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Fuzzy filters of MTL-algebras
Information Sciences 175 (2005) 120–138 www.elsevier.com/locate/ins
Fuzzy filters of MTL-algebras Young Bae Jun
a,*
, Yang Xu b, Xiao Hong Zhang
c
a
Department of Mathematics Education, Gyeongsang National University, Chinju 660-701, South Korea b Department of Applied Mathematics, Southwest Jiaotong University, Chengdu 610031, Sichuan, China c Department of Mathematics, The Faculty of Science, Ningbo University, Ningbo 315211, Zhejiang, China Received 19 August 2004; received in revised form 14 November 2004; accepted 15 November 2004
Abstract Characterizations of fuzzy filters in MTL-algebras are given. The fuzzy filter generated by a fuzzy set is considered. The notion of Boolean fuzzy filters and MV-fuzzy filters are introduced and related properties are investigated. A condition for a fuzzy filter to be Boolean is provided. A characterization of a Boolean fuzzy filter is given. A congruence relation on a MTL-algebra induced by a fuzzy filter is established, and we show that the set of all congruence relations induced by a fuzzy filter is a completely distributive lattice. 2004 Elsevier Inc. All rights reserved. Keywords: MTL-algebra; (Boolean) fuzzy filter; MV-fuzzy filter
*
Corresponding author. Tel.: +82 55 751 5674; fax: +82 55 751 6117. E-mail addresses: [email protected] (Y.B. Jun), [email protected] (Y. Xu), zxhonghz@263. net (X.H. Zhang). 0020-0255/$ - see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2004.11.004
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1. Introduction The logic MTL, Monoidal t-norm based logic, was introduced by Esteva and Godo [1]. This logic is very interesting from many points of view. From the logic point of view, it can be regarded as a weak system of Fuzzy Logic. Indeed, it arises from Ha´jekÕs Basic Logic BL [2] by replacing the axiom ^ ! BÞÞ $ ðA ^ BÞ ðA^ðA by the weaker axiom ^ ! BÞÞ ! ðA ^ BÞ: ðA^ðA In connection with the logic MTL, Esteva and Godo [1] introduced a new algebra, called a MTL-algebra, and studied several basic properties. They also introduced the notion of (prime) filters in MTL-algebras. Zhang [6] studied further properties of filters in MTL-algebras, and introduced the notion of Boolean filters and MV-filters. Based on the fuzzy set theory, Kim et al. [3] studied the fuzzy structure of filters in MTL-algebras. As a continuation of the paper [3], we give characterizations of fuzzy filters in MTL-algebras, and investigate further properties of fuzzy filters in MTL-algebras. We consider the fuzzification of Boolean filters and MV-filters, and study related properties. We discuss the fuzzy filter generated by a fuzzy set. We provide a condition for a fuzzy filter to be Boolean. Using a fuzzy filter, we give a congruence relation on a MTL-algebra, and show that the set of all congruence relations induced by a fuzzy filter is a completely distributive lattice.
2. Preliminaries By a residuated lattice we shall mean a lattice L = (L, 6, ^, _, , !, 0, 1) containing the least element 0 and the largest element 1, and endowed with two binary operations (called product) and ! (called residuum) such that • is associative, commutative and isotone. • ("x 2 L) (x 1 = x). • The Galois correspondence holds, that is, ð8x; y; z 2 LÞðx y 6 z () x 6 y ! zÞ: In a residuated lattice, the following are true (see [5]): (a1) x 6 y () x ! y ¼ 1. (a2) 0 ! x = 1, 1 ! x = x, x ! (y ! x) = 1. (a3) y 6 (y ! x) ! x.
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(a4) (a5) (a6) (a7)
x ! (y ! z) = (x y) ! z = y ! (x ! z). x ! y 6 (z ! x) ! (z ! y), (x ! y) 6 (y ! z) ! (x ! z). y 6 x ) x ! z 6 y ! z, z ! y 6 z ! x. (¤i 2 Cyi) ! x = §i2C(yi ! x).
We define x* = ¤{y 2 Ljx y = 0}, equivalently, x* = x ! 0. Then (a8) 0* = 1, 1* = 0, x 6 x**, and x* = x***. Based on the Ha´jekÕs results [2], Axioms of MTL and Formulas which are provable in MTL, Esteva and Godo [1] defined the algebras, so called MTLalgebras, corresponding to the MTL-logic in the following way. Definition 2.1. A MTL-algebra is a residuated lattice L = (L, 6, ^, _, , !, 0, 1) satisfying the pre-linearity equation: ðx ! yÞ _ ðy ! xÞ ¼ 1: In a MTL-algebra, the following are true (see [6]): (a9) x ! (y _ z) = (x ! y) _ (x ! z). (a10) x y 6 x ^ y. Definition 2.2 [1]. Let L be a MTL-algebra. A nonempty subset F of L is called a filter of L if it satisfies (b1) ("x, y 2 F) (x y 2 F). (b2) ("x 2 F) ("y 2 L) (x 6 y ) y 2 F). Since ^ is not definable from and ! in a MTL-algebra, one could consider that the further condition (b3) ("x, y 2 F) (x ^ y 2 F) should be also required for a filter. However the condition (b3) is indeed redundant because it is a consequence of conditions (b1) and (b2). Namely, since x y 6 x ^ y, if x, y 2 F the x y 2 F and thus x ^ y 2 F as well. Proposition 2.3 [6]. A nonempty subset F of a MTL-algebra L is a filter of L if and only if it satisfies: (b4) 1 2 F. (b5) ("x 2 F) ("y 2 L) (x ! y 2 F ) y 2 F).
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Proof. Let F be a nonempty subset of L satisfying conditions (b4) and (b5). Let x, y 2 L be such that x 6 y and x 2 F. Then x ! y = 1 2 F by (b4), and so y 2 F by (b5). Thus (b2) is valid. Let x, y 2 F. Then x ! ðy ! ðx yÞÞ ¼ ðx yÞ ! ðx yÞ ¼ 1 2 F ; which implies from (b5) that x y 2 F. Hence (b1) holds. Conversely, suppose that F is a filter of L. Obviously 1 2 F by (b2). Let x, y 2 L be such that x 2 F and x ! y 2 F. Then x (x ! y) 2 F by (b1). Using (a3), we have x 6 (x ! y) ! y, i.e., x ! ((x ! y) ! y) = 1 which implies from (a4) that (x (x ! y)) ! y = 1, i.e., (x (x ! y)) 6 y. It follows from (b2) that y 2 F so that (b5) is valid. This completes the proof. h
3. Fuzzy filters In what follows let L denote a MTL-algebra unless otherwise specified. In [3], Kim et al. defined the fuzzy filter in MTL-algebras as follows. Definition 3.1 [3]. A fuzzy set l in L is called a fuzzy filter of L if it satisfies (b6) ("x, y 2 L) (l(x y) P min{l(x), l(y)}). (b7) l is order-preserving, that is, ð8x; y 2 LÞðx 6 y ) lðxÞ 6 lðyÞÞ: Example 3.2. (1) Let L = [0, 1] and define a product and a residuum ! on L as follows: 1 if x 6 y; x ^ y if x þ y > 12 ; x y :¼ x ! y :¼ ð0:5 xÞ _ y if x > y; 0 otherwise; for all x, y 2 L. Then L is a MTL-algebra. Let l be a fuzzy set in L given by a if x 2 ð0:5; 1; lðxÞ :¼ b otherwise; where a > b in [0, 1]. Then it is routine to verify that l is a fuzzy filter of L. (2) Let L = [0, 1] and define a product and a residuum ! on L as follows: x ^ y if x þ y > 1; 1 if x 6 y; x y :¼ x ! y :¼ 0 otherwise; ð1 xÞ _ y otherwise; for all x, y 2 L. Then L is a MTL-algebra. Let l1 and l2 be fuzzy sets in L given by
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l1 ðxÞ :¼ l2 ðxÞ :¼
a
if x 2 ½0; 0:5;
a þ 0:5x2
if x 2 ð0:5; 1;
a
if x 2 ½0; 0:5;
a þ 0:3x if x 2 ð0:5; 1;
where a 2 [0, 0.5). Then l1 and l2 are fuzzy filters of L. We first give characterizations of a fuzzy filter. Theorem 3.3. A fuzzy set l in L is a fuzzy filter of L if and only if it satisfies (b8) ("x 2 L) (l(1) P l(x)). (b7) ("x, y 2 L) (l(y) P min{l(x), l(x ! y)}). Proof. Assume that l satisfies conditions (b8) and (b9). Let x, y 2 L be such that x 6 y. Then x ! y = 1, and so lðyÞ P minflðxÞ; lð1Þg ¼ lðxÞ by (b8) and (b9). Using (a4), we know that x ! ðy ! ðx yÞÞ ¼ ðx yÞ ! ðx yÞ ¼ 1: It follows from (b8) and (b9) that lðx yÞ P minflðyÞ; lðy ! ðx yÞÞg P minflðyÞ; minflðxÞ; lðx ! ðy ! ðx yÞÞÞgg ¼ minflðyÞ; minflðxÞ; lð1Þgg ¼ minflðxÞ; lðyÞg: Thus (b6) is valid. Suppose that l is a fuzzy filter of L. Since x 6 1 for all x 2 L it follows from (b7) that l(1) P l(x) for all x 2 L. Let x, y 2 L. Since x 6 (x ! y) ! y, we have x (x ! y) 6 y, by the Galois correspondence. Hence lðyÞ P lðx ðx ! yÞÞ P minflðxÞ; lðx ! yÞg by (b7) and (b6). This completes the proof.
h
Theorem 3.4. A fuzzy set l in L is a fuzzy filter of L if and only if it satisfies ð8a; b; c 2 LÞða 6 b ! c ) lðcÞ P minflðaÞ; lðbÞgÞ:
ð1Þ
Proof. Suppose that l is a fuzzy filter of L. Let a, b, c 2 L be such that a 6 b ! c. Then l(a) 6 l(b ! c) by (b7), and so lðcÞ P minflðbÞ; lðb ! cÞg P minflðbÞ; lðaÞg
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by (b9). Conversely let l be a fuzzy set in L satisfying (1). Since x 6 x ! 1 for all x 2 L, it follows from (1) that lð1Þ P minflðxÞ; lðxÞg ¼ lðxÞ for all x 2 L. Since x ! y 6 x ! y for all x, y 2 L, we have lðyÞ P minflðxÞ; lðx ! yÞg for all x, y 2 L. Hence l is a fuzzy filter of L. h Corollary 3.5. A fuzzy set l in L is a fuzzy filter of L if and only if it satisfies the following assertion: lðxÞ P minflða1 Þ; . . . ; lðan Þg
ð2Þ
whenever an ! ( ! (a1 ! x) ) = 1 for every a1, , an 2 L. Proof. It can be easily proved by induction.
