Abstract
Fuzzy Inf. Eng. (2010) 2: 187-200 DOI 10.1007/s12543-010-0041-x ORIGINAL ARTICLE Conjugate Families of Mappings in Pointwise Metric Fuzzy Lattices Xing-hua Zhu · Jian-zhong Xiao Received: 27 September 2009/ Revised: 8 April 2010/ Accepted: 6 May 2010/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China 2010 Abstract In this paper some relations among the axioms of pointwise metric for fuzzy lattices introduced by Shi are studied. By means of the conjugate families of mappings, some representation results are presented concerning metric, pseudo- metric and pseudo-quasi-metric fuzzy lattices. These results reconcile the metric lat- tice theory generated via concept of neighborhood with the one generated via concept of remote-neighborhood. Keywords Fuzzy lattice· Pointwise metric· Conjugate mapping 1. Introduction and Preliminaries The theory of metric lattice in this paper is a sort of asymmetrical metric theory or quasi-metric theory. It was originated with the concept of Pompeiu-Hausdorﬀ metric on a family of sets and the concept of fuzzy metric on a family of fuzzy sets (see [3, 14, 23, 28]). In the last three decades, there have been many creative and remarkable researches done on it (see [2, 4, 6-7, 9, 12-13, 18-25, 28] etc.). In [20] it is pointed out that the theory of metric lattice has some applications in theoretic computer science. The distance between two programs can be calculated by means of the metric of lattices. In fact, if we have a lattice L whose elements repre- sent “pieces of information”, then the order in L can be intended with respect to the informative content. In other words, saying that a ≤ b means that b represents more complete information than a. In accordance with this interpretation, quasi-metrics or asymmetric distances for lattices have been used in several places in information science. For examples, the tools of quasi-metrics can be used in finding fixed point semantics of logic programs. Also, they have many applications to complexity anal- ysis of programs and algorithms (see [10-11, 15-17, 22] etc.). Xing-hua Zhu · Jian-zhong Xiao () College of Mathematics and Physics, Nanjing University of Information Science and Technology, Nanjing 210044, P.R.China email: xiaojz@nuist.edu.cn 188 Xing-hua Zhu · Jian-zhong Xiao (2010) The main purpose of this paper is to deal with the mathematical setting of quasi- metric lattices. For the works of pointwise metrics on fuzzy lattices, some of them dealt with concept of neighborhood (for example, see Peng [12-13]), and others did that of the remote-neighborhood (for example, see Shi [18-21]). In [27] it is pointed out that there exists a kind of conjugacy between neighborhood and remote- neighborhood, and the topology induced by a pseudo-metric is determined by the conjugate families of mappings. Some fundamental question arises: Are the met- ric, pseudo-metric and pseudo-quasi-metric determined by the conjugate families of mappings? What