ON FUZZY FANTASTIC FILTERS OF LATTICE IMPLICATION ALGEBRAS

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1 J. Appl. Math. & Computing Vol. 14(2004), No. 1-2, pp ON FUZZY FANTASTIC FILTERS OF LATTICE IMPLICATION ALGEBRAS YOUNG BAE JUN AND SEOK ZUN SONG Abstract. Fuzzification of a fantastic filter in a lattice implication algebra is considered. Relations among a fuzzy filter, a fuzzy fantastic filter, and fuzzy positive implicative filter are stated. Conditions for a fuzzy filter to be a fuzzy fantastic filter are given. Using the notion of level set, a characterization of a fuzzy fantastic filter is considered. Extension property for fuzzy fantastic filters is established. The notion of normal/maximal fuzzy fantastic filters and complete fuzzy fantastic filters is introduced, and some related properties are investigated. AMS Mathematics Subject Classification : 03G10, 03B05, 06B10, 04A72. Key words and phrases : Lattice implication algebra, fuzzy (implicative, positive implicative, fantastic) filter, level fantastic filter, maximal fuzzy fantastic filter, complete fuzzy fantastic filter. 1. Introduction In the field of many-valued logic, lattice-valued logic plays an important role for two aspects: Firstly, it extends the chain-type truth-value field of some wellknown presented logic [1] (such as two-valued logic, three-valued logics introduced by Lukasiewicz, Bochvar, Kleene, Heyting, Finn, Hallden, Segerberg, Slupecki and Sobociński, n-valued logics introduced by Lukasiewicz, Post, Slupecki, Sobociński and Gödel, as well as the Lukasiewicz logic with truth value in the interval [0, 1] or Zadeh s infinite-valued logic, etc.) to some relatively general lattices. Secondly, the incompletely comparable property of truth value characterized by general lattice can more efficiently reflect the uncertainty of people s thinking, judging and decision. Hence, lattice-valued logic is becoming a research field which strongly influences the development of Algebraic Logic, Computer Science and Artificial Intelligence Technology. In 1969, Goguen proposed the Received February 10, This work was supported by grant R from the Korea Science and Engineering Foundation. c 2004 Korean Society for Computational & Applied Mathematics and Korean SIGCAM. 137

2 138 Young Bae Jun and Seok Zun Song first lattice-valued logic formal system based on complete-lattice-ordered semigroups [2], where the author did not provide a syntax associated with the given semantics. However, the concept of enriched residuated lattice introduced by Goguen provided a new idea and approach to study the lattice-valued logic. So, in 1979, Pavelka proposed a lattice-valued propositional logic system based on enriched residuated lattices [13]. Although this logic is based on relatively general lattice, its main results are limited to the interval [0, 1] or the finite chain of truth value. In spite of such limitation, these results reflect some fundamental characteristics of fuzzy logic. Pavelka s work is concerned only with propositional fuzzy logic. In 1982, Novak extended it to first-order fuzzy logic based on the interval [0, 1] or a finite chain [12], especially including some additional generalized quantifiers, and proved the soundness theorem and completeness theorem of this formal system. In order to establish a logic system with truth value in a relatively general lattice, in 1990, during the study of the project The Study of Abstract Fuzzy Logic granted by National Natural Science Foundation in China, Xu firstly established the lattice implication algebra by combining lattice and implication algebra, and investigated many useful structures [10, 11, 14, 15, 16]. Lattice implication algebra provided the foundation to establish the corresponding logic system from the algebraic viewpoint. For the general development of lattice implication algebras, the filter theory plays an important role. Xu and Qin [16] introduced the notions of filter and implicative filter in a lattice implication algebra, and investigated their properties. The first author (together with Xu, Qin, Kim and Roh) studied several filters in lattice implication algebras [3, 4, 5, 8, 9]. In particular, Jun [3] gave an equivalent condition of a filter, and provided some equivalent conditions for a filter to be an implicative filter in a lattice implication algebra. In this paper, we consider the fuzzification of the notion of fantastic filters in a lattice implication algebra. We give relations among a fuzzy filter, a fuzzy fantastic filter, and a fuzzy positive implicative filter. We provide conditions for a fuzzy filter to be a fuzzy fantastic filter. Using the notion of level set, we establish a characterization of a fuzzy fantastic filter. We construct extension property for fuzzy fantastic filters. Using a collection of fantastic filters with additional conditions, we establish a fuzzy fantastic filter. With relation to the ascending chain of fantastic filters, we state a characterization for the set of values of any fuzzy fantastic filter to be a well ordered subset of [0, 1]. We introduce the concept of normal/maximal fuzzy fantastic filters and complete fuzzy fantastic filters, and then we investigate some related properties. 2. Preliminaries Definition 2.1. (Xu [14]) A lattice implication algebra is defined to be a bounded lattice (L,,, 0, 1) with order-reversing involution and a binary operation satisfying the following axioms:

3 On fuzzy fantastic filters of Lattice Implication Algebras 139 (I1) x (y z) =y (x z), (I2) x x =1, (I3) x y = y x, (I4) x y = y x =1 x = y, (I5) (x y) y =(y x) x, (L1) (x y) z =(x z) (y z), (L2) (x y) z =(x z) (y z), for all x, y, z L Example 2.2. (Xu and Qin [16]) Let L := {0, a, b, c, 1}. Define the partially ordered relation on L as 0 <a<b<c<1, and define x y := min{x, y}, x y := max{x, y} for all x, y L and and as follows: x x 0 a b c a c a c b b b b c c a c a b c a b c 1 Then (L,,,, ) is a lattice implication algebra. In the sequel the binary operation will be denoted by juxtaposition. We can define a partial ordering on a lattice implication algebra L by x y if and only if xy =1. In a lattice implication algebra L, the following hold (see [14]): (P1) 0x =1,1x = x and x1 =1. (P2) xy (yz)(xz). (P3) x y implies yz xz and zx zy. (P4) x = x0. (P5) x y =(xy)y. (P6) ((yx)y ) = x y =((xy)x ). (P7) x (xy)y. Definition 2.3. (Xu and Qin [16]) A subset F of a lattice implication algebra L is called a filter of L if it satisfies for all x, y L, (f1) 1 F, (f2) x F and xy F imply y F. Definition 2.4. (Xu and Qin [16]) A subset F of a lattice implication algebra L is called an implicative filter of L if it satisfies (f1) and

4 140 Young Bae Jun and Seok Zun Song (f3) x(yz) F and xy F imply xz F for all x, y, z L. Definition 2.5. (Jun et al. [8]) A nonempty subset F of a lattice implication algebra L is called a positive implicative filter of L if it satisfies (f1) and (f4) x((yz)y) F and x F imply y F for all x, y, z L. Definition 2.6. (Jun [4]) A nonempty subset F of a lattice implication algebra L is called a fantastic filter of L if it satisfies (f1) and (f5) z(yx) F and z F imply ((xy)y)x F for all x, y, z L. Note from [4, Theorem 3.4] that a filter F of a lattice implication algebra L is fantastic if and only if it satisfies: (f6) yx F implies ((xy)y)x F for all x, y L. Definition 2.7. (Xu and Qin [17]) A fuzzy set µ in a lattice implication algebra L is called a fuzzy filter of L if it satisfies (f7) µ(1) µ(x), x L, (f8) µ(y) min{µ(x),µ(xy)}, x, y L. Definition 2.8. (Xu and Qin [17]) A fuzzy set µ in a lattice implication algebra L is called a fuzzy implicative filter of L if it satisfies (f7) and { ( ) } (f9) µ(xz) min µ x(yz),µ(xy), x, y, z L. Definition 2.9. (Jun [6]) A fuzzy set µ in a lattice implication algebra L is called a fuzzy positive implicative filter of L if it satisfies (f7) and { ( ) } (f10) µ(y) min µ x((yz)y),µ(x), x, y, z L. Note that every fuzzy filter is order preserving, and every fuzzy positive implicative filter is a fuzzy filter. 3. Fuzzy fantastic filters In what follows, let L denote a lattice implication algebra unless otherwise specified. We begin with the fuzzification of a fantastic filter. Definition 3.1. A fuzzy set µ in L is called a satisfies (f7) and fuzzy fantastic filter of L if it

5 On fuzzy fantastic filters of Lattice Implication Algebras 141 ( ) { ( ) } (f11) µ ((xy)y)x min µ z(yx),µ(z), x, y, z L. Example 3.2. Let L := {0, a, b, c, d, 1} be a set with the following Hasse diagram as a partial ordering. Define a unary operation and a binary operation denoted by juxtaposition on L as follows: 1 x x 0 a b c d a b a c a c 1 b c b 1 d b d b d a 1 b a 1 c c a c a a 1 1 a 1 d b d b 1 1 b a b c d 1 Define - and -operations on L as follows: x y := (xy)y and x y := ((x y )y ) for all x, y L. Then L is a lattice implication algebra. It is easy to see that a fuzzy set µ in L defined by µ(b) =µ(c) =µ(1) = 0.8 and µ(0) = µ(a) = µ(d) =0.3 is a fuzzy fantastic filter of L, and the fuzzy set ν in L defined by ν(a) =ν(1) >ν(x) for all x L \{a, 1} is a fuzzy fantastic filter of L. Theorem 3.3. Every fuzzy fantastic filter is a fuzzy filter. Proof. Let µ be a fuzzy fantastic filter of L. Taking y = 1 in (f11) and using (P1), we get µ(x) =µ(((x1)1)x) min{µ((z(1x)),µ(z)} = min{µ(zx),µ(z)} for all x, z L. Hence µ is a fuzzy filter of L The following is an equivalent condition for a fuzzy filter to be a fuzzy fantastic filter. Theorem 3.4. A fuzzy filter µ of L is fuzzy fantastic if and only if it satisfies the inequality: (f12) µ(((xy)y)x) µ(yx) for all x, y L. Proof. Assume that µ is a fuzzy fantastic filter of L. Taking z = 1 in (f11) and using (P1) and (f7), we get µ(((xy)y)x) min{µ(1(yx)),µ(1)} = µ(yx)