h
Theorem 3.6. For a filter F of L and a 2 L, let l be a fuzzy set in L defined by s if x 2 fz 2 Lja _ z 2 F g; lðxÞ :¼ t otherwise; for all x 2 L where s > t in [0, 1]. Then l is a fuzzy filter of L. Proof. Since a _ 1 2 F, we have 1 2 {z 2 Lja _ z 2 F} and so l(1) = s P l(x) for all x 2 L. Now if y 2 {z 2 Lja _ z 2 F}, then clearly lðyÞ ¼ s P minflðxÞ; lðx ! yÞg: Suppose that y 62 {z 2 Lja_z 2 F}. Then at least one of x and x ! y does not belong to {z 2 Lja _ z 2 F}. Hence l(y) = min{l(x), l(x ! y)}, and therefore l is a fuzzy filter of L. h Lemma 3.7 [3]. A fuzzy set l in L is a fuzzy filter of L if and only if the level set l½t :¼ fx 2 LjlðxÞ P t; t 2 ½0; 1g is either empty or a filter of L. Theorem 3.8. If l is a fuzzy filter of L, then the set Xa :¼ fx 2 LjlðxÞ P lðaÞg is a filter of L for every a 2 L.
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Proof. Since l(1) P (x) for all x 2 L. we have 1 2 Xa. Let x, y 2 L be such that x 2 Xa and x ! y 2 Xa. Then l(x) P l(a) and l(x ! y) P l(a). Since l is a fuzzy filter of L; it follows from (b9) that lðyÞ P minflðxÞ; lðx ! yÞg P lðaÞ so that y 2 Xa. Hence Xa is a filter of L.
h
Theorem 3.9. Let a 2 L and let l be a fuzzy set in L. Then (i) If Xa is a filter of L, then l satisfies the following implication ð8x; y 2 LÞðlðaÞ 6 minflðx ! yÞ; lðxÞg ) lðaÞ 6 lðyÞÞ:
ð3Þ
(ii) If l satisfies (b8) and (3), then Xa is a filter of L. Proof. (i) Assume that Xa is a filter of L. Let x, y 2 L be such that lðaÞ 6 minflðx ! yÞ; lðxÞg: Then x ! y 2 Xa and x 2 Xa. Using (b5), we have y 2 Xa and so l(y) P l(a). (ii) Suppose that l satisfies (b8) and (3). From (b8) it follows that 1 2 Xa. Let x, y 2 L be such that x 2 Xa and x ! y 2 Xa. Then l(a) 6 l(x) and l(a) 6 l(x ! y), which imply that l(a) 6 min{l(x), l(x ! y)}. Thus l(a) 6 l(y) by (3), and so y 2 Xa. Therefore Xa is a filter of L. h Proposition 3.10. Let l be a fuzzy filter of L. Then the following are equivalent. (i) ("x, y, z 2 L) (l (x ! z) P min{l (x ! (y ! z)), l(x ! y)}). (ii) ("x, y 2 L) (l(x ! y) P l(x ! (x ! y))). (iii) ("x, y, z 2 L) (l((x ! y) ! (x ! z)) P (x ! (y ! z))). Proof. (i) ) (ii) Suppose that l satisfies the condition (i). Taking z = y and y = x in (i) and using (b8), we have lðx ! yÞ P minflðx ! ðx ! yÞÞ; lðx ! xÞg ¼ minflðx ! ðx ! yÞÞ; lð1Þg ¼ lðx ! ðx ! yÞÞ for all x, y, z 2 L. (ii) ) (iii) Suppose that l satisfies the condition (ii) and let x, y, z 2 L. Since x ! (y ! z) 6 x ! ((x ! y) ! (x ! z)), it follows that
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lðx ! yÞ ! ðx ! zÞ ¼ lðx ! ððx ! yÞ ! zÞÞ P lðx ! ðx ! ððx ! yÞ ! zÞÞÞ ¼ lðx ! ððx ! yÞ ! ðx ! zÞÞÞ P lðx ! ðy ! zÞÞ: (iii) ) (i) If l satisfies the condition (iii), then lðx ! yÞ P minflððx ! yÞ ! ðx ! zÞÞ; lðx ! yÞg P minflðx ! ðy ! zÞÞ; lðx ! yÞg: h
This completes the proof.
A fuzzy filter l of L is said to be Boolean if it satisfies the following equality ðb10Þ ð8x 2 LÞðlðx _ x Þ ¼ lð1ÞÞ: Proposition 3.11. Let l and m be fuzzy filters of L such that l 6 m and l(1) = m(1). If l is Boolean, then so is m. Proof. Straightforward.
h
Proposition 3.12. Every Boolean fuzzy filter l of L satisfies the following inequality: ð8x; y; z 2 LÞðlðx ! zÞ P minflðx ! ðz ! yÞÞ; lðy ! zÞgÞ: Proof. Using (a5), we have y ! z 6 ðz ! yÞ ! ðz ! zÞ 6 ðx ! ðz ! yÞÞ ! ðx ! ðz ! zÞÞ: It follows from (b7) that lðy ! zÞ 6 lððx ! ðz ! yÞÞ ! ðx ! ðz ! zÞÞÞ so from (b9) that lðx ! ðz ! zÞÞ P minflðx ! ðz ! yÞÞ; lððx ! ðz ! yÞÞ ! ðx ! ðz ! zÞÞÞg P minflðx ! ðz ! yÞÞ; lðy ! zÞg: Since z _ z ¼ ððz ! zÞ ! zÞ ^ ððz ! z Þ ! z Þ 6 ðz ! zÞ ! z; we have l((z* ! z) ! z) P l(z*_z) = l(1). Since x ! ðz ! zÞ 6 ððz ! zÞ ! zÞ ! ðx ! zÞ;
ð4Þ
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it follows from (b7) that lðx ! ðz ! zÞÞ 6 lðððz ! zÞ ! zÞ ! ðx ! zÞÞ: Thus lðx ! zÞ P minflððz ! zÞ ! zÞ; lðððz ! zÞ ! zÞ ! ðx ! zÞÞg P minflð1Þ; lðx ! ðz ! zÞÞg ¼ lðx ! ðz ! zÞÞ P minflðx ! ðz ! yÞÞ; lðy ! zÞg: This completes the proof.
h
Proposition 3.13. If a fuzzy filter l of L satisfies the following inequality ð8x; y 2 LÞðlðxÞ P lððx ! yÞ ! xÞÞ;
ð5Þ
then it is Boolean. Proof. Using (a2), (a4) and (a5), we have
1 ¼ x ! ððx ! xÞ ! xÞ 6 ððx ! xÞ ! xÞ ! x 6 ðx ! xÞ ! ðððx ! xÞ ! xÞ ! xÞ
¼ ððx ! xÞ ! xÞ ! ððx ! xÞ ! xÞ ¼ ðððx ! xÞ ! xÞ ! 0Þ ! ððx ! xÞ ! xÞ: It follows from (b7), (b8) and (5) that lððx ! xÞ ! xÞ P lððððx ! xÞ ! xÞ ! 0Þ ! ððx ! xÞ ! xÞÞ ¼ lð1Þ: Using (a7) and (a9), since ðx ! xÞ ! x 6 ððx ! xÞ ! xÞ _ ððx ! xÞ ! x Þ ¼ ðx ! xÞ ! ðx _ x Þ ¼ ð1 ^ ðx ! xÞÞ ! ðx _ x Þ ¼ ððx ! xÞ ^ ðx ! xÞÞ ! ðx _ x Þ ¼ ððx _ x Þ ! xÞ ! ðx _ x Þ; we get l(1) = l((x* ! x) ! x) 6 l(((x_x*) ! x) ! (x_x*)) 6 l(x_x*), and so l(x_x*) = l(1). Therefore l is Boolean. h Proposition 3.14. Let l be a fuzzy filter of L that satisfies the condition (4). Then l satisfies the condition (5).
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Proof. Since (x ! y) ! x 6 x* ! x and l is order-preserving, it follows that lðxÞ ¼ lð1 ! xÞ P minflð1 ! ðx ! x ÞÞ; lðx ! xÞg P minflð1Þ; lððx ! yÞ ! xÞg ¼ lððx ! yÞ ! xÞ: Hence l satisfies the condition (5). h Combining Propositions 3.12, 3.13 and 3.14, we have a characterization of a Boolean fuzzy filter. Theorem 3.15. Let l be a fuzzy filter of L. Then the following assertions are equivalent: (i) l is Boolean. (ii) ("x, y, z 2 L) (l(x ! z) P min{l(x ! (z* ! y)), l(y ! z)}). (iii) ("x, y 2 L) (l(x) P l((x ! y) ! x)). Proposition 3.16. If a fuzzy filter l of L satisfies (5), then it satisfies the following inequality ð8x; y; z 2 LÞðlðx ! zÞ P minflðx ! ðy ! zÞÞ; lðx ! yÞgÞ:
ð6Þ
Proof. Since x ! (y ! z) = y ! (x ! z) 6 (x ! y) ! (x ! (x ! z)), it follows from (b7) that lðx ! ðy ! zÞÞ 6 lððx ! yÞ ! ðx ! ðx ! zÞÞÞ so from (b9) that lðx ! ðx ! zÞÞ P minflðx ! yÞ; lððx ! yÞ ! ðx ! ðx ! zÞÞÞg P minflðx ! yÞ; lðx ! ðy ! zÞÞg: Since x ! ðx ! zÞ 6 x ! ðððx ! zÞ ! zÞ ! zÞ ¼ ððx ! zÞ ! zÞ ! ðx ! zÞ; we have lðx ! zÞ P lðððx ! zÞ ! zÞ ! ðx ! zÞÞ P lðx ! ðx ! zÞÞ P minflðx ! yÞ; lðx ! ðy ! zÞÞg by using (b7) and (4). This completes the proof.
