F )(b)∧ [( F )(x)∨ ( F )(¬b⊕ y)] i i i i∈I i∈I i∈I = [( F )(b)∧ ( F )(x)]∨ [( F )(b)∧ ( F )(¬b⊕ y)] i i i i i∈I i∈I i∈I i∈I ≤ ( F )(x)∨ [( F )(b)∧ ( F )(b → y)] i i i i∈I i∈I i∈I ≤ ( F )(x)∨ ( F )(y). i i i∈I i∈I Therefore ( F )(x⊕ y) ≤ ( F )(x)∨ ( F )(y). This shows that F is i∈I i i∈I i i∈I i i∈I i a fuzzy prime Boolean filter. In order to prove that F is a fuzzy prime Boolean filter, we need to introduce i∈I i some notion and prove some lemma. For x ∈ M, denote x = {y ∈ M| x⊕ y = 1} = {y ∈ M| y≥¬x}. For each fuzzy set F of M, a new fuzzy set of M, denoted it by F , is defined as follows: F (x) = 1−∧{F(y)| y ∈ x } = 1−∧{F(y)| y≥¬x}. Fuzzy Inf. Eng. (2009) 4: 401-419 411 Lemma 4.4 Let F and G be two fuzzy sets of M. Then (1) F (1) ≥ F (x) for any x ∈ M, (2) if x ≤ y then F (x) ≤ F (y), (3) if F ≤ G then G ⊆ F , (4) F ⊆ F, where F means (F ) . Proof It is trivial to verify (1),(2),(3). We prove only (4). For any x ∈ M,wehave F (x) = 1−∧{F (y)| y≥¬x} = 1−∧{1−∧{F(z)| z≥¬y}| y≥¬x} ≤ 1−∧{1−F(x)| y≥¬x} = 1−(1−F(x)) = F(x). Hence F ⊆ F. Lemma 4.5 Let F be a fuzzy filter of M. Then F = F. Proof It is suffices to show that F ⊆ F .If F ⊆ F does not hold, then there exists x ∈ M such that F (x) < F(x). Hence there exists a real number r such that 1−∧{F (y)| y ≥¬x} < r < F(x), that is {F (y)| y ≥¬x} > 1 − r. This implies that F (¬x) > 1− r. Thus 1−∧{F(z)| z≥¬¬x} > 1− r, that is ∧{F(z)| z ≥ x} < r. Hence there exists z ≥ x such that F(z ) < r. This shows that F(x) ≤ F(z ) < r. This 0 0 0 contradicts the fact that F(x) > r. Therefore F = F. Lemma 4.6 If F is a fuzzy filer of M, then F (x) = 1− F(¬x) for all x ∈ M. Proof Since F is a fuzzy filter, we have F(z) ≥ F(¬x)if z≥¬x. Hence F (x) = 1−∧{F(z)| z≥¬x} = 1− F(¬x). Theorem 4.7 If F is a fuzzy prime Boolean filter of M, then F is also a fuzzy prime Boolean filter. Proof Let x, y ∈ M. By Lemma 4.6, we have F (x⊗ y) = 1− F(¬(x⊗ y)) = 1− F(¬x⊕¬y) = 1−F(¬x)∨F(¬y) = (1−F(¬x))∧(1−F(¬y)) = F (x)∧F (y). F (x⊕ y) = 1− F(¬(x⊕ y)) = 1− F(¬x⊗¬y) ≤ 1−F(¬x)∧F(¬y) = (1−F(¬x))∨(1−F(¬y)) = F (x)∨F (y). Combining Lemma 4.4 (1)(2) and Theorem 3.4 (3), we obtain that F is a fuzzy prime Boolean filter. From Theorem 4.7 we know that is an order-reversing involution on setPB(M). 412 Jia-lu Zhang (2009) Example 4.8 The IMT L-algebra is defined as Example 2.8. Define a fuzzy set H of M by 0.8, for x∈{1, b, c}, H(x) = 0.5, for x∈{0, a, d}. Then H is a fuzzy prime Boolean filter of M, and H is 0.5, for x∈{1, bc}, H (x) = 0.2, for x∈{0, a, d}. H is also a fuzzy prime Boolean filter of M. Lemma 4.9 Let 0 be a prime element of M. If a ⊗ a ⊗···⊗ a = 0, then there exist 1 2 m a and a positive integer k such that a = 0. Proof When m = 2, if a ⊗ a = 0 then a ≤¬a . Hence 1 2 1 2 ¬(a ⊗ a ∧ a ⊗ a ) = (¬a ⊕¬a )∨ (¬a ⊕¬a ) = (a →¬a )∨ (a →¬a ) 1 1 2 2 1 1 2 2 1 1 2 2 ≥ (¬a →¬a )∨(a → a ) = (a → a )∨(a → a ) = 1. 