diﬀerences are between their metric axioms? We devote this paper to answer the above questions. In Section 2, we study the relations among the axioms of pointwise metric for fuzzy lattices introduced by Shi. In Section 3, we present some representations results concerning metric, pseudo- metric and pseudo-quasi-metric fuzzy lattices in terms of the conjugate families of mappings. Our results will further reveal the conjugate feature of the metric lattices, reconcile the metric lattice theory generated via concept of neighborhood with the one generated via concept of remote-neighborhood. Throughout this paper, L always denotes a fuzzy lattice (or a molecular lattice, see [23-24]), i.e., L is a completely distributive lattice equipped with an order-reversing involution ; and M(L) stands for the set of all nonzero ∨-irreducible elements (or molecules) in L. Let a, A ∈ L. We say that a is way below A in symbols a A if for any directed subset D ⊂ L the relation A ≤ D always implies existence of b ∈ D with a ≤ b (see [5-6,12-13,29]). A mapping f from M(L) into L is denoted by f : M(L) −→ L.If F, G : M(L) −→ L, then F = G indicates F(a) = G(a) for all a ∈ M(L); F ◦ G and F G mean (F ◦ G)(a) = {F(b) | b ≤ G(a)} and (F G)(a) = {F(b) | b G(a)} for a, b ∈ M(L), respectively. We appoint that Ø = 0, in the cases of a lattice and the set of nonnegative real numbers. Other unexplained terminologies and notations and further details can be found in [9,12-13,18-24]. 2. Relations Among the Metric Axioms In this section, we show some relations among the axioms of a pointwise metric for a fuzzy lattice introduced by Shi. Definition 2.1 [19,12] Let d be a mapping from M(L)× M(L) into [0,+∞). (1) d is called a pointwise metric and (L, d) a pointwise metric fuzzy lattice (brief ly, PMFL) if the following axioms are satisf ied: (M ) ∀a, b ∈ M(L),d(a, b) ∈ [0,+∞); d(a, b) = 0 iﬀ a ≤ b. (M ) ∀a, b, c ∈ M(L),d(a, c) ≤ d(a, b)+ d(b, c). (M ) ∀a, b ∈ M(L),d(a, b) = d(a, c). cb (M ) ∀a, b ∈ M(L), d(x, b) = d(y, a). xa yb Fuzzy Inf. Eng. (2010) 2: 187-200 189 (2) d is called a pointwise pseudo-quasi-metric (brief ly, p. q. metric, see [2,6])if it satisf ies (WM ) and (M ), 1 2 where (WM ) ∀a, b ∈ M(L),d(a, b) ∈ [0,+∞); d(a, b) = 0 if a ≤ b. (3) d is called a Shi’s pseudo-quasi-metric (brief ly, Shi’s p. q. metric) if it is a p. q. metric with (M ). (4) d is called a Shi’s pseudo-metric (brief ly, Shi’s p. metric) and (L, d) a pointwise pseudo-metric fuzzy lattice (brief ly, PPMFL) if it satisf ies (WM ), (M ), (M ) and 1 2 3 (M ). Example 2.1 Let X be any set, and d a distance function. Then (X, d ) is a usual 0 0 metric space. Let I = [0, 1], i.e., the real unit interval. Then L = I is a fuzzy lattice, and M(L) = {a | a ∈ X,λ ∈ (0, 1]} (see [23-24]), where a is a fuzzy point. Define λ λ d (a , b ) = max{λ−μ, 0} and d(a , b ) = d (a, b)+ max{λ−μ, 0}. Proving similarly 1 λ μ λ μ 0 the Example 3.2 in [19], we assert that (L, d ) is a PPMFL and (L, d) is a PMFL. From Definition 2.1 we get the following lemma. Lemma 2.1 Let d be a p. q. metric, and let λ ∈ (0,+∞), and a, b, a , a , b , b ∈ 1 2 1 2 M(L). Then (1) a ≤ a implies d(a , b) ≤ d(a , b); and b ≤ b implies d(a, b ) ≥ d(a, b ). 