h
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Proposition 3.17. Every Boolean fuzzy filter l of L satisfies the following inequality. ð8x; y; z 2 LÞðlðx ! zÞ P minflðx ! ðy ! zÞÞ; lðx ! yÞgÞ: Proof. Since x ! (y ! z) = y ! (x ! z) 6 (x ! y) ! (x ! (x ! z)) and x ! (x ! z) 6 x ! (((x ! z) ! z) ! z) = ((x ! z) ! z) ! (x ! z), it follows from (b7), (b9) and Theorem 3.15 (iii) that lðx ! zÞ P lðððx ! zÞ ! zÞ ! ðx ! zÞÞ P lðx ! ðx ! zÞÞ P minflðx ! yÞ; lððx ! yÞ ! ðx ! ðx ! zÞÞÞg P minflðx ! yÞ; lðx ! ðy ! zÞÞg: This completes the proof.
h
Definition 3.18. A fuzzy set l in L is called a MV-fuzzy filter of L if it is a fuzzy filter of L that satisfies the following inequality. ðb11 Þ ð8x; y 2 LÞðlðx ! yÞ 6 lðððy ! xÞ ! xÞ ! yÞÞ: Theorem 3.19. In a MV-algebra, every fuzzy filter is a MV-fuzzy filter. Proof. Let l be a fuzzy filter of a MV-algebra L. Since x ! y 6 ððx ! yÞ ! yÞ ! y ¼ ððy ! xÞ ! xÞ ! y; we have l(x ! y) 6 l(((y ! x) ! x) ! y), and so l is a MV-fuzzy filter of L. h Theorem 3.20. Every Boolean fuzzy filter is a MV-fuzzy filter. Proof. Let l be a Boolean fuzzy filter of L. Since y 6 ((y ! x) ! x) ! y, we have ðððy ! xÞ ! xÞ ! yÞ ! x 6 y ! x
ð7Þ
by (a6). Using (a4), (a5), (a6) and (7), we get x ! y 6 ððy ! xÞ ! xÞ ! ððy ! xÞ ! yÞ ¼ ðy ! xÞ ! ðððy ! xÞ ! xÞ ! yÞ 6 ððððy ! xÞ ! xÞ ! yÞ ! xÞ ! ðððy ! xÞ ! xÞ ! yÞ
and so lðððy ! xÞ ! xÞ ! yÞ P lðððððy ! xÞ ! xÞ ! yÞ ! xÞ ! ðððy ! xÞ ! xÞ ! yÞÞ P lðx ! yÞ by Theorem 3.15 (iii) and (b7). Therefore l is a MV-fuzzy filter of L.
h
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Lemma 3.21. Suppose that L satisfies the following assertions. a ! ðb ! xÞ ¼ 1;
ð8Þ
an ! ð ! ða1 ! aÞ Þ ¼ 1;
ð9Þ
bm ! ð ! ðb1 ! bÞ Þ ¼ 1:
ð10Þ
Then bm ! ( ! (b1 ! (an ! ( ! (a1 ! a) ))) ) = 1. Proof. The condition (8) implies a 6 b ! x. Using (a6) and (9), we know that 1 ¼ an ! ð ! ða1 ! aÞ Þ 6 an ! ð ! ða1 ! ðb ! xÞÞ Þ; and so 1 ¼ an ! ð ! ða1 ! ðb ! xÞÞ Þ ¼ b ! ðan ! ð ! ða1 ! xÞ ÞÞ; that is, b 6 (an ! ( ! (a1 ! x) ). It follows from (a6) and (10) that 1 ¼ bm ! ð ! ðb1 ! bÞ Þ 6 bm ! ð ! ðb1 ! ðan ! ð ! ða1 ! xÞ ÞÞÞ Þ so that bm ! ( ! (b1 ! (an ! ( ! (a1 ! x) ))) ) = 1. This completes the proof. h Note that 1L is always a fuzzy filter of L containing any fuzzy set in L, and the intersection of any family of fuzzy filters of L is also a fuzzy filter of L. Hence we can define a fuzzy filter generated by a fuzzy set as follows. Let c be a fuzzy set in L. A fuzzy filter l of L is said to be generated by c, denoted by hci, if it satisfies: • c 6 l, • (8m 2 FðLÞ) (c 6 m ) l 6 m) where FðLÞ is the set of all fuzzy filters of L. For every fuzzy sets l and m in L, we have • l 2 FðLÞ ) hli ¼ l. • l 6 m)hli 6 hmi.
Theorem 3.22. For a fuzzy set l in L, if we define a fuzzy set m in L by an ! ð ! ða1 ! xÞ Þ ¼ 1; mðxÞ :¼ sup minflða1 Þ; . . . ; lðan Þg a1 ; . . . ; an 2 L then m = hli.
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Proof. Let x, a, b 2 L be such that a 6 b ! x. For every e > 0, we can take a1, . . . , an, b1, . . . , bm 2 L such that an ! ð ! ða1 ! aÞ Þ ¼ 1; bm ! ð ! ðb1 ! bÞ Þ ¼ 1; mðaÞ e < minflða1 Þ; . . . ; lðan Þg; mðbÞ e < minflðb1 Þ; . . . ; lðbm Þg: Using Lemma 3.21, we have bm ! ð ! ðb1 ! ðan ! ð ! ða1 ! aÞ ÞÞÞ Þ ¼ 1; and so mðxÞ P minflða1 Þ; . . . ; lðan Þ; lðb1 Þ; . . . ; lðbm Þg ¼ minfminflða1 Þ; . . . ; lðan Þg; minflðb1 Þ; . . . ; lðbm Þgg > minfmðaÞ e; mðbÞ eg ¼ minfmðaÞ; mðbÞg e: Since e is arbitrary, it follows that m(x) P min{m(a), m(b)}. Hence m is a fuzzy filter of L by Theorem 3.4. Since x ! x = 1 for all x 2 L, we have l(x) 6 m(x), i.e., l 6 m. Let c be a fuzzy filter of L such that l 6 c. Then ( ) an ! ð ! ða1 ! xÞ Þ ¼ 1; mðxÞ ¼ sup minflða1 Þ; . . . ; lðan Þg a1 ; . . . ; an 2 L ( ) an ! ð ! ða1 ! xÞ Þ ¼ 1; 6 sup minfcða1 Þ; . . . ; cðan Þg a1 ; . . . ; an 2 L 6 supfcðxÞg by Corollary 3.5 ¼ cðxÞ for all x 2 L, that is, m 6 c. Hence m = hli. h For any li 2 FðLÞ, where i 2 J and J is any index set, we define fli ji 2 J g ¼ hsupfli ji 2 J gi:
Theorem 3.23. For any l; m 2 FðLÞ and x 2 L, we have ðfl; mgÞðxÞ :¼ supffminflðaÞ; mðbÞg : a ! ðb ! xÞ ¼ 1g [flðaÞ : a ! x ¼ 1g [ fmðbÞ : b ! x ¼ 1gg:
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Proof. Using Theorem 3.22, we know that ðfl; mgÞðxÞ ¼ supfminfsupflða1 Þ; mða1 Þg; . . . ; supflðan Þ; mðan Þg: an ! ð ! ða1 ! xÞ Þ ¼ 1; a1 ; . . . ; an 2 Lg Let e > 0 be arbitrary and suppose that there exist a1, . . . , an, b1, . . . , bm 2 L such that bm ! ð ! ðb1 ! ðan ! ð ! ða1 ! xÞ ÞÞÞ Þ ¼ 1; ðfl; mgÞðxÞ < e þ minfsupflða1 Þ; mða1 Þg; . . . ; supflðan Þ; mðan Þg; supflðb1 Þ; mðb1 Þg; . . . ; supflðbm Þ; mðbm Þgg; lðai Þ P mðai Þ;
lðbj Þ 6 mðbj Þ;
i ¼ 1; . . . ; n; j ¼ 1; . . . ; m:
ð11Þ ð12Þ ð13Þ
Then ðfl; mgÞðxÞ < e þ minflða1 Þ; . . . ; lðan Þ; mðb1 Þ; . . . ; mðbm Þg: If we take b = an ! ( ! (a1 ! x) ) and a = b ! x, then bm ! ð ! ðb1 ! bÞ Þ ¼ 1; an ! ð ! ða1 ! aÞ Þ ¼ 1; b ! ða ! xÞ ¼ 1: It follows from Corollary 3.5 that mðbÞ P minfmðb1 Þ; . . . ; mðbm Þg
and
lðaÞ P minflða1 Þ; . . . ; lðan Þg
so that ({l, m})(x) < e + min{l(a), m(b)}. Now assume that there exist a1, . . . , an 2 L such that an ! ð ! ða1 ! xÞ Þ ¼ 1; ðfl; mgÞðxÞ < e þ minfsupflða1 Þ; mða1 Þg; . . . ; supflðan Þ; mðan Þgg; lðai Þ P mðai Þ; i ¼ 1; . . . ; n: Then ðfl; mgÞðxÞ < e þ minflða1 Þ; . . . ; lðan Þg: Since l(x) P min{l(a1), . . . , l(an)} by Corollary 3.5, we have ðfl; mgÞðxÞ < e þ lðxÞ: Finally suppose that there exist b1, . . . , bm 2 L such that bm ! ð. . . ! ðb1 ! xÞ . . .Þ ¼ 1;
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ðfl; mgÞðxÞ < e þ minfsupflðb1 Þ; mðb1 Þg; . . . ; supflðbm Þ; mðbm Þgg; lðbi Þ 6 mðbi Þ; i ¼ 1; . . . ; m: Then ðfl; mgÞðxÞ < e þ minfmðb1 Þ; . . . ; mðbm Þg 6 e þ mðxÞ: Summarizing the above results induces ðfl; mgÞðxÞ 6 supffminflðaÞ; mðbÞg : a ! ðb ! xÞ ¼ 1g [ flðaÞ : a ! x ¼ 1g [ fmðbÞ : b ! x ¼ 1gg: Next since minflðaÞ; mðbÞg 6 minfsupflðaÞ; mðaÞg; supflðbÞ; mðbÞgg; we have supfminflðaÞ; mðbÞg : a ! ðb ! xÞ ¼ 1g 6 supfminfsupflða1 Þ; mða1 Þg; . . . ; supflðan Þ; mðan Þgg : an ! ð ! ða1 ! xÞ Þ ¼ 1g: Similarly we get supflðaÞ : a ! x ¼ 1g 6 supfminfsupflða1 Þ; mða1 Þg; . . . ; supflðan Þ; mðan Þgg : an ! ð ! ða1 ! xÞ Þ ¼ 1g and supfmðbÞ : b ! x ¼ 1g 6 supfminfsupflða1 Þ; mða1 Þg; . . . ;
supflðan Þ; mðan Þgg : an
! ð. . . ! ða1 ! xÞ . . .Þ ¼ 1g: Therefore supffminflðaÞ;mðbÞg : a ! ðb ! xÞ ¼ 1g [ flðaÞ : a ! x ¼ 1g [ fmðbÞ : b ! x ¼ 1gg 6 supfminfsupflða1 Þ;mða1 Þg;...;supflðan Þ;mðan Þgg : an ! ð ! ða1 ! xÞÞ ¼ 1g ¼ ðfl;mgÞðxÞ: This completes the proof.