6 142 Young Bae Jun and Seok Zun Song for all x, y L. Conversely, let µ be a fuzzy filter of L satisfying (f12). Let x, y, z L. Then by (f12) and (f8), we have µ(((xy)y)x) µ(yx) min{µ(z(yx)),µ(z)}. Hence µ is a fuzzy fantastic filter of L. The first author [4] posed an open problem as follows: Is every fantastic filter of L positive implicative? We now provide a negative answer. Consider a lattice implication algbera L := {0, a, b, c, 1} as in Example 2.2. We know that {1} is a fantastic filter of L, but it is not a positive implicative filter (see [8, Remark 3.1]). In connection with this situation, we can induce that a fuzzy fantastic filter may not be a fuzzy positive implicative filter. Lemma 3.5 (Jun [6, Theorem 3.4]). implicative if and only if it satisfies: (f13) µ(x) µ((xy)x), x, y L. A fuzzy filter µ of L is fuzzy positive Lemma 3.6 (Jun and Song [7, Theorem 3.9]). Let µ be a fuzzy filter of L. Then the following are equivalent: (i) µ is a fuzzy implicative filter of L. (ii) µ(xy) ( µ(x(xy)), ) ( x, y ) L. (iii) µ (xy)(xz) µ x(yz), x, y, z L. Theorem 3.7. A fuzzy set µ in L is a fuzzy positive implicative filter of L if and only if it is both a fuzzy implicative filter and a fuzzy fantastic filter of L. Proof. Suppose that µ is a fuzzy positive implicative filter of L. Then µ is a fuzzy implicative filter of L (see [6, Theorem 3.6]). Note that x ((xy)y)x, and so (((xy)y)x)y xy, x, y L. (3.1) Putting b =((xy)y)x, then ( )( ) (by)b = (((xy)y)x)y ((xy)y)x ( ) (xy) ((xy)y)x by (3.1) and (P3) = ((xy)y)((xy)x) by (I1) yx. by (I1) and (P2) Since µ is order preserving, it follows from Lemma 3.5 that ( ) ( ) µ ((xy)y)x = µ(b) µ (by)b µ(yx). Hence, by Theorem 3.4, µ is a fuzzy fantastic filter of L. Conversely, suppose that µ is both a fuzzy implicative filter and a fuzzy fantastic filter of L. Combining

7 On fuzzy fantastic filters of Lattice Implication Algebras 143 (P7) and (P3), we have (xy)x (xy)((xy)y) for all x, y L. It follows from Lemma 3.6 that ( ) ( ) ( ) µ (xy)y µ (xy)((xy)y) µ (xy)x (3.2) for all x, y L. On the other hand, since (xy)x yx for all x, y L, it follows from Theorem 3.4 that ( ) ( ) µ ((xy)y)x µ(yx) µ (xy)x so from (f8) and (3.2) that { ( ) ( )} µ(x) min µ (xy)y,µ ((xy)y)x ( ) ( ) = µ (xy)y µ (xy)x for all x, y L. Therefore, by Lemma 3.5, µ is a fuzzy positive implicative filter of L Lemma 3.8. Every fuzzy filter µ of L satisfies the following implication: x yz µ(z) min{µ(x), µ(y)}. Proof. Let x, y, z L be such that x yz. Since µ is order preserving, it follows from (f8) that µ(z) min{µ(y), µ(yz)} min{µ(x), µ(y)}. This completes the proof. Theorem 3.9. (Extension property for fuzzy fantastic filters). Let µ and ν be fuzzy filters of L such that µ(1) = ν(1) and µ ν, that is, µ(x) ν(x) for all x L. Ifµ is a fuzzy fantastic filter of L, then so is ν. Proof. For any x, y L, we have ( ) ν ((((yx)x)y)y)((yx)x) ( ) µ ((((yx)x)y)y)((yx)x) ( ) because µ ν µ y((yx)x) ( ) by Theorem 3.4 = µ (yx)(yx) by (I1) = µ(1) by (I2) = ν(1). by assumption (3.3)

8 144 Young Bae Jun and Seok Zun Song Since ( )( ) ((((yx)x)y)y)x ((xy)y)x ( )( ) (xy)y (((yx)x)y)y ( ) ((yx)x)y (xy) x((yx)x) =(yx)(xx) =(yx)1=1 for all x, y L, it follows that ( ) { ( )} ν ((xy)y)x min ν(1), ν ((((yx)x)y)y)x by Lemma 3.8 ( ) = ν ((((yx)x)y)y)x by (f7) { ( ) } min ν (yx)(((((yx)x)y)y)x),ν(yx) by (f8) { ( ) } = min ν ((((yx)x)y)y)((yx)x),ν(yx) by (I1) { } min ν(1), ν(yx) by (3.3) = ν(yx). by (f7) Hence, by Theorem 3.4, ν is a fuzzy fantastic filter of L. Let µ be a fuzzy set in L. For any a L, we consider the set µ(a) :={x L µ(a) µ(x)}. Note that if µ is a fuzzy filter of L, then µ(a) is a filter of L (see [7, Theorem 3.5]). Theorem Let a L. Ifµ is a fuzzy fantastic filter of L, then µ(a) is a fantastic filter of L. Proof. If µ is a fuzzy fantastic filter of L, then µ is a fuzzy filter of L (see Theorem 3.3), and so µ(a) is a filter of L for every a L. Let x, y L be such that yx µ(a). Then µ(a) µ(yx) µ(((xy)y)x), and thus ((xy)y)x µ(a). Hence µ(a) is a fantastic filter of L. Corollary If µ is a fuzzy fantastic filter of L, then µ(1) is a fantastic filter of L. Theorem A fuzzy set µ in L is a fuzzy fantastic filter of L if and only if its nonempty level set µ[t] :={x L µ(x) t}, t [0, 1], is a fantastic filter of L, which is called a level fantastic filter of µ.