1 2 1 2 1 2 1 2 (2) (M ) is equivalent to (M ) , 4 4 where (M ) ∃x a ,d(x, b)<λ iﬀ ∃y b , d(y, a)<λ. (3) d(e, b) ≤ d(e, c) ≤ d(e, c) ≤ d(a, c). ea ea cb cb ea cb (4) (M ) implies (M )[1, 27], where (M ) ∀a, b ∈ M(L),d(a, b) = d(e, b). ea ∧∨ (5) (M ) implies (M ), where ∧∨ (M ) ∀a, b ∈ M(L),d(a, b) = d(e, c). cb ea ∨ ∧∨ (6) (M ) and (M ) implies (M ). 3 3 Proof (1) If a ≤ a , then 1 2 d(a , b) ≤ d(a , a )+ d(a , b) = d(a , b). 1 1 2 2 2 If b ≤ b , then 1 2 d(a, b ) ≤ d(a, b )+ d(b , b ) = d(a, b ). 2 1 1 2 1 (2) It is straightforward. (3) Suppose e a and c b. Then, d(e, c) ≤ d(e, c). ea Thus, d(e, c) ≤ d(e, c),∀e a. cb cb ea 190 Xing-hua Zhu · Jian-zhong Xiao (2010) Hence, d(e, c) ≤ d(e, c). ea ea cb cb On the other hand, by (1) we have d(e, b) ≤ d(e, c), ∀c b; and d(e, c) ≤ d(a, c), ∀e a. Then, d(e, b) ≤ d(e, c) and d(e, c) ≤ d(a, c). Hence, ea cb d(e, b) ≤ d(e, c) ea ea cb and d(e, c) ≤ d(a, c). cb ea cb (4) See the proof of Lemma 8 in [1] or the proof Lemma 2.1 in [27]. (5) If (M ) holds, then by (4) we have (M ) holds. Hence, d(a, b) = d(a, c) = d(e, c). ea cb cb ∨ ∧∨ (6) If (M ) and (M ) hold, then 3 3 d(a, b) = d(e, c) = d(a, c). ea cb cb Remark 2.1 In [12− 13], Peng gave the following simplif ication of Erceg’s pseudo- metric for a fuzzy lattice. Definition A [12-13] A mapping d : M(L) × M(L) → [0,+∞) is called a Erceg- Peng’s p. metric if the following axioms are satisf ied: (P )∀a, b ∈ M(L),b ≤ a implies d(a, b) = 0. (P )∀a, b, c ∈ M(L),d(a, c) ≤ d(a, b)+ d(b, c). (P )∀a, b ∈ M(L),d(a, b) = d(c, e). ca eb (P )∃x a ,d(b, x)<λ iﬀ ∃y b , d(a, y)<λ. ˜ ˜ ˜ If we redef ine d as Erceg-Peng’s p. metric function by d(a, b) = d(b, a), then d satisf ies the following axioms by Def inition A: (P ) ∀a, b ∈ M(L),a ≤ b implies d(a, b) = 0. ˜ ˜ ˜ (P ) ∀a, b, c ∈ M(L), d(a, c) ≤ d(a, b)+ d(b, c). ˜ ˜ (P ) ∀a, b ∈ M(L), d(a, b) = d(c, e). ca eb ˜ ˜ (P ) ∃x a , d(x, b)<λ iﬀ ∃y b , d(y, a)<λ. Clearly, the diﬀerence between Shi’s p. metric d and Erceg-Peng’s p. metric dis ∗ ∗ ∗ at (M ) and (P ) . But (P ) is not equivalent to (M ). Namely, (M ) implies (P ) by 3 3 3 3 3 3 Lemma 2.1(4), and (P ) does not imply (M ). This can be seen from the following 3 3 counterexample. Counterexample [1,17] Let L = [0, 1] and the order ≤ in L be the usual order of reals, and a = 1− a for each a ∈ [0, 1]. Then we have M(L) = (0, 1] and a bif and only if a < b. Def ine d :(0, 1]× (0, 1] → [0,+∞) as follows: Fuzzy Inf. Eng. (2010) 2: 187-200 191 0, a ≤ b; d(a, b) = 1, a > b. ∗ ∗ ∨ Then it is easy to check that d satisf ies (P ) -(P ) and (M ). But it satisf ies neither 1 4 ∧∨ (M ) nor (M ) (see [1]). From the fact that condition (M ) implies (P ) we have that each Shi’s p. metric 3 3 is an Erceg-Peng’s p. metric. In the light of the above counterexample we conclude that the converse is not true. By Def inition 2.1 and Lemma 2.1, the following statements hold: a pointwise metric is a Shi’s p. metric; if d is a Shi’s p. metric with (M ), then d is a pointwise metric. If d is a Shi’s p. q. metric, then d is a p. q. metric with (M ); and the contrary ∨ ∧∨ is false; d is accurately equivalent to a p. q. metric with (M ) and (M ). 