h
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Theorem 3.24. For a fixed element a 2 L, let la be a fuzzy set in L defined by t if a 6 x; la ðxÞ :¼ s otherwise; where t > s in [0, 1]. Then la is a fuzzy filter of L if and only if it satisfies the following inequality. ð8x; y 2 LÞða 6 y ! x; a 6 y ) a 6 xÞ:
ð14Þ
Proof. Assume that la is a filter of L and let x, y 2 L be such that a 6 y ! x and a 6 y. Then la(y ! x) = t = la(y), and thus la ðxÞ P minfla ðy ! xÞ; la ðyÞg ¼ t which implies that la(x) = t, that is, a 6 x. Conversely, suppose that (14) is valid. Note that la[s] = L and la[t] = {x 2 Lja 6 x}. Obviously 1 2 la[t]. Let x, y 2 L be such that x 2 la[t] and x ! y 2 la[t]. Then a 6 x and a 6 x ! y, which imply from the hypothesis that a 6 y, that is, y 2 la[t]. Hence la[t] is a filter of L. It follows from Lemma 3.7 that la is a fuzzy filter of L. h Theorem 3.25. For a fuzzy set l in L, let lsup be a fuzzy set in L defined by lsup ðxÞ :¼ supft 2 ½0; 1jx 2 hl½tig for all x 2 L. Then lsup is the least fuzzy filter of L that contains l, where hl[t]i means the least filter of L containing l[t]. Proof. For any s 2 Im(lsup), let sn ¼ s 1n for some n 2 N. Let x 2 lsup[s]. Then lsup(x) P s, which implies that supft 2 ½0; 1jx 2 hl½tig P s > s
1 ¼ sn ; 8n 2 N: n
Hence there exists r 2 {t 2 [0, 1]jx 2 hl[t]i} such that r > sn. Thus T l[r] l[sn], and so x 2 hl[r]i hl[sn]i for all n 2 N. Consequently x 2 n2N hl½sn i. On T the other hand, if x 2 n2N hl½sn i, then sn 2 {t 2 [0, 1]jx 2 hl[t]i} for any n 2 N. Therefore s
1 ¼ sn 6 supft 2 ½0; 1jx 2 hl½tig ¼ lsup ðxÞ; 8n 2 N: n
Since n is T arbitrary, it follows that s < lsup(x) so that x 2 lsup[s]. Hence lsup ½s ¼ n2N hl½sn i, which is a filter of L. Therefore we conclude that lsup is a fuzzy filter of L by Lemma 3.7. We now prove that lsup contains l. For any x 2 L, let s 2 {t 2 [0, 1]jx 2 l[t]}. Then x 2 l[s] and thus x 2 hl[s]i. Therefore s 2 {t 2 [0, 1]jx 2 hl[t]i}, which implies that ft 2 ½0; 1jx 2 l½tg ft 2 ½0; 1jx 2 hl½tig:
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It follows that lðxÞ ¼ supft 2 ½0; 1jx 2 l½tg 6 supft 2 ½0; 1jx 2 hl½tig ¼ lsup ðxÞ; which shows that lsup contains l. Finally let m be a fuzzy filter of L containing l. Let x 2 L. If lsup(x) =T 0, then clearly lsup(x) 6 m(x). Assume that lsup(x) = s 5 0. Then x 2 lsup ½s ¼ n2N hl½sn i, and so x 2 hl[sn]i for all n 2 N. It follows that mðxÞ P lðxÞ P sn ¼ s 1n for every n 2 N so that m(x) P s = lsup(x) since n is arbitrary. This shows that lsup m. This completes the proof. h Let l be a fuzzy filter of L and a 2 [0, 1]. Consider a relation on L as follows. la :¼ fðx; yÞjlðx ! yÞ > a; lðy ! xÞ > ag: Lemma 3.26. Let l be a fuzzy filter of L and a 2 [0, 1]. If la 5 ;, then l(1) > a. Proof. If la 5 ;, then l(x ! y) > a for some (x, y) 2 L · L. It follows from (b8) that l(1) P l(x ! y) > a. This completes the proof. h Proposition 3.27. Let l be a fuzzy filter of L and a 2 [0, 1). Then either la = ;, or la is an equivalence relation on L. Proof. Assume that la 5 ;. Since l(x ! x) = l(1) > a for all x 2 L, we have (x, x) 2 la, that is, la is reflexive. Obviously, la is symmetric. Let x, y, z 2 L be such that (x, y) 2 la and (y, z) 2 la. Then l(x ! y) > a, l(y ! x) > a, l(y ! z) > a and l(z ! y) > a. Since x ! y 6 (y ! z) ! (x ! z), it follows from (b7) that l((y ! z) ! (x ! z)) P l(x ! y) > a so from (b9) that lðx ! zÞ P minflðy ! zÞ; lððy ! zÞ ! ðx ! zÞÞg > a: Similarly we get l(z ! x) > a, and so (x, z) 2 la. Therefore la is transitive.
h
Proposition 3.28. Let l be a fuzzy filter of L and a 2 [0, 1) be such that la 5 ;. Then (i) ("x, y, z 2 L) ((x, y) 2 la )(x ! z, y ! z) 2 la,(z ! x, z ! y) 2 la). (ii) ("x, y, a, b 2 L) ((x, y) 2 la,(a, b) 2 la ) (x ! a, y ! b) 2 la). Proof. (i) Let x, y, z 2 L be such that (x, y) 2 la. Since y ! x 6 (x ! z) ! (y ! z), it follows from (b7) that l((x ! z) ! (y ! z)) P l(y ! x) > a.
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Similarly we get l((y ! z) ! (x ! z)) > a. Hence (x ! z, y ! z) 2 la. Since x ! y 6 (z ! x) ! (z ! y), (b7) implies that a < lðx ! yÞ 6 lððz ! xÞ ! ðz ! yÞÞ: Similarly we have l((z ! y) ! (z ! x)) > a. Therefore (z ! x, z ! y) 2 la. (ii) follows from (i) and the transitivity of la. h Proposition 3.29. Let l be a fuzzy filter of L and a 2 [0, 1). If l(1) > a, then la is a congruence relation on L. Proof. Assume that l(1) > a for a 2 [0,1). Then l(x ! x) = l(1) > a for all x 2 L, and so (x, x) 2 la. Hence la 5 ;, which implies from Propositions 3.27 and 3.28 that la is a congruence relation on L. h Corollary 3.30. Let l be a fuzzy filter of L and a 2 [0, 1). Then either la = 0, or la is a congruence relation on L. We call la a congruence relation on L induced by l. Let l be a fuzzy filter of L and let CR(l) denote the set of all congruence relations on L induced by l, that is, CRðlÞ :¼ fla ja 2 ½0; 1Þg: Then CR(l) is a poset under the set inclusion . Denote by (CR(l), _, ^) the lattice induced by the partial order . The following proposition is straightforward. Proposition 3.31. For every fuzzy filter l of L, we have the following assertions. ("a, b 2 [0, l(1)]) (a P b ) la lb). (CR(l), _, ^) is a chain. ("a, b 2 [0, l(1)]) (la _ lb = lmin{a, b}, la ^ lb = lmax{a, b}). If l(x) 5 0 for all x 2 L, then la = L · L for every a 2 [0, 1] with a < inf{l(x) jx 2 L}. (v) If l(x) 5 1 for all x 2 L, then lb = ; for every b 2 [0, 1] with b P sup{l(x)jx 2 L}.
(i) (ii) (iii) (iv)
Proposition 3.32. Let l be a fuzzy filter of L and C [0, l(1)]. Then linf(C) = sup{laja 2 C} and lsup(C) = inf{lbjb 2 C}. Proof. Since infC 6 a for any a 2 C, we have la linf(C) for all a 2 C. Hence linf(C) is an upper bound of {laja 2 C}. Let c 2 [0, l(1)] be such that lc linf(C). Then there exists (x, y) 2 L · L such that (x, y) 2 linf(C)nlc, and so
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l(x ! y) > inf(C) and l(y ! x) > inf(C), but l(x ! y) 6 c or l(y ! x) 6 c. It follows that there exists d 2 C such that infðCÞ 6 d < minflðx ! yÞ; lðy ! xÞg so that (x, y) 2 ld. Since (x, y) 62 lc, it follows that lc ld. This shows that lc is not an upper bound of {laja 2 C}. Hence sup{laja 2 C} = linf(C). Similarly we can prove that the equality lsup(C) = inf{lbjb 2 C}. This completes the proof. h Corollary 3.33. (CR(l), _, ^) is a completely distributive lattice.
4. Conclusions We gave characterizations of fuzzy filters, and investigated further properties of fuzzy filters in MTL-algebras. We considered the fuzzification of Boolean filters and MV-filters, and studied related properties. We provided a condition for a fuzzy filter to be Boolean. We discussed a method to generate a fuzzy filter by a fuzzy set. Given a fuzzy filter, we stated a congruence relation and showed that the set of all congruence relations induced by a fuzzy filter is a completely distributive lattice. Future research will focus on constructing a quotient MTL-algebra by using a fuzzy filter, on studying prime fuzzy filter, on establishing a fuzzy prime spectrum of a MTL-algebra, and on finding intuitionistic and/or intervalvalued fuzzy structures of a (prime) filter in MTL-algebras.
Acknowledgments The first author, Y. B. Jun, was supported by Korea Research Foundation Grant (KRF-2003-005-C00013). The authors are highly grateful to referees for their valuable comments and suggestions for improving the paper.