2 1 2 1 1 2 2 1 This means that a ⊗ a ∧ a ⊗ a = 0. Since 0 is a prime element, we get a ⊗ a = 0 1 1 2 2 1 1 or a ⊗ a = 0. Assume that when m ≤ i the conclusion holds, then when m = i+ 1, 2 2 from that a ⊗···⊗ a ⊗ a = 0, we have a ⊗···⊗ a ≤¬a . Similar to the above 1 i i+1 1 i i+1 proof, we obtain (a ⊗···⊗ a )⊗ (a ⊗···⊗ a ) = 0or a ⊗ a = 0. 1 i 1 i i+1 i+1 2 2 2 If (a ⊗···⊗a )⊗ (a ⊗···⊗a ) = a ⊗···⊗a = 0, then by assuming, there exists a and 1 i 1 i 1 i j 2 k 2k k such that (a ) = 0, i.e. a = 0. Therefore when m = i+1 the conclusion also holds. j j Lemma 4.10 Let 0 be a prime element of M and F (i ∈ I) are fuzzy prime Boolean filters. Then ( F )(0) = ( F )(0) = F (0). i i i i∈I i∈I i∈I Proof ( F )(0) = { ( F )(a )| 0 = a ⊗···⊗ a }. By Lemma 4.8, there i i j 1 m i∈I j=1 i∈I exist k and a such that a = 0. Since F (i ∈ I) are fuzzy prime Boolean filters, j i we know F (0) = F (a ) ≥ F (a ), it follows that F (0) = F (a )(i ∈ I). Hence i i i j i i j ( F )(a ) ≤ F (0), it follows that ( F )(0) ≤ F (0). Obviously, i∈I i j i∈I i i∈I i i∈I i j=1 ( F )(0) ≥ F (0). Therefore ( F )(0) = F (0). i i i i i∈I i∈I i∈I i∈I The following example shows the condition that “0 is a prime element” in Lemma 4.10 is indispensable. Example 4.11 Let L = {0, a, b, 1} be a set with Fig. 2 as a partial order. Define two Fuzzy Inf. Eng. (2009) 4: 401-419 413 binary operations “⊗ ” and “ → ”on M as Table 3 and Table 4, respectively: a b Fig. 2: Partial order set in L = {0, a, b, 1}. Table 3: Two binary operation “⊗ ”. ⊗ 0 ab 1 0 000 0 a 0 a 0 a b 00 bb 1 0 ab 1 Table 4: Two binary operation “ → ”. → 0 ab 1 0 111 1 a b 1 b 1 b aa 11 1 0 ab 1 The operation “¬” is defined as¬x = x → 0. Then M is an IMT L-algebra and “0” is not a prime element”. Define two fuzzy sets F and G of M as follows: 0.8, for x∈{a, 1}, F(x) = 0.4, for x∈{0, b}, 0.9, for x∈{b, 1}, G(x) = 0.5, for x∈{0, a}. One can easily check that they are fuzzy prime Boolean filters of M. The F ∨ G is expressed by 0.9, for x∈{b, 1}, (F ∨ G)(x) = 0.8, for x∈{0, a}. Obviously, (F ∨ G)(0) F(0)∨ G(0). Theorem 4.12 Let 0 be a prime element of M and F (i ∈ I) are fuzzy prime Boolean filters. Then ( F ) = F , ( F ) = F i i i i i∈I i∈I i∈I i∈I 414 Jia-lu Zhang (2009) Proof Firstly, we prove ( F ) = F . It follows from Lemma 4.4 that i∈I i i∈I ( F ) ≤ F (i ∈ I) and ( F ) ≤ F . By Lemma 4.6, i∈I i i∈I i i∈I i i ( F ) (x) = 1− ( F )(¬x) = 1− { ( F )(a )|¬x ≥ a ⊗···⊗ a }. i i i j 1 m i∈I i∈I j=1 i∈I Putting b = a ⊗···⊗ a , then b⊗ a ≤¬x. Similar to the proof of Theorem 4.3, 1 m−1 m we obtain ( F )(a ) ≤ ( F )(b)∧ ( F )(¬b⊕¬x) i j i i j=1 i∈I i∈I i∈I = ( F )(b)∧ [ F (¬b⊕¬x)] i i i∈I i∈I = ( F )(b)∧ [ F (¬b)∨ F (¬x)] i i i i∈I i∈I i∈I = [( F )(b)∧ F (¬b)]∨ [( F )(b)∧ F (¬x)] i i i i i∈I i∈I i∈I i∈I ≤ [( F )(b)∧ ( F )(¬b)]∨ ( F )(¬x) i i i i∈I i∈I i∈I ≤ ( F )(b⊗¬b)∨ ( F )(¬x) = ( F )(0)∨ ( F )(¬x) i i i i i∈I i∈I i∈I i∈I = ( F )(0)∨ ( F )(¬x) = ( F )(¬x). i i i i∈I i∈I i∈I Hence ( F ) (x) = 1− { ( F )(a )|¬x ≥ a ⊗···⊗ a } i i j 1 m i∈I j=1 i∈I ≥ 1− ( F )(¬x) = 1− F (¬x) = (1− F (¬x)) i i i i∈I i∈I i∈I = F (x) = ( F )(x). i i i∈I i∈I This means that ( F ) ≥ F . Therefore ( F ) = F . i∈I i i∈I i∈I i i∈I i i Secondly, recall that F = F = ( F ) , i i i∈I i∈I i∈I we obtain ( F ) = F . i∈I i∈I If F (i ∈ I) are fuzzy prime Boolean filters, then from F = ( F ) and i i i∈I i∈I Theorem 4.3, 4.12, we have F is also a fuzzy prime Boolean filter. Hence i∈I i PB(M) are closed under the operations “ ∨ ” and “ ∧ ”. Using Theorem 4.2, 4.3, Fuzzy Inf. Eng. (2009) 4: 401-419 415 4.7, 4.12, we obtain the following theorem. Theorem 4.13 Denote PB(M) = PB(M) ∪ {∅}.If 0 is a prime element of M, then (PB(M); ∧, ∨, ) is a complete bounded distributive lattice with an order-reversing involution (a complete quasi-Boolean algebra). 5. An Adjoint Pair on the LatticePB(M) In this section, we introduce an adjoint pair on the lattice PB(M) and prove that it is a residuated lattice. Definition 5.1 For F, G ∈PB(M), define two fuzzy sets F⊗ G and F ⇒ Gof M as follows: ∀x ∈ M, (F⊗ G)(x) = { (F(a )⊗ G(a ))| x ≥ a ⊗ a ⊗···⊗ a }, i i 1 2 m i=1 (F ⇒ G)(x) = [F(z) → G(z⊕ x)], z∈M where ⊗ is a left-continuous t-norm with an order-reversing involution and → is its residuum. Theorem 5.2 If 0 is a prime element of M. Then F⊗ G is a fuzzy prime Boolean filter. Proof Obviously, F⊗ G is a fuzzy filter. In the following we prove that F⊗ G is a fuzzy prime Boolean filter. Let x, y ∈ M, (F⊗ G)(x∨ y) = { (F(a )⊗ G(a ))| x∨ y ≥ a ⊗ a ⊗···⊗ a }. i i 1 2 m i=1 For any a , a , ··· , a , x∨ y ≥ a ⊗ a ⊗···⊗ a , 1 2 m 1 2 m we have a ⊗ a ⊗···⊗ a → x∨ y = 1. 1 2 m Hence ¬(a ⊗ a ⊗···⊗ a → x∨ y) 1 2 m = ¬(a ⊗ a ⊗···⊗ a → x)∧¬(a ⊗ a ⊗···⊗ a → y) = 0. 1 2 m 1 2 m Since 0 is a prime element of M, we know that ¬(a ⊗ a ⊗···⊗ a → x) = 0or ¬(a ⊗ a ⊗···⊗ a → y) = 0, 1 2 m 1 2 m i.e., a ⊗ a ⊗···⊗ a → x = 1or a ⊗ a ⊗···⊗ a → y = 1. That is 1 2 m 1 2 m a ⊗ a ⊗···⊗ a ≤ x or a ⊗ a ⊗···⊗ a ≤ y. 1 2 m 1 2 m 416 Jia-lu Zhang (2009) Thus (F⊗ G)(x∨ y) = { (F(a )⊗ G(a ))| x∨ y ≥ a ⊗ a ⊗···⊗ a } i i 1 2 m i=1 ≤ { (F(a )⊗ G(a ))| x ≥ a ⊗ a ⊗···⊗ a } i i 1 2 m i=1 { (F(a )⊗ G(a ))| y ≥ a ⊗ a ⊗···⊗ a } i i 1 2 m i=1 = (F⊗ G)(x)∨ (F⊗ G)(y). This means that F⊗ G is a fuzzy prime filter. For any x ∈ M, (F⊗ G)(x∨¬x) ≥ F(x∨¬x)⊗ G(x∨¬x) = F(1)⊗ G(1) ≥ { (F(a )⊗ G(a ))| 1 ≥ a ⊗ a ⊗···⊗ a } = (F⊗ G)(1). i i 1 2 m i=1 Hence (F⊗ G)(x∨¬x) = (F⊗ G)(1). This shows that F⊗ G is also a fuzzy Boolean filter. The following example shows the condition that “0 is a prime element” in Theo- rem 5.2 is indispensable. Example 5.3 Consider the filters F, G in Example 4.11. If we get nilpotent minimum t-norm in Definition 5.1 then F⊗ G is expressed by 0.8, for x = 1, (F⊗ G)(x) = 0.4, for x∈{0, a, b}. One can easily check that F⊗ G is not a fuzzy prime Boolean filter. Theorem 5.4 F ⇒ G is a fuzzy prime Boolean filter. Proof Firstly, we prove that F → G is a fuzzy filter. (1) (F ⇒ G)(1)= [F(z) → G(z⊕ 1)] = [F(z) → G(1)] z∈M z∈M ≥ [F(z) → G(z⊕ x)] = (F → G)(x). z∈M (2) If x ≤ y, then (F ⇒ G)(y) = [F(z) → G(z⊕ y)] z∈M ≥ [F(z) → G(z⊕ x)] = (F → G)(x). z∈M (3) (F ⇒ G)(x⊗ y)= [F(z) → G(z⊕ (x⊗ y))] z∈M = [F(z) → G(z)∨ (G(x)∧ G(y))] z∈M = [F(z) → (G(z)∨ G(x))∧ (G(z)∨ G(y))] z∈M = [F(z) → G(z⊕ x)∧ G(z⊕ y)] z∈M = [(F(z) → G(z⊕ x))∧ (F(z) → G(z⊕ y))] z∈M = [F(z) → G(z⊕ x)]∧ [F(z) → G(z⊕ y)] z∈M z∈M = (F ⇒ G)(x)∧ (F ⇒ G)(y) Fuzzy Inf. Eng. (2009) 4: 401-419 417 Secondly, we prove that F ⇒ G is a fuzzy prime Boolean filter. For x, y ∈ M, (F ⇒ G)(x⊕ y) = [F(z) → G(z⊕ (x⊕ y))] z∈M ≥ [F(z) → G(z⊕ x)] [F(z) → G(z⊕ y)]. z∈M z∈M If the above inequality strictly holds, then there exists r ∈ (0, 1) such that [F(z) → G(z⊕ (x⊕ y))] > r z∈M > [F(z) → G(z⊕ x)] [F(z) → G(z⊕ y)]. z∈M z∈M Thus, on one hand, ∀z ∈ M, F(z) → G(z⊕ x⊕ y) > r. On the other hand, since [F(z) → G(z⊕ x)] [F(z) → G(z⊕ y)] < r, z∈M z∈M we obtain that there are z , z such that 1 2 F(z ) → G(z ⊕ x) < r, F(z ) → G(z ⊕ y) < r. 1 1 2 2 Hence F(z ⊕ z ) → G(z ⊕ z ⊕ x⊕ y) = F(z )∨ F(z ) → G(z ⊕ x)∨ G(z ⊕ y) 1 2 1 2 1 2 1 2 = [F(z )∨ F(z ) → G(z ⊕ x)]∨ [F(z )∨ F(z ) → G(z ⊕ y)] 1 2 1 1 2 2 ≤ [F(z ) → G(z ⊕ x)]∨ [F(z ) → G(z ⊕ y)] < r. 1 1 2 2 This is a contradiction. Thus (F ⇒ G)(x⊕ y) = [F(z) → G(z⊕ (x⊕ y))] z∈M = [F(z) → G(z⊕ x)] [F(z) → G(z⊕ y)] z∈M z∈M = (F ⇒ G)(x)∨ (F ⇒ G)(y). Therefore F ⇒ G ∈PB(M). Theorem 5.5 If 0 is a prime element of M, then (PB(M), ∅, ∧, ∨, ⊗, ⇒) is a resid- uated lattice. Proof By Theorem 4.12, 5.2, 5.4, it suffices to show that (⊗, ⇒) is an adjoint pair on F (M). Clearly ⊗ is monotone increasing, ⇒ is monotone decreasing with respect to the first variable and is monotone increasing with respect to the second variable. In the following we will prove that F⊗ G ≤ H if and only if F ≤ G ⇒ H. 418 Jia-lu Zhang (2009) Suppose that F⊗ G ≤ H. Then∀x ∈ M, F(x)⊗ G(x) ≤ (F ⊗ G)(x) ≤ H(x). Hence (G ⇒ H)(x)= [G(z) → H(z⊕ x)] ≥ [G(z⊕ x) → H(z⊕ x)] z∈M z∈M ≥ [G(z⊕ x) → F(z⊕ x)⊗ G(z⊕ x)] ≥ F(z⊕ x) ≥ F(x). z∈M z∈M This means that F ≤ G ⇒ H. Conversely, suppose that F ≤ G ⇒ H. Then ∀x ∈ M, [G(z) → H(z⊕ x)]. F(x) ≤ (G ⇒ H)(x) = z∈M Hence F(x) ≤ G(x) → H(x⊕ x) = G(x) → H(x)∨ H(x) = G(x) → H(x). It follows that ∀x ∈ M, F(x)⊗ G(x) ≤ H(x). Thus (F⊗ G)(x) = { (F(a )⊗ G(a ))| x ≥ a ⊗ a ⊗···⊗ a } i i 1 2 m i=1 ≤ { (H(a )| x ≥ a ⊗ a ⊗···⊗ a } i 1 2 m i=1 = {(H(a ⊗ a ⊗···⊗ a )| x ≥ a ⊗ a ⊗···⊗ a } 1 2 m 1 2 m ≤ H(x). This means that F⊗ G ≤ H. Therefore (PB(M), ∅, ∧, ∨, ⊗, ⇒) is a residuated lat- tice. 6. Conclusion In this paper, we studied the fuzzy prime Boolean filters of IMT L-algebra. The lat- tice operations and the order-reversing involution on the setPB(M) of all fuzzy prime Boolean filters of IMT L-algebra are defined. It is showed that the set PB(M) en- dowed with these operations is a complete quasi-Boolean algebra. We derived some characterizations of fuzzy prime Boolean filters of IMT L-algebras. It is given that the algebra M/F, which is the set of all cosets of F, is isomorphic to the Boolean algebra {0, 1} if F is a fuzzy prime Boolean filter. By introducing an adjoint pair on PB(M), it is proved that the set PB(M) is also a residuated lattice. This paper only investigated the algebra properties of the set of all fuzzy prime Boolean filters of IMT L-algebras. To study the properties of fuzzy prime Boolean filters set of other logical algebras systems and the applications of fuzzy prime Boolean filter lattices in fuzzy logic and fuzzy reasoning is a further topic. Acknowledgments The work of this paper has been supported by the construction program of the key dis- cipline in Hunan Province and the aid program for science and technology innovative research team in higher educational institutions of Hunan Province. Fuzzy Inf. Eng. (2009) 4: 401-419 419 The author would like to express his sincere thanks to the referees for their valu- able suggestions and comments. References 1. Gorz ¨ G, Holldobler ¨ S (1998) Advances in Artificial Intelligence. Lecture Notes in Artificial Intelli- gence. New York: Springer-Verlag 2. Shi C Y, Huang C N, Wang J Q (1998) Principle of Artificial Intelligence. Beijing: Tsinghua Univer- sity Press 3. Esteva F, Godo L (2001) Monoidal t-norm based logic: towards a logic for left-continuous t-norms. Fuzzy Sets and Systems 124: 271-288 4. Hajek ´ P (1998) Metamathematics of Fuzzy Logic. Dordrecht: Kluwer Adacemic Publishers 5. Wang G J (1997) A formal deductive system for fuzzy propositional calculus. 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Journal Fuzzy Mathematics 16(2): 457-467
Fuzzy Information and Engineering – Taylor & Francis
Published: Dec 1, 2009
Keywords: IMTL -algebra; Fuzzy filter; Fuzzy prime Boolean filter; Residuated lattice; Quasi-Boolean algebra
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