9 On fuzzy fantastic filters of Lattice Implication Algebras 145 Proof. Note that µ is a fuzzy filter of L if and only if µ[t] ( ) is a filter of L for all t [0, 1] (see [17, Theorem 7]). Assume that µ is a fuzzy fantastic ( filter ) of L. Let x, y L be such that yx µ[t]. Then t µ(yx) µ ((xy)y)x, and so ((xy)y)x µ[t]. Hence µ[t] is a fantastic filter of L. Conversely suppose that µ is not ( a fuzzy ) fantastic filter of L. Then we can take a, b L such that µ(ba) >µ ((ab)b)a. Putting t 0 := 1 ( ) µ(ba)+µ(((ab)b)a), 2 ( ) we get µ ((ab)b)a <t 0 <µ(ba). It follows that ba µ[t 0 ], but ((ab)b)a / µ[t 0 ], so( that µ[t 0 )] is not a fantastic filter. Hence µ satisfies the inequality µ(yx) µ ((xy)y)x for all x, y L, that is, µ is a fuzzy fantastic filter of L. Theorem Let {F k : k Λ}, where Λ [0, 1], be a collection of fantastic filters of L such that (i) L = F k, k Λ (ii) k, r Λ, k>rif and only if F k F r. Then a fuzzy set µ in L defined by µ(x) = sup{r Λ:x F r } is a fuzzy fantastic filter of L. Proof. Let k [0, 1] be such that µ[k]. Note that either k = sup{r Λ:r<k} = sup{r Λ:F k F r } or k sup{r Λ:r<k} = sup{r Λ:F k F r }. The first case implies that µ[k] = F r which is a fantastic filter, because k>r x µ[k] x F r for all r<k x k>r F r. For the second case there exists ε>0 such that (k ε, k) Λ=. We prove that µ[k] = F r, which is a fantastic filter. Indeed, if x F r then x F r k r k r for some r k. Hence µ(x) r k, and so x µ[k]. Now if x/ F r, then x/ F r for all r k. Therefore x/ F r for all r>k ε, which shows that if x F r then r k ε. Thus µ(x) k ε, and so x / µ[k]. It follows from Theorem 3.12 that µ is a fuzzy fantastic filter of L k r

10 146 Young Bae Jun and Seok Zun Song Theorem Let µ be a fuzzy set in L with Im(µ) ={t 1,t 2, t n }, where t i <t j whenever i>j.let{f k k =1, 2,,n} be a family of fantastic filters of L such that (i) F 1 F 2 F n = L (ii) µ( F k )=t k, where F k = F k \ F k 1,F 0 = for k =1, 2,,n. Then µ is a fuzzy fantastic filter of L. Proof. Since µ[t k ]=F k for all k =1, 2,,n, it is straightforward by Theorem Corollary Let µ be a fuzzy set in L and let Im(µ) ={t 1,t 2,,t n }, where t 1 >t 2 > >t n. If F 1 F 2 F n = L are fantastic filters of L such that µ(f k ) t k for k =1, 2,,n, then µ is a fuzzy fantastic filter of L. Corollary If µ is a fuzzy fantastic filter of L with Im(µ) ={t 1,t 2,,t n }, where t 1 >t 2 > >t n, then µ[t k ],k=1, 2,,n, are fantastic filters of L such that µ(µ[t 1 ]) = t 1 and µ(µ[t k ] \ µ[t k 1 ]) = t k for k =2, 3,,n. Proof. By Theorem 3.12, µ[t k ],k=1, 2,,n, are fantastic filters of L. Obviously, µ(µ[t 1 ]) = t 1. Since µ(µ[t 2 ]) t 2, we get µ(x) = t 1 for x µ[t 1 ]; and µ(x) =t 2 for x µ[t 2 ] \ µ[t 1 ]. Repeating this procedure, we conclude that µ(µ[t k ] \ µ[t k 1 ]) = t k for k =2, 3,,n. Theorem Let {F k k N} be a family of fantastic filters of L which is nested, that is, F 1 F 2. Let µ be a fuzzy set in L defined by k k+1 for x F k \ F k+1, k =0, 1, 2,, µ(x) = 1 for x F k k=0 for all x L, where F 0 stands for L. Then µ is a fuzzy fantastic filter of L. Proof. Clearly µ(1) µ(x) for all x L. Let x, y, z L. Suppose that z(yx) F k \ F k+1 and z F r \ F r+1 for k =0, 1, 2, ; r =0, 1, 2,. Without loss of generality, we may assume that k r. Then obviously z F k. Since F k is a fantastic filter, it follows from (f5) that ((xy)y)x F k so that µ(((xy)y)x) k = min {µ(z(yx)),µ(z)}. k +1 If z(yx) F k and z F k, then ((xy)y)x F k. Hence k=0 k=0 k=0 µ(((xy)y)x) =1=min{µ(z(yx)),µ(z)}.