References [1] F. Esteva, L. Godo, Monoidal t-norm based logic: towards a logic for left-continuous t-norms, Fuzzy Sets and Systems 124 (2001) 271–288. [2] P. Ha´jek, Metamathematics of Fuzzy Logic, Kluwer Academic Press, Dordrecht, 1998. [3] K.H. Kim, Q. Zhang, Y.B. Jun, On fuzzy filters of MTL-algebras, Journal of Fuzzy Mathematics 10 (4) (2002) 981–989. [5] E. Turunen, BL-algebras of basic fuzzy logic, Mathware and Soft Computing 6 (1999) 49–61. [6] X.H. Zhang, On filters in MTL-algebras, submitted for publication.
Fuzzy filters of MTL-algebras Young Bae Jun
a,*
, Yang Xu b, Xiao Hong Zhang
c
a
Department of Mathematics Education, Gyeongsang National University, Chinju 660-701, South Korea b Department of Applied Mathematics, Southwest Jiaotong University, Chengdu 610031, Sichuan, China c Department of Mathematics, The Faculty of Science, Ningbo University, Ningbo 315211, Zhejiang, China Received 19 August 2004; received in revised form 14 November 2004; accepted 15 November 2004
Abstract Characterizations of fuzzy filters in MTL-algebras are given. The fuzzy filter generated by a fuzzy set is considered. The notion of Boolean fuzzy filters and MV-fuzzy filters are introduced and related properties are investigated. A condition for a fuzzy filter to be Boolean is provided. A characterization of a Boolean fuzzy filter is given. A congruence relation on a MTL-algebra induced by a fuzzy filter is established, and we show that the set of all congruence relations induced by a fuzzy filter is a completely distributive lattice. 2004 Elsevier Inc. All rights reserved. Keywords: MTL-algebra; (Boolean) fuzzy filter; MV-fuzzy filter
*
Corresponding author. Tel.: +82 55 751 5674; fax: +82 55 751 6117. E-mail addresses: [email protected] (Y.B. Jun), [email protected] (Y. Xu), zxhonghz@263. net (X.H. Zhang). 0020-0255/$ - see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2004.11.004
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1. Introduction The logic MTL, Monoidal t-norm based logic, was introduced by Esteva and Godo [1]. This logic is very interesting from many points of view. From the logic point of view, it can be regarded as a weak system of Fuzzy Logic. Indeed, it arises from Ha´jekÕs Basic Logic BL [2] by replacing the axiom ^ ! BÞÞ $ ðA ^ BÞ ðA^ðA by the weaker axiom ^ ! BÞÞ ! ðA ^ BÞ: ðA^ðA In connection with the logic MTL, Esteva and Godo [1] introduced a new algebra, called a MTL-algebra, and studied several basic properties. They also introduced the notion of (prime) filters in MTL-algebras. Zhang [6] studied further properties of filters in MTL-algebras, and introduced the notion of Boolean filters and MV-filters. Based on the fuzzy set theory, Kim et al. [3] studied the fuzzy structure of filters in MTL-algebras. As a continuation of the paper [3], we give characterizations of fuzzy filters in MTL-algebras, and investigate further properties of fuzzy filters in MTL-algebras. We consider the fuzzification of Boolean filters and MV-filters, and study related properties. We discuss the fuzzy filter generated by a fuzzy set. We provide a condition for a fuzzy filter to be Boolean. Using a fuzzy filter, we give a congruence relation on a MTL-algebra, and show that the set of all congruence relations induced by a fuzzy filter is a completely distributive lattice.
2. Preliminaries By a residuated lattice we shall mean a lattice L = (L, 6, ^, _, , !, 0, 1) containing the least element 0 and the largest element 1, and endowed with two binary operations (called product) and ! (called residuum) such that • is associative, commutative and isotone. • ("x 2 L) (x 1 = x). • The Galois correspondence holds, that is, ð8x; y; z 2 LÞðx y 6 z () x 6 y ! zÞ: In a residuated lattice, the following are true (see [5]): (a1) x 6 y () x ! y ¼ 1. (a2) 0 ! x = 1, 1 ! x = x, x ! (y ! x) = 1. (a3) y 6 (y ! x) ! x.
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(a4) (a5) (a6) (a7)
x ! (y ! z) = (x y) ! z = y ! (x ! z). x ! y 6 (z ! x) ! (z ! y), (x ! y) 6 (y ! z) ! (x ! z). y 6 x ) x ! z 6 y ! z, z ! y 6 z ! x. (¤i 2 Cyi) ! x = §i2C(yi ! x).
We define x* = ¤{y 2 Ljx y = 0}, equivalently, x* = x ! 0. Then (a8) 0* = 1, 1* = 0, x 6 x**, and x* = x***. Based on the Ha´jekÕs results [2], Axioms of MTL and Formulas which are provable in MTL, Esteva and Godo [1] defined the algebras, so called MTLalgebras, corresponding to the MTL-logic in the following way. Definition 2.1. A MTL-algebra is a residuated lattice L = (L, 6, ^, _, , !, 0, 1) satisfying the pre-linearity equation: ðx ! yÞ _ ðy ! xÞ ¼ 1: In a MTL-algebra, the following are true (see [6]): (a9) x ! (y _ z) = (x ! y) _ (x ! z). (a10) x y 6 x ^ y. Definition 2.2 [1]. Let L be a MTL-algebra. A nonempty subset F of L is called a filter of L if it satisfies (b1) ("x, y 2 F) (x y 2 F). (b2) ("x 2 F) ("y 2 L) (x 6 y ) y 2 F). Since ^ is not definable from and ! in a MTL-algebra, one could consider that the further condition (b3) ("x, y 2 F) (x ^ y 2 F) should be also required for a filter. However the condition (b3) is indeed redundant because it is a consequence of conditions (b1) and (b2). Namely, since x y 6 x ^ y, if x, y 2 F the x y 2 F and thus x ^ y 2 F as well. Proposition 2.3 [6]. A nonempty subset F of a MTL-algebra L is a filter of L if and only if it satisfies: (b4) 1 2 F. (b5) ("x 2 F) ("y 2 L) (x ! y 2 F ) y 2 F).
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Proof. Let F be a nonempty subset of L satisfying conditions (b4) and (b5). Let x, y 2 L be such that x 6 y and x 2 F. Then x ! y = 1 2 F by (b4), and so y 2 F by (b5). Thus (b2) is valid. Let x, y 2 F. Then x ! ðy ! ðx yÞÞ ¼ ðx yÞ ! ðx yÞ ¼ 1 2 F ; which implies from (b5) that x y 2 F. Hence (b1) holds. Conversely, suppose that F is a filter of L. Obviously 1 2 F by (b2). Let x, y 2 L be such that x 2 F and x ! y 2 F. Then x (x ! y) 2 F by (b1). Using (a3), we have x 6 (x ! y) ! y, i.e., x ! ((x ! y) ! y) = 1 which implies from (a4) that (x (x ! y)) ! y = 1, i.e., (x (x ! y)) 6 y. It follows from (b2) that y 2 F so that (b5) is valid. This completes the proof. h
3. Fuzzy filters In what follows let L denote a MTL-algebra unless otherwise specified. In [3], Kim et al. defined the fuzzy filter in MTL-algebras as follows. Definition 3.1 [3]. A fuzzy set l in L is called a fuzzy filter of L if it satisfies (b6) ("x, y 2 L) (l(x y) P min{l(x), l(y)}). (b7) l is order-preserving, that is, ð8x; y 2 LÞðx 6 y ) lðxÞ 6 lðyÞÞ: Example 3.2. (1) Let L = [0, 1] and define a product and a residuum ! on L as follows: 1 if x 6 y; x ^ y if x þ y > 12 ; x y :¼ x ! y :¼ ð0:5 xÞ _ y if x > y; 0 otherwise; for all x, y 2 L. Then L is a MTL-algebra. Let l be a fuzzy set in L given by a if x 2 ð0:5; 1; lðxÞ :¼ b otherwise; where a > b in [0, 1]. Then it is routine to verify that l is a fuzzy filter of L. (2) Let L = [0, 1] and define a product and a residuum ! on L as follows: x ^ y if x þ y > 1; 1 if x 6 y; x y :¼ x ! y :¼ 0 otherwise; ð1 xÞ _ y otherwise; for all x, y 2 L. Then L is a MTL-algebra. Let l1 and l2 be fuzzy sets in L given by
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l1 ðxÞ :¼ l2 ðxÞ :¼
a
if x 2 ½0; 0:5;
a þ 0:5x2
if x 2 ð0:5; 1;
a
if x 2 ½0; 0:5;
a þ 0:3x if x 2 ð0:5; 1;
where a 2 [0, 0.5). Then l1 and l2 are fuzzy filters of L. We first give characterizations of a fuzzy filter. Theorem 3.3. A fuzzy set l in L is a fuzzy filter of L if and only if it satisfies (b8) ("x 2 L) (l(1) P l(x)). (b7) ("x, y 2 L) (l(y) P min{l(x), l(x ! y)}). Proof. Assume that l satisfies conditions (b8) and (b9). Let x, y 2 L be such that x 6 y. Then x ! y = 1, and so lðyÞ P minflðxÞ; lð1Þg ¼ lðxÞ by (b8) and (b9). Using (a4), we know that x ! ðy ! ðx yÞÞ ¼ ðx yÞ ! ðx yÞ ¼ 1: It follows from (b8) and (b9) that lðx yÞ P minflðyÞ; lðy ! ðx yÞÞg P minflðyÞ; minflðxÞ; lðx ! ðy ! ðx yÞÞÞgg ¼ minflðyÞ; minflðxÞ; lð1Þgg ¼ minflðxÞ; lðyÞg: Thus (b6) is valid. Suppose that l is a fuzzy filter of L. Since x 6 1 for all x 2 L it follows from (b7) that l(1) P l(x) for all x 2 L. Let x, y 2 L. Since x 6 (x ! y) ! y, we have x (x ! y) 6 y, by the Galois correspondence. Hence lðyÞ P lðx ðx ! yÞÞ P minflðxÞ; lðx ! yÞg by (b7) and (b6). This completes the proof.
h
Theorem 3.4. A fuzzy set l in L is a fuzzy filter of L if and only if it satisfies ð8a; b; c 2 LÞða 6 b ! c ) lðcÞ P minflðaÞ; lðbÞgÞ:
ð1Þ
Proof. Suppose that l is a fuzzy filter of L. Let a, b, c 2 L be such that a 6 b ! c. Then l(a) 6 l(b ! c) by (b7), and so lðcÞ P minflðbÞ; lðb ! cÞg P minflðbÞ; lðaÞg
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by (b9). Conversely let l be a fuzzy set in L satisfying (1). Since x 6 x ! 1 for all x 2 L, it follows from (1) that lð1Þ P minflðxÞ; lðxÞg ¼ lðxÞ for all x 2 L. Since x ! y 6 x ! y for all x, y 2 L, we have lðyÞ P minflðxÞ; lðx ! yÞg for all x, y 2 L. Hence l is a fuzzy filter of L. h Corollary 3.5. A fuzzy set l in L is a fuzzy filter of L if and only if it satisfies the following assertion: lðxÞ P minflða1 Þ; . . . ; lðan Þg
ð2Þ
whenever an ! ( ! (a1 ! x) ) = 1 for every a1, , an 2 L. Proof. It can be easily proved by induction.
h
Theorem 3.6. For a filter F of L and a 2 L, let l be a fuzzy set in L defined by s if x 2 fz 2 Lja _ z 2 F g; lðxÞ :¼ t otherwise; for all x 2 L where s > t in [0, 1]. Then l is a fuzzy filter of L. Proof. Since a _ 1 2 F, we have 1 2 {z 2 Lja _ z 2 F} and so l(1) = s P l(x) for all x 2 L. Now if y 2 {z 2 Lja _ z 2 F}, then clearly lðyÞ ¼ s P minflðxÞ; lðx ! yÞg: Suppose that y 62 {z 2 Lja_z 2 F}. Then at least one of x and x ! y does not belong to {z 2 Lja _ z 2 F}. Hence l(y) = min{l(x), l(x ! y)}, and therefore l is a fuzzy filter of L. h Lemma 3.7 [3]. A fuzzy set l in L is a fuzzy filter of L if and only if the level set l½t :¼ fx 2 LjlðxÞ P t; t 2 ½0; 1g is either empty or a filter of L. Theorem 3.8. If l is a fuzzy filter of L, then the set Xa :¼ fx 2 LjlðxÞ P lðaÞg is a filter of L for every a 2 L.
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Proof. Since l(1) P (x) for all x 2 L. we have 1 2 Xa. Let x, y 2 L be such that x 2 Xa and x ! y 2 Xa. Then l(x) P l(a) and l(x ! y) P l(a). Since l is a fuzzy filter of L; it follows from (b9) that lðyÞ P minflðxÞ; lðx ! yÞg P lðaÞ so that y 2 Xa. Hence Xa is a filter of L.
h
Theorem 3.9. Let a 2 L and let l be a fuzzy set in L. Then (i) If Xa is a filter of L, then l satisfies the following implication ð8x; y 2 LÞðlðaÞ 6 minflðx ! yÞ; lðxÞg ) lðaÞ 6 lðyÞÞ:
ð3Þ
(ii) If l satisfies (b8) and (3), then Xa is a filter of L. Proof. (i) Assume that Xa is a filter of L. Let x, y 2 L be such that lðaÞ 6 minflðx ! yÞ; lðxÞg: Then x ! y 2 Xa and x 2 Xa. Using (b5), we have y 2 Xa and so l(y) P l(a). (ii) Suppose that l satisfies (b8) and (3). From (b8) it follows that 1 2 Xa. Let x, y 2 L be such that x 2 Xa and x ! y 2 Xa. Then l(a) 6 l(x) and l(a) 6 l(x ! y), which imply that l(a) 6 min{l(x), l(x ! y)}. Thus l(a) 6 l(y) by (3), and so y 2 Xa. Therefore Xa is a filter of L. h Proposition 3.10. Let l be a fuzzy filter of L. Then the following are equivalent. (i) ("x, y, z 2 L) (l (x ! z) P min{l (x ! (y ! z)), l(x ! y)}). (ii) ("x, y 2 L) (l(x ! y) P l(x ! (x ! y))). (iii) ("x, y, z 2 L) (l((x ! y) ! (x ! z)) P (x ! (y ! z))). Proof. (i) ) (ii) Suppose that l satisfies the condition (i). Taking z = y and y = x in (i) and using (b8), we have lðx ! yÞ P minflðx ! ðx ! yÞÞ; lðx ! xÞg ¼ minflðx ! ðx ! yÞÞ; lð1Þg ¼ lðx ! ðx ! yÞÞ for all x, y, z 2 L. (ii) ) (iii) Suppose that l satisfies the condition (ii) and let x, y, z 2 L. Since x ! (y ! z) 6 x ! ((x ! y) ! (x ! z)), it follows that
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lðx ! yÞ ! ðx ! zÞ ¼ lðx ! ððx ! yÞ ! zÞÞ P lðx ! ðx ! ððx ! yÞ ! zÞÞÞ ¼ lðx ! ððx ! yÞ ! ðx ! zÞÞÞ P lðx ! ðy ! zÞÞ: (iii) ) (i) If l satisfies the condition (iii), then lðx ! yÞ P minflððx ! yÞ ! ðx ! zÞÞ; lðx ! yÞg P minflðx ! ðy ! zÞÞ; lðx ! yÞg: h
This completes the proof.
A fuzzy filter l of L is said to be Boolean if it satisfies the following equality ðb10Þ ð8x 2 LÞðlðx _ x Þ ¼ lð1ÞÞ: Proposition 3.11. Let l and m be fuzzy filters of L such that l 6 m and l(1) = m(1). If l is Boolean, then so is m. Proof. Straightforward.
h
Proposition 3.12. Every Boolean fuzzy filter l of L satisfies the following inequality: ð8x; y; z 2 LÞðlðx ! zÞ P minflðx ! ðz ! yÞÞ; lðy ! zÞgÞ: Proof. Using (a5), we have y ! z 6 ðz ! yÞ ! ðz ! zÞ 6 ðx ! ðz ! yÞÞ ! ðx ! ðz ! zÞÞ: It follows from (b7) that lðy ! zÞ 6 lððx ! ðz ! yÞÞ ! ðx ! ðz ! zÞÞÞ so from (b9) that lðx ! ðz ! zÞÞ P minflðx ! ðz ! yÞÞ; lððx ! ðz ! yÞÞ ! ðx ! ðz ! zÞÞÞg P minflðx ! ðz ! yÞÞ; lðy ! zÞg: Since z _ z ¼ ððz ! zÞ ! zÞ ^ ððz ! z Þ ! z Þ 6 ðz ! zÞ ! z; we have l((z* ! z) ! z) P l(z*_z) = l(1). Since x ! ðz ! zÞ 6 ððz ! zÞ ! zÞ ! ðx ! zÞ;
ð4Þ
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it follows from (b7) that lðx ! ðz ! zÞÞ 6 lðððz ! zÞ ! zÞ ! ðx ! zÞÞ: Thus lðx ! zÞ P minflððz ! zÞ ! zÞ; lðððz ! zÞ ! zÞ ! ðx ! zÞÞg P minflð1Þ; lðx ! ðz ! zÞÞg ¼ lðx ! ðz ! zÞÞ P minflðx ! ðz ! yÞÞ; lðy ! zÞg: This completes the proof.
h
Proposition 3.13. If a fuzzy filter l of L satisfies the following inequality ð8x; y 2 LÞðlðxÞ P lððx ! yÞ ! xÞÞ;
ð5Þ
then it is Boolean. Proof. Using (a2), (a4) and (a5), we have
1 ¼ x ! ððx ! xÞ ! xÞ 6 ððx ! xÞ ! xÞ ! x 6 ðx ! xÞ ! ðððx ! xÞ ! xÞ ! xÞ
¼ ððx ! xÞ ! xÞ ! ððx ! xÞ ! xÞ ¼ ðððx ! xÞ ! xÞ ! 0Þ ! ððx ! xÞ ! xÞ: It follows from (b7), (b8) and (5) that lððx ! xÞ ! xÞ P lððððx ! xÞ ! xÞ ! 0Þ ! ððx ! xÞ ! xÞÞ ¼ lð1Þ: Using (a7) and (a9), since ðx ! xÞ ! x 6 ððx ! xÞ ! xÞ _ ððx ! xÞ ! x Þ ¼ ðx ! xÞ ! ðx _ x Þ ¼ ð1 ^ ðx ! xÞÞ ! ðx _ x Þ ¼ ððx ! xÞ ^ ðx ! xÞÞ ! ðx _ x Þ ¼ ððx _ x Þ ! xÞ ! ðx _ x Þ; we get l(1) = l((x* ! x) ! x) 6 l(((x_x*) ! x) ! (x_x*)) 6 l(x_x*), and so l(x_x*) = l(1). Therefore l is Boolean. h Proposition 3.14. Let l be a fuzzy filter of L that satisfies the condition (4). Then l satisfies the condition (5).
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Proof. Since (x ! y) ! x 6 x* ! x and l is order-preserving, it follows that lðxÞ ¼ lð1 ! xÞ P minflð1 ! ðx ! x ÞÞ; lðx ! xÞg P minflð1Þ; lððx ! yÞ ! xÞg ¼ lððx ! yÞ ! xÞ: Hence l satisfies the condition (5). h Combining Propositions 3.12, 3.13 and 3.14, we have a characterization of a Boolean fuzzy filter. Theorem 3.15. Let l be a fuzzy filter of L. Then the following assertions are equivalent: (i) l is Boolean. (ii) ("x, y, z 2 L) (l(x ! z) P min{l(x ! (z* ! y)), l(y ! z)}). (iii) ("x, y 2 L) (l(x) P l((x ! y) ! x)). Proposition 3.16. If a fuzzy filter l of L satisfies (5), then it satisfies the following inequality ð8x; y; z 2 LÞðlðx ! zÞ P minflðx ! ðy ! zÞÞ; lðx ! yÞgÞ:
ð6Þ
Proof. Since x ! (y ! z) = y ! (x ! z) 6 (x ! y) ! (x ! (x ! z)), it follows from (b7) that lðx ! ðy ! zÞÞ 6 lððx ! yÞ ! ðx ! ðx ! zÞÞÞ so from (b9) that lðx ! ðx ! zÞÞ P minflðx ! yÞ; lððx ! yÞ ! ðx ! ðx ! zÞÞÞg P minflðx ! yÞ; lðx ! ðy ! zÞÞg: Since x ! ðx ! zÞ 6 x ! ðððx ! zÞ ! zÞ ! zÞ ¼ ððx ! zÞ ! zÞ ! ðx ! zÞ; we have lðx ! zÞ P lðððx ! zÞ ! zÞ ! ðx ! zÞÞ P lðx ! ðx ! zÞÞ P minflðx ! yÞ; lðx ! ðy ! zÞÞg by using (b7) and (4). This completes the proof.