11 On fuzzy fantastic filters of Lattice Implication Algebras 147 If z(yx) / F k and z F k, then there exists i N such that z(yx) k=0 k=0 F i \ F i+1. It follows that ((xy)y)x F i so that µ(((xy)y)x) i = min {µ(z(yx)),µ(z)}. i +1 Finally, assume that z(yx) F k and z/ F k. Then z F j \F j+1 for some k=0 j N. Hence ((xy)y)x F j, and thus k=0 µ(((xy)y)x) j = min {µ(z(yx)),µ(z)}. j +1 Consequently, µ is a fuzzy fantastic filter of L. Corollary If every fuzzy fantastic filter µ of L has finite image, then every descending chain of fantastic filters of L terminates after a finite number of steps. Theorem Assume that every descending chain F 1 F 2 of fantastic filters of L terminates after a finite number of steps, that is, there exists r N such that F r = F k for all k r. Letµ be a fuzzy fantastic filter of L in which a sequence of elements of Im(µ) is strictly increasing. Then µ has finite number of values. Proof. Assume that Im(µ) is not finite. Let {t k } be a strictly increasing sequence of elements of Im(µ), that is, 0 t 1 <t 2 < 1. Then by Theorem 3.12, µ[t r ] is a fantastic filter of L for all r N, and we get a strictly descending chain µ[t 1 ] µ[t 2 ] µ[t 3 ] of fantastic filters of L which is not terminating. This is a contradiction, and so µ has finite number of values. Theorem Every ascending chain of fantastic filters of L terminates after a finite number of steps if and only if the set of values of any fuzzy fantastic filter in L is a well ordered subset of [0, 1]. Proof. Assume that the set of values of a fuzzy fantastic filter µ of L is not well ordered. Then there exists a strictly decreasing sequence {t n } such that t n = µ(x n ) for some x n L. But in this case the family {µ[t n ]} of level fantastic filters from a strictly ascending chain, which is a contradiction. To prove the converse, suppose that there exists a strictly ascending chain F 1 F 2 F 3

12 148 Young Bae Jun and Seok Zun Song of fantastic filters of L. Then F := F n is a fantastic filter of L. Define a n N fuzzy set µ in L by { 0 for x/ F, µ(x) = 1 k where k = min{n N : x F n }. Since 1 F n for all n N, we have µ(1) = 1 µ(x) for all x L. For any x, y, z L, ifz(yx) / F or z/ F then obviously µ ( ((xy)y)x ) 0 = min{µ(z(yx)), µ(z)}. Now let x, y, z L be such that z(yx) F and z F. If z(yx),z F n \ F n 1 for some n N, then ((xy)y)x F n. Hence µ ( ((xy)y)x ) 1 = min{µ(z(yx)), µ(z)}. n If z(yx) F n \ F n 1 and z / F n \ F n 1, then either z F n 1 F n or z F m \ F m 1 for some m>n. The first case implies that ((xy)y)x F n and so µ ( ((xy)y)x ) 1 = µ(z(yx)) min{µ(z(yx)), µ(z)}. n For the second case we have ((xy)y)x F m and thus µ(z(yx)) = 1 n > 1 m = µ(z). Therefore µ ( ((xy)y)x ) 1 = min{µ(z(yx)), µ(z)}. m In a similar way we get µ ( ((xy)y)x ) min{µ(z(yx)), µ(z)} for z(yx) / F n \ F n 1 and z F n \ F n 1. This proves that µ is a fuzzy fantastic filter of L. Since the chain of fantastic filters F 1 F 2 F 3 is not terminating, µ has a strictly descending sequence of values. This contradicts that the value set of any fuzzy fantastic filter is well ordered. This completes the proof. Theorem Let µ be a fuzzy fantastic filter of L with Im(µ) ={t i : i Λ} and let Ω:={µ[t] :t Im(µ)}. Then (i) there exists a unique t 0 Im(µ) such that t 0 t for all t Im(µ). (ii) L is the set-theoretic union of all µ[t] Ω. (iii) the members of Ω form a chain. (iv) Ω contains all level fantastic filters of µ if and only if µ attains its infimum on all fantastic filters of L.