h
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Proposition 3.17. Every Boolean fuzzy filter l of L satisfies the following inequality. ð8x; y; z 2 LÞðlðx ! zÞ P minflðx ! ðy ! zÞÞ; lðx ! yÞgÞ: Proof. Since x ! (y ! z) = y ! (x ! z) 6 (x ! y) ! (x ! (x ! z)) and x ! (x ! z) 6 x ! (((x ! z) ! z) ! z) = ((x ! z) ! z) ! (x ! z), it follows from (b7), (b9) and Theorem 3.15 (iii) that lðx ! zÞ P lðððx ! zÞ ! zÞ ! ðx ! zÞÞ P lðx ! ðx ! zÞÞ P minflðx ! yÞ; lððx ! yÞ ! ðx ! ðx ! zÞÞÞg P minflðx ! yÞ; lðx ! ðy ! zÞÞg: This completes the proof.
h
Definition 3.18. A fuzzy set l in L is called a MV-fuzzy filter of L if it is a fuzzy filter of L that satisfies the following inequality. ðb11 Þ ð8x; y 2 LÞðlðx ! yÞ 6 lðððy ! xÞ ! xÞ ! yÞÞ: Theorem 3.19. In a MV-algebra, every fuzzy filter is a MV-fuzzy filter. Proof. Let l be a fuzzy filter of a MV-algebra L. Since x ! y 6 ððx ! yÞ ! yÞ ! y ¼ ððy ! xÞ ! xÞ ! y; we have l(x ! y) 6 l(((y ! x) ! x) ! y), and so l is a MV-fuzzy filter of L. h Theorem 3.20. Every Boolean fuzzy filter is a MV-fuzzy filter. Proof. Let l be a Boolean fuzzy filter of L. Since y 6 ((y ! x) ! x) ! y, we have ðððy ! xÞ ! xÞ ! yÞ ! x 6 y ! x
ð7Þ
by (a6). Using (a4), (a5), (a6) and (7), we get x ! y 6 ððy ! xÞ ! xÞ ! ððy ! xÞ ! yÞ ¼ ðy ! xÞ ! ðððy ! xÞ ! xÞ ! yÞ 6 ððððy ! xÞ ! xÞ ! yÞ ! xÞ ! ðððy ! xÞ ! xÞ ! yÞ
and so lðððy ! xÞ ! xÞ ! yÞ P lðððððy ! xÞ ! xÞ ! yÞ ! xÞ ! ðððy ! xÞ ! xÞ ! yÞÞ P lðx ! yÞ by Theorem 3.15 (iii) and (b7). Therefore l is a MV-fuzzy filter of L.
h
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Lemma 3.21. Suppose that L satisfies the following assertions. a ! ðb ! xÞ ¼ 1;
ð8Þ
an ! ð ! ða1 ! aÞ Þ ¼ 1;
ð9Þ
bm ! ð ! ðb1 ! bÞ Þ ¼ 1:
ð10Þ
Then bm ! ( ! (b1 ! (an ! ( ! (a1 ! a) ))) ) = 1. Proof. The condition (8) implies a 6 b ! x. Using (a6) and (9), we know that 1 ¼ an ! ð ! ða1 ! aÞ Þ 6 an ! ð ! ða1 ! ðb ! xÞÞ Þ; and so 1 ¼ an ! ð ! ða1 ! ðb ! xÞÞ Þ ¼ b ! ðan ! ð ! ða1 ! xÞ ÞÞ; that is, b 6 (an ! ( ! (a1 ! x) ). It follows from (a6) and (10) that 1 ¼ bm ! ð ! ðb1 ! bÞ Þ 6 bm ! ð ! ðb1 ! ðan ! ð ! ða1 ! xÞ ÞÞÞ Þ so that bm ! ( ! (b1 ! (an ! ( ! (a1 ! x) ))) ) = 1. This completes the proof. h Note that 1L is always a fuzzy filter of L containing any fuzzy set in L, and the intersection of any family of fuzzy filters of L is also a fuzzy filter of L. Hence we can define a fuzzy filter generated by a fuzzy set as follows. Let c be a fuzzy set in L. A fuzzy filter l of L is said to be generated by c, denoted by hci, if it satisfies: • c 6 l, • (8m 2 FðLÞ) (c 6 m ) l 6 m) where FðLÞ is the set of all fuzzy filters of L. For every fuzzy sets l and m in L, we have • l 2 FðLÞ ) hli ¼ l. • l 6 m)hli 6 hmi.
Theorem 3.22. For a fuzzy set l in L, if we define a fuzzy set m in L by an ! ð ! ða1 ! xÞ Þ ¼ 1; mðxÞ :¼ sup minflða1 Þ; . . . ; lðan Þg a1 ; . . . ; an 2 L then m = hli.
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Proof. Let x, a, b 2 L be such that a 6 b ! x. For every e > 0, we can take a1, . . . , an, b1, . . . , bm 2 L such that an ! ð ! ða1 ! aÞ Þ ¼ 1; bm ! ð ! ðb1 ! bÞ Þ ¼ 1; mðaÞ e < minflða1 Þ; . . . ; lðan Þg; mðbÞ e < minflðb1 Þ; . . . ; lðbm Þg: Using Lemma 3.21, we have bm ! ð ! ðb1 ! ðan ! ð ! ða1 ! aÞ ÞÞÞ Þ ¼ 1; and so mðxÞ P minflða1 Þ; . . . ; lðan Þ; lðb1 Þ; . . . ; lðbm Þg ¼ minfminflða1 Þ; . . . ; lðan Þg; minflðb1 Þ; . . . ; lðbm Þgg > minfmðaÞ e; mðbÞ eg ¼ minfmðaÞ; mðbÞg e: Since e is arbitrary, it follows that m(x) P min{m(a), m(b)}. Hence m is a fuzzy filter of L by Theorem 3.4. Since x ! x = 1 for all x 2 L, we have l(x) 6 m(x), i.e., l 6 m. Let c be a fuzzy filter of L such that l 6 c. Then ( ) an ! ð ! ða1 ! xÞ Þ ¼ 1; mðxÞ ¼ sup minflða1 Þ; . . . ; lðan Þg a1 ; . . . ; an 2 L ( ) an ! ð ! ða1 ! xÞ Þ ¼ 1; 6 sup minfcða1 Þ; . . . ; cðan Þg a1 ; . . . ; an 2 L 6 supfcðxÞg by Corollary 3.5 ¼ cðxÞ for all x 2 L, that is, m 6 c. Hence m = hli. h For any li 2 FðLÞ, where i 2 J and J is any index set, we define fli ji 2 J g ¼ hsupfli ji 2 J gi:
Theorem 3.23. For any l; m 2 FðLÞ and x 2 L, we have ðfl; mgÞðxÞ :¼ supffminflðaÞ; mðbÞg : a ! ðb ! xÞ ¼ 1g [flðaÞ : a ! x ¼ 1g [ fmðbÞ : b ! x ¼ 1gg:
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Proof. Using Theorem 3.22, we know that ðfl; mgÞðxÞ ¼ supfminfsupflða1 Þ; mða1 Þg; . . . ; supflðan Þ; mðan Þg: an ! ð ! ða1 ! xÞ Þ ¼ 1; a1 ; . . . ; an 2 Lg Let e > 0 be arbitrary and suppose that there exist a1, . . . , an, b1, . . . , bm 2 L such that bm ! ð ! ðb1 ! ðan ! ð ! ða1 ! xÞ ÞÞÞ Þ ¼ 1; ðfl; mgÞðxÞ < e þ minfsupflða1 Þ; mða1 Þg; . . . ; supflðan Þ; mðan Þg; supflðb1 Þ; mðb1 Þg; . . . ; supflðbm Þ; mðbm Þgg; lðai Þ P mðai Þ;
lðbj Þ 6 mðbj Þ;
i ¼ 1; . . . ; n; j ¼ 1; . . . ; m:
ð11Þ ð12Þ ð13Þ
Then ðfl; mgÞðxÞ < e þ minflða1 Þ; . . . ; lðan Þ; mðb1 Þ; . . . ; mðbm Þg: If we take b = an ! ( ! (a1 ! x) ) and a = b ! x, then bm ! ð ! ðb1 ! bÞ Þ ¼ 1; an ! ð ! ða1 ! aÞ Þ ¼ 1; b ! ða ! xÞ ¼ 1: It follows from Corollary 3.5 that mðbÞ P minfmðb1 Þ; . . . ; mðbm Þg
and
lðaÞ P minflða1 Þ; . . . ; lðan Þg
so that ({l, m})(x) < e + min{l(a), m(b)}. Now assume that there exist a1, . . . , an 2 L such that an ! ð ! ða1 ! xÞ Þ ¼ 1; ðfl; mgÞðxÞ < e þ minfsupflða1 Þ; mða1 Þg; . . . ; supflðan Þ; mðan Þgg; lðai Þ P mðai Þ; i ¼ 1; . . . ; n: Then ðfl; mgÞðxÞ < e þ minflða1 Þ; . . . ; lðan Þg: Since l(x) P min{l(a1), . . . , l(an)} by Corollary 3.5, we have ðfl; mgÞðxÞ < e þ lðxÞ: Finally suppose that there exist b1, . . . , bm 2 L such that bm ! ð. . . ! ðb1 ! xÞ . . .Þ ¼ 1;
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ðfl; mgÞðxÞ < e þ minfsupflðb1 Þ; mðb1 Þg; . . . ; supflðbm Þ; mðbm Þgg; lðbi Þ 6 mðbi Þ; i ¼ 1; . . . ; m: Then ðfl; mgÞðxÞ < e þ minfmðb1 Þ; . . . ; mðbm Þg 6 e þ mðxÞ: Summarizing the above results induces ðfl; mgÞðxÞ 6 supffminflðaÞ; mðbÞg : a ! ðb ! xÞ ¼ 1g [ flðaÞ : a ! x ¼ 1g [ fmðbÞ : b ! x ¼ 1gg: Next since minflðaÞ; mðbÞg 6 minfsupflðaÞ; mðaÞg; supflðbÞ; mðbÞgg; we have supfminflðaÞ; mðbÞg : a ! ðb ! xÞ ¼ 1g 6 supfminfsupflða1 Þ; mða1 Þg; . . . ; supflðan Þ; mðan Þgg : an ! ð ! ða1 ! xÞ Þ ¼ 1g: Similarly we get supflðaÞ : a ! x ¼ 1g 6 supfminfsupflða1 Þ; mða1 Þg; . . . ; supflðan Þ; mðan Þgg : an ! ð ! ða1 ! xÞ Þ ¼ 1g and supfmðbÞ : b ! x ¼ 1g 6 supfminfsupflða1 Þ; mða1 Þg; . . . ;
supflðan Þ; mðan Þgg : an
! ð. . . ! ða1 ! xÞ . . .Þ ¼ 1g: Therefore supffminflðaÞ;mðbÞg : a ! ðb ! xÞ ¼ 1g [ flðaÞ : a ! x ¼ 1g [ fmðbÞ : b ! x ¼ 1gg 6 supfminfsupflða1 Þ;mða1 Þg;...;supflðan Þ;mðan Þgg : an ! ð ! ða1 ! xÞÞ ¼ 1g ¼ ðfl;mgÞðxÞ: This completes the proof.
h
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Theorem 3.24. For a fixed element a 2 L, let la be a fuzzy set in L defined by t if a 6 x; la ðxÞ :¼ s otherwise; where t > s in [0, 1]. Then la is a fuzzy filter of L if and only if it satisfies the following inequality. ð8x; y 2 LÞða 6 y ! x; a 6 y ) a 6 xÞ:
ð14Þ
Proof. Assume that la is a filter of L and let x, y 2 L be such that a 6 y ! x and a 6 y. Then la(y ! x) = t = la(y), and thus la ðxÞ P minfla ðy ! xÞ; la ðyÞg ¼ t which implies that la(x) = t, that is, a 6 x. Conversely, suppose that (14) is valid. Note that la[s] = L and la[t] = {x 2 Lja 6 x}. Obviously 1 2 la[t]. Let x, y 2 L be such that x 2 la[t] and x ! y 2 la[t]. Then a 6 x and a 6 x ! y, which imply from the hypothesis that a 6 y, that is, y 2 la[t]. Hence la[t] is a filter of L. It follows from Lemma 3.7 that la is a fuzzy filter of L. h Theorem 3.25. For a fuzzy set l in L, let lsup be a fuzzy set in L defined by lsup ðxÞ :¼ supft 2 ½0; 1jx 2 hl½tig for all x 2 L. Then lsup is the least fuzzy filter of L that contains l, where hl[t]i means the least filter of L containing l[t]. Proof. For any s 2 Im(lsup), let sn ¼ s 1n for some n 2 N. Let x 2 lsup[s]. Then lsup(x) P s, which implies that supft 2 ½0; 1jx 2 hl½tig P s > s
1 ¼ sn ; 8n 2 N: n
Hence there exists r 2 {t 2 [0, 1]jx 2 hl[t]i} such that r > sn. Thus T l[r] l[sn], and so x 2 hl[r]i hl[sn]i for all n 2 N. Consequently x 2 n2N hl½sn i. On T the other hand, if x 2 n2N hl½sn i, then sn 2 {t 2 [0, 1]jx 2 hl[t]i} for any n 2 N. Therefore s
1 ¼ sn 6 supft 2 ½0; 1jx 2 hl½tig ¼ lsup ðxÞ; 8n 2 N: n
Since n is T arbitrary, it follows that s < lsup(x) so that x 2 lsup[s]. Hence lsup ½s ¼ n2N hl½sn i, which is a filter of L. Therefore we conclude that lsup is a fuzzy filter of L by Lemma 3.7. We now prove that lsup contains l. For any x 2 L, let s 2 {t 2 [0, 1]jx 2 l[t]}. Then x 2 l[s] and thus x 2 hl[s]i. Therefore s 2 {t 2 [0, 1]jx 2 hl[t]i}, which implies that ft 2 ½0; 1jx 2 l½tg ft 2 ½0; 1jx 2 hl½tig:
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It follows that lðxÞ ¼ supft 2 ½0; 1jx 2 l½tg 6 supft 2 ½0; 1jx 2 hl½tig ¼ lsup ðxÞ; which shows that lsup contains l. Finally let m be a fuzzy filter of L containing l. Let x 2 L. If lsup(x) =T 0, then clearly lsup(x) 6 m(x). Assume that lsup(x) = s 5 0. Then x 2 lsup ½s ¼ n2N hl½sn i, and so x 2 hl[sn]i for all n 2 N. It follows that mðxÞ P lðxÞ P sn ¼ s 1n for every n 2 N so that m(x) P s = lsup(x) since n is arbitrary. This shows that lsup m. This completes the proof. h Let l be a fuzzy filter of L and a 2 [0, 1]. Consider a relation on L as follows. la :¼ fðx; yÞjlðx ! yÞ > a; lðy ! xÞ > ag: Lemma 3.26. Let l be a fuzzy filter of L and a 2 [0, 1]. If la 5 ;, then l(1) > a. Proof. If la 5 ;, then l(x ! y) > a for some (x, y) 2 L · L. It follows from (b8) that l(1) P l(x ! y) > a. This completes the proof. h Proposition 3.27. Let l be a fuzzy filter of L and a 2 [0, 1). Then either la = ;, or la is an equivalence relation on L. Proof. Assume that la 5 ;. Since l(x ! x) = l(1) > a for all x 2 L, we have (x, x) 2 la, that is, la is reflexive. Obviously, la is symmetric. Let x, y, z 2 L be such that (x, y) 2 la and (y, z) 2 la. Then l(x ! y) > a, l(y ! x) > a, l(y ! z) > a and l(z ! y) > a. Since x ! y 6 (y ! z) ! (x ! z), it follows from (b7) that l((y ! z) ! (x ! z)) P l(x ! y) > a so from (b9) that lðx ! zÞ P minflðy ! zÞ; lððy ! zÞ ! ðx ! zÞÞg > a: Similarly we get l(z ! x) > a, and so (x, z) 2 la. Therefore la is transitive.
h
Proposition 3.28. Let l be a fuzzy filter of L and a 2 [0, 1) be such that la 5 ;. Then (i) ("x, y, z 2 L) ((x, y) 2 la )(x ! z, y ! z) 2 la,(z ! x, z ! y) 2 la). (ii) ("x, y, a, b 2 L) ((x, y) 2 la,(a, b) 2 la ) (x ! a, y ! b) 2 la). Proof. (i) Let x, y, z 2 L be such that (x, y) 2 la. Since y ! x 6 (x ! z) ! (y ! z), it follows from (b7) that l((x ! z) ! (y ! z)) P l(y ! x) > a.
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Similarly we get l((y ! z) ! (x ! z)) > a. Hence (x ! z, y ! z) 2 la. Since x ! y 6 (z ! x) ! (z ! y), (b7) implies that a < lðx ! yÞ 6 lððz ! xÞ ! ðz ! yÞÞ: Similarly we have l((z ! y) ! (z ! x)) > a. Therefore (z ! x, z ! y) 2 la. (ii) follows from (i) and the transitivity of la. h Proposition 3.29. Let l be a fuzzy filter of L and a 2 [0, 1). If l(1) > a, then la is a congruence relation on L. Proof. Assume that l(1) > a for a 2 [0,1). Then l(x ! x) = l(1) > a for all x 2 L, and so (x, x) 2 la. Hence la 5 ;, which implies from Propositions 3.27 and 3.28 that la is a congruence relation on L. h Corollary 3.30. Let l be a fuzzy filter of L and a 2 [0, 1). Then either la = 0, or la is a congruence relation on L. We call la a congruence relation on L induced by l. Let l be a fuzzy filter of L and let CR(l) denote the set of all congruence relations on L induced by l, that is, CRðlÞ :¼ fla ja 2 ½0; 1Þg: Then CR(l) is a poset under the set inclusion . Denote by (CR(l), _, ^) the lattice induced by the partial order . The following proposition is straightforward. Proposition 3.31. For every fuzzy filter l of L, we have the following assertions. ("a, b 2 [0, l(1)]) (a P b ) la lb). (CR(l), _, ^) is a chain. ("a, b 2 [0, l(1)]) (la _ lb = lmin{a, b}, la ^ lb = lmax{a, b}). If l(x) 5 0 for all x 2 L, then la = L · L for every a 2 [0, 1] with a < inf{l(x) jx 2 L}. (v) If l(x) 5 1 for all x 2 L, then lb = ; for every b 2 [0, 1] with b P sup{l(x)jx 2 L}.
(i) (ii) (iii) (iv)
Proposition 3.32. Let l be a fuzzy filter of L and C [0, l(1)]. Then linf(C) = sup{laja 2 C} and lsup(C) = inf{lbjb 2 C}. Proof. Since infC 6 a for any a 2 C, we have la linf(C) for all a 2 C. Hence linf(C) is an upper bound of {laja 2 C}. Let c 2 [0, l(1)] be such that lc linf(C). Then there exists (x, y) 2 L · L such that (x, y) 2 linf(C)nlc, and so
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l(x ! y) > inf(C) and l(y ! x) > inf(C), but l(x ! y) 6 c or l(y ! x) 6 c. It follows that there exists d 2 C such that infðCÞ 6 d < minflðx ! yÞ; lðy ! xÞg so that (x, y) 2 ld. Since (x, y) 62 lc, it follows that lc ld. This shows that lc is not an upper bound of {laja 2 C}. Hence sup{laja 2 C} = linf(C). Similarly we can prove that the equality lsup(C) = inf{lbjb 2 C}. This completes the proof. h Corollary 3.33. (CR(l), _, ^) is a completely distributive lattice.
4. Conclusions We gave characterizations of fuzzy filters, and investigated further properties of fuzzy filters in MTL-algebras. We considered the fuzzification of Boolean filters and MV-filters, and studied related properties. We provided a condition for a fuzzy filter to be Boolean. We discussed a method to generate a fuzzy filter by a fuzzy set. Given a fuzzy filter, we stated a congruence relation and showed that the set of all congruence relations induced by a fuzzy filter is a completely distributive lattice. Future research will focus on constructing a quotient MTL-algebra by using a fuzzy filter, on studying prime fuzzy filter, on establishing a fuzzy prime spectrum of a MTL-algebra, and on finding intuitionistic and/or intervalvalued fuzzy structures of a (prime) filter in MTL-algebras.
Acknowledgments The first author, Y. B. Jun, was supported by Korea Research Foundation Grant (KRF-2003-005-C00013). The authors are highly grateful to referees for their valuable comments and suggestions for improving the paper.
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