13 On fuzzy fantastic filters of Lattice Implication Algebras 149 Proof. (i) is by the fact that t 0 = µ(1) µ(x) for all x L. (ii) If x L, then µ(x) =t x Im(µ). This implies x µ[t x ] µ[t] L, t Im(µ) which proves (ii). (iii) Note that µ[t i ] µ[t j ] t i t j for i, j Λ. Hence Ω is totally ordered by inclusion. (iv) Suppose that Ω contains all level fantastic filters of µ. Let F be a fantastic filter of L. If µ is constant on F, then we are done. Assume that µ is not constant on F. We consider two cases: F = L and F L. For the case F = L, let β = inf Im(µ). Then β t Im(µ), that is, µ[β] µ[t] for all t Im(µ). But µ[0] = L Ω because Ω contains all level fantastic filters of µ. Hence there exists α Im(µ) such that µ[α] =L. It follows that L = µ[α] µ[β] so that µ[β] =µ[α] =L because every level fantastic filter of µ is a fantastic filter of L. Now it is sufficient to show that β = α. If β<α, then there exists γ Im(µ) such that β γ < α. Thus µ[γ] µ[α] = L, a contradiction. Therefore β = α Im(µ). In the case F L we consider the fuzzy set µ F in L defined by { δ 0 for x F, µ F (x) = 0 for x L \ F. It is routine to check that µ F is a fuzzy fantastic filter of L. Let J := {i Λ:µ(y) =t i for some y F } and Ω F := {µ F [t i ]:i J}. Noticing that Ω F contains all level fantastic filters, then there exists x 0 F such that µ(x 0 ) = inf{µ F (x) :x F }, which implies that µ(x 0 )={µ(x) :x F }. This proves that µ attains its infimum on all fantastic filters of L. To prove the converse let µ[α] be a level fantastic filter of µ. Ifα = t for some t Im(µ), then clearly µ[α] Ω. If α t for all t Im(µ), then there does not exist x L such that µ(x) =α. Let F = {x L : µ(x) >α}. Obviously 1 F. Now let x, y, z L be such that z(yx) F and z F. Then µ ( z(yx) ) >αand µ(z) >α. It follows from (f11) that ( ) { µ ((xy)y)x min µ ( z(yx) ) },µ(z) >α so that ((xy)y)x F. Hence F is a fantastic filter of L. By hypothesis there exists y F such that µ(y) = inf{µ(x) :x F }. But µ(y) Im(µ) implies µ(y) =s for some s Im(µ). Hence inf{µ(x) :x F } = s>α. Note that there does not exist z L such that α µ(z) <s. This gives µ[α] =µ[s], and so µ[α] Ω. Thus Ω contains all level fantastic filters of µ. This completes the proof. 4. Normalization of Fuzzy Fantastic Filters

14 150 Young Bae Jun and Seok Zun Song Definition 4.1. A fuzzy fantastic filter µ of L is said to be normal if there exists a L such that µ(a) =1. We note that if µ is a normal fuzzy fantastic filter of L, then µ(1) = 1, hence a fuzzy fantastic filter µ of L is normal if and only if µ(1) = 1. Let F N (L) denote the set of all normal fuzzy fantastic filters of L. Theorem 4.2. Let µ be a fuzzy fantastic filter of L and let µ + be a fuzzy set in L given by µ + (x) =µ(x) +1 µ(1) for all x L. Then µ + F N (L) and µ µ +. Proof. For any x, y, z L we have µ + (1) = µ(1) + 1 µ(1) = 1 µ + (x), and min{µ + (z(yx)), µ + (z)} = min{µ(z(yx)) + 1 µ(1), µ(z)+1 µ(1)} = min{µ(z(yx)), µ(z)} +1 µ(1) µ(((xy)y)x) +1 µ(1) = µ + (((xy)y)x). Hence µ + F N (L). Obviously, µ µ +. This completes the proof. Corollary 4.3. If µ is a fuzzy fantastic filter of L satisfying µ + (a) =0for some a L, then µ(a) =0also. It is clear that a fuzzy fantastic filter µ of L is normal if and only if µ + = µ, and for any fuzzy fantastic filter µ of L we have (µ + ) + = µ +. Hence if µ is a normal fuzzy fantastic filter of L, then (µ + ) + = µ. Theorem 4.4. Let µ be a fuzzy fantastic filter of L. If there exists a fuzzy fantastic filter ν of L satisfying ν + µ, then µ F N (L). Proof. Suppose that there exists a fuzzy fantastic filter ν of L such that ν + µ. Then 1 = ν + (1) µ(1), whence µ(1) = 1. Hence µ F N (L). Theorem 4.5. Let µ be a fuzzy fantastic filter of L and let φ :[0,µ(1)] [0, 1] be an increasing function. Let µ φ be a fuzzy set in L defined by µ φ (x) =φ(µ(x)) for all x L. Then µ φ is a fuzzy fantastic filter of L. Moreover, if φ(µ(1)) = 1 then µ φ F N (L); and if φ(t) t for all t [0,µ(1)] then µ µ φ.

15 On fuzzy fantastic filters of Lattice Implication Algebras 151 Proof. Since φ is increasing and µ(1) µ(x) for all x L, we have µ φ (1) = φ(µ(1)) φ(µ(x)) = µ φ (x), x L. Let x, y, z L. Then min{µ φ (z(yx)), µ φ (z)} = min{φ(µ(z(yx))), φ(µ(z))} = φ(min{µ(z(yx)), µ(z)}) φ(µ(((xy)y)x). Hence µ φ is a fuzzy fantastic filter of L. If φ(µ(1)) = 1 then obviously µ φ is normal, and so µ φ F N (L). Assume that φ(t) t for all t [0,µ(1)]. Then µ φ (x) =φ(µ(x)) µ(x) for all x L, which proves that µ µ φ. Theorem 4.6. Let µ F N (L) be a non-constant maximal element of the poset (F N (L), ). Then µ takes only the values 0 and 1. Proof. Note that µ(1) = 1. Let x L be such that µ(x) 1. It is sufficient to show that µ(x) = 0. Indeed, if not, then there exists a L such ) that 0 <µ(a) < 1. Let ν be a fuzzy set in L defined by ν(x) = 2( 1 µ(x)+µ(a) for all x L. Then clearly ν is well defined, and we have ν(1) = 1 ( ) 1( ) µ(1) + µ(a) µ(x)+µ(a) = ν(x), x L. 2 2 Let x, y, z L. Then ν(((xy)y)x) = 1 ( ) µ(((xy)y)x)+µ(a) 2 1 ( ) min{µ(z(yx)), µ(z)} + µ(a) 2 { 1 = min 2 (µ(z(yx)) + µ(a)), 1 } 2 (µ(z)+µ(a)) = min{ν(z(yx)), ν(z)}. Hence ν is a fuzzy fantastic filter of L, and so ν + F N (L) by Theorem 4.2. Noticing that ν + (1) = 1 >ν + (a) = 2( 1 ) µ(a)+1 >µ(a), we know that ν + is non-constant. From ν + (a) >µ(a) it follows that µ is not maximal in (F N (L), ). This is a contradiction and the proof is complete. For a fantastic filter F of L, if we define a fuzzy set µ F in L by { 1 for x F, µ F (x) := 0 otherwise, for all x L, then µ F F N (L) and µ F (1) = F. Let F(L) denote the set of all fantastic filters of L. We define mappings Φ : F(L) F N (L) andψ:

16 152 Young Bae Jun and Seok Zun Song F N (L) F(L) byφ(f )=µ F and Ψ(µ) = µ(1), respectively, for all F F(L) and µ F N (L). Then ΨΦ = 1 F(L) and ΦΨ(µ) =Φ( µ(1)) = µ µ(1) µ. Theorem 4.7. If F, G F(L), then µ F G = µ F µ G, that is, Φ(F G) = Φ(F ) Φ(G). If µ, ν F N (L), then (µ ν)(1) = µ(1) ν(1), that is, Ψ(µ ν) =Ψ(µ) Ψ(ν). Proof. Let x L. Ifx F G, then µ F G (x) = 1 and µ F (x) =1=µ G (x). It follows that µ F G (x) = 1 = min{µ F (x),µ G (x)} = ( ) µ F µ G (x). If x/ F G, then x/ F or x/ G, and thus µ F G (x) = 0 = min{µ F (x),µ G (x)} = ( ) µ F µ G (x). Therefore µ F G = µ F µ G. Now let µ, ν F N (L). Obviously µ ν F N (L) and so Ψ(µ ν) = (µ ν)(1) = { x L :(µ ν)(x) =(µ ν)(1) } = { x L : min{µ(x),ν(x)} =1 } = { x L : µ(x) =1=ν(x) } = { x L : µ(x) =1 } { x L : ν(x) =1 } = { x L : µ(x) =µ(1) } { x L : ν(x) =ν(1) } = µ(1) ν(1) =Ψ(µ) Ψ(ν). This completes the proof. Definition 4.8. A fuzzy fantastic filter µ of L is said to be maximal if it satisfies: (f14) µ is non-constant, (f15) µ + is a maximal element of (F N (L), ). Theorem 4.9. If µ is a maximal fuzzy fantastic filter of L, then (i) µ is normal, (ii) µ + takes only the values 0 and 1, (iii) µ µ(1) = µ, (iv) µ(1) is a maximal fantastic filter of L.

17 On fuzzy fantastic filters of Lattice Implication Algebras 153 Proof. Let µ be a maximal fuzzy fantastic filter of L. Then µ + is a non-constant maximal element of the poset (F N (L), ). It follows from Theorem 4.6 that µ + takes only the values 0 and 1. Note that µ + (x) = 1 if and only if µ(x) =µ(1); and µ + (x) = 0 if and only if µ(x) =µ(1) 1. Using Corollary 4.3, we get µ(x) = 0, that is, µ(1) = 1. Hence µ is normal. This proves (i) and (ii). (iii) Obviously µ µ(1) µ and µ µ(1) takes only the values 0 and 1. Let x L. Ifµ(x) = 0, then clearly µ µ µ(1). If µ(x) = 1, then x µ(1) and so µ µ(1) (x) =1. This shows that µ µ µ(1). (iv) Since µ is non-constant, µ(1) is a proper fantastic filter of L. Let G be a fantastic filter of L such that µ(1) G. Noticing that µ F µ G if and only if F G for every fantastic filters F and G of L, using (iii) we have µ = µ µ(1) µ G. Since µ, µ G F N (L) and µ = µ + is a maximal element of F N (L), it follows that either µ = µ G or µ G = 1 where 1 : L [0, 1] is a fuzzy set defined by 1(x) = 1 for all x L. Ifµ = µ G, then µ(1) = µ G (1) = G. The later case implies that G = L. Hence µ(1) is a maximal fantastic filter of L. This completes the proof. Definition A fuzzy fantastic filter µ of L is said to be complete if it is normal and there exists z L such that µ(z) =0. Note that µ F is a complete fuzzy fantastic filter of L for every fantastic filter F of L. Theorem Let µ be a fuzzy fantastic filter of L and let w be a fixed element of L such that µ(1) µ(w). Define a fuzzy set µ in L by µ (x) = µ(x) µ(w) µ(1) µ(w) for all x L. Then µ is a complete fuzzy fantastic filter of L. Proof. For any x L, we have µ µ(1) µ(w) (1) = µ(1) µ(w) =1 µ (x). Let x, y, z L. Then min{µ ( z(yx) ),µ (z)} { µ(z(yx)) µ(w) µ(z) µ(w) } = min, µ(1) µ(w) µ(1) µ(w) 1 = (min { µ(z(yx)) µ(w), µ(z) µ(w) }) µ(1) µ(w) 1 ( ) = min{µ(z(yx)), µ(z)} µ(w) µ(1) µ(w)

18 154 Young Bae Jun and Seok Zun Song 1 ( µ ( ((xy)y)x ) ) µ(w) µ(1) µ(w) = µ ( ((xy)y)x ). Hence µ F N (L). Since µ (w) = 0, we conclude that µ is a complete fuzzy fantastic filter of L. Denote by C(L) the set of all complete fuzzy fantastic filters of L. Note that C(L) F N (L) and the restriction of the partial ordering off N (L) gives a partial ordering of C(L). Note that if µ C(L), then µ = µ. Theorem Every non-constant maximal element of (F N (L), ) is also a maximal element of (C(L), ). Proof. Let µ be a non-constant maximal element of (F N (L), ). Then µ takes only the values 0 and 1 (see Theorem 4.6), and in fact, µ(1) = 1 and µ(w) = 0 for some w( 1) L. Hence µ is complete. Assume that there exists ν C(L) such that µ ν. It follows that µ ν in F N (L). Since µ is maximal in (F N (L), ) and since ν is non-constant, we have µ = ν. Thus µ is a maximal element of (C(L), ). Theorem Every maximal fuzzy fantastic filter of L is complete. Proof. Let µ be a maximal fuzzy fantastic filter of L. Then µ is normal and µ = µ + takes only the values 0 and 1. Since µ is non-constant and µ(1) = 1, it is clear that there exists w( 1) L such that µ(w) = 0. Hence µ is complete. References 1. L. Bolc and P. Borowik, Many-Valued Logic, Springer, Berlin, J. A. Goguen, The logic of inexact concepts, Synthese 19 (1969), Y. B. Jun, Implicative filters of lattice implication algebras, Bull. Korean Math. Soc. 34(2) (1997), Y. B. Jun, Fantastic filters of lattice implication algebras, Internat. J. Math. & Math. Sci. 24(4) (2000), Y. B. Jun, On n-fold implicative filters of lattice implication algebras, Internat. J. Math. & Math. Sci. 26(11) (2001), Y. B. Jun, Fuzzy positive implicative and fuzzy associative filters of lattice implication algebras, Fuzzy Sets and Systems 121 (2001), Y. B. Jun and S. Z. Song, On fuzzy implicative filters of lattice implication algebras, J. Fuzzy Math. 10(4) (2002),

19 On fuzzy fantastic filters of Lattice Implication Algebras Y. B. Jun, Y. Xu and K. Y. Qin, Positive implicative and associative filters of lattice implication algebras, Bull. Korean Math. Soc. 35(1) (1998), S. Y. Kim, E. H. Roh and Y. B. Jun, On ultra filters of lattice implication algebras, Scientiae Mathematicae 2(2) (1999), J. Liu and Y. Xu, On certain filters in lattice implication algebras, Chinese Quarterly J. Math. 11(4) (1996), J. Liu and Y. Xu, Filters and structure of lattice implication algebras, Chinese Science Bulletin 42(18) (1997), V. Novak, First order fuzzy logic, Studia Logica 46(1) (1982), J. Pavelka, On fuzzy logic I, II, III, Zeit. Math. Logik u. Grundl. Math. 25 (1979), 45 52, , Y. Xu, Lattice implication algebras, J. Southwest Jiaotong Univ. 1 (1993), Y. Xu and K. Y. Qin, Lattice H implication algebras and lattice implication algebra classes, J. Hebei Mining and Civil Engineering Institute 3 (1992), Y. Xu and K. Y. Qin, On filters of lattice implication algebras, J. Fuzzy Math. 1(2) (1993), Y. Xu and K. Y. Qin, Fuzzy lattice implication algebras, J. Southwest Jiaotong Univ. 30(2) (1995), Y. B. Jun has been an educator and research mathematician since 1982, mostly at the Gyeongsang National University; and a member of the editorial board of Far East Journal of Mathematical Science (India) since 1998, and Quasigroups and Related Systems (Moldova) since He did postdoctoral work (one year, , supported by KOSEF) at the University of Albert in Albert, Canada; and worked for one year ( ) as a visiting professor at the Northwest University in Xian, China (supported by LG Yonam Foundation). His research interests focus on the structure theory of BCK/BCIalgebras, Hilbert algebras, (lattice) implication algebras and negatively partially ordered semigroups, and fuzzy and hyper theory of such algebraic structures. Jun is a co-author of the text BCK-algebras with J. Meng which is an approachable introduction to BCK/BCIalgebras. Department of Mathematics Education, Gyeongsang National University, Chinju (Jinju) , Korea. ybjun@nongae.gsnu.ac.kr S. Z. Song has been an educator and research mathematician since 1983, mostly at the Cheju National University; and a visiting professor at Utah State University in USA(1990), Hokkaido University in Japan(1994), Bielefeld University in Germany(1996), Warsaw University in Poland(1998) and Indian Statistical Institute at Delhi(1999). He researches in linear algebra and matrix theory. Department of Mathematics, Cheju National University, Cheju , Korea. szsong@cheju.ac.kr