3 3 Example 2.2 Let L = [0, 1] and the order ≤ in L be the usual order of reals, and a = 1 − a for each a ∈ [0, 1]. Then d satisfies (WM ), (M ) and (M ). But it does 1 2 4 not satisfy (M ). This shows that the axiom (M ) is independent. 3 3 Now we define d :(0, 1]× (0, 1] → [0,+∞) as follows: 0, a ≤ b; d(a, b) = a− b, b < a ≤ 2b; b, a > 2b. 1 2 Then it is easy to check that d satisfies (WM ), (M ) and (M ). If a = , b = , then 1 2 3 9 3 2 1 d(x, b) = d(x, b) = , d(y, a) = d(y, a) = , 9 9 x>1−a y>1−b xa yb i.e., d does not satisfy (M ). This shows that the axiom (M ) is independent and d is 4 4 a Shi’s p. q. metric function. Definition 2.2 [27] Let d be a p. q. metric, and f a mapping from M(L) into L. Then f is called a neighborhood mapping if for each a ∈ M(L),a ≤ f (a); and f is called a remote-neighborhood mapping if for each a ∈ M(L),a f (a). The sets of all neighborhood mappings and of all remote-neighborhood mappings are denoted by N (L) and R(L), respectively. For each a, b ∈ M(L) andε,λ ∈ (0,+∞), we def ine U ,B ∈ N (L) and P ,W ∈ R(L) by ε ε λ λ U (b) = {c ∈ M(L)| d(c, b)<ε}, B (b) = {c ∈ M(L) | d(c, b) ≤ ε}; ε ε P (a) = {c ∈ M(L) | d(a, c) ≥ λ}, W (a) = {c ∈ M(L) | d(a, c)>λ}; λ λ which are called the neighborhood balls and the remote-neighborhood balls respec- tively. By Definition 2.1 and 2.2 we have the following Lemma 2.2-2.7. The proofs of Lemma 2.2-2.4 are similar to the given ones in lemmas in Section 2 in [27], so we omit them. Lemma 2.2 Let ε,λ ∈ (0,+∞), and a, b, c ∈ M(L). (i) If d is a p. q. metric, then (1) U (b) ≤ B (b); ε ε 192 Xing-hua Zhu · Jian-zhong Xiao (2010) (2) W (a) ≤ P (a); λ λ (3) c U (b) implies d(c, b)<ε; (4) c W (a) implies d(a, c)>λ. (ii) If d is a p. q. metric with (M ), then (1)-(6) hold, where (5) c ≤ B (b) iﬀ d(c, b) ≤ ε; (6) c ≤ U (b) implies d(c, b) ≤ ε. (iii) If d is a Shi’s p. q. metric, then (1)-(8) hold, where (7) c ≤ P (a) iﬀ d(a, c) ≥ λ; (8) c ≤ W (a) implies d(a, c) ≥ λ. Lemma 2.3 Letε,λ ∈ (0,+∞) and a, b ∈ M(L). (i) If d is a p. q. metric, then (1) U (b) = U (b); ε δ δ<ε (2) U (b) = B (b). ε δ δ<ε (3) W (a) = W (a); λ μ λ<μ (4) W (a) = P (a). λ μ λ<μ (ii) If d is a p. q. metric with (M ), then (1)-(6) hold, where (5) B (b) = B (b); ε δ ε<δ (6) B (b) = U (b). ε δ ε<δ (iii) If d is a Shi’s p. q. metric, then (1)-(8) hold, where (7) P (a) = P (a); λ μ μ<λ (8) P (a) = W (a). λ μ μ<λ Lemma 2.4 Letε,λ ∈ (0,+∞) and a, b ∈ M(L). (i) If d is a p. q. metric, then (1) U (c) ≤ U (b); ε ε cb (2) B (c) ≤ B (b); ε ε cb (3) W (e) ≤ W (a); λ λ ea (4) P (e) ≤ P (a). λ λ ea (ii) If d is a Shi’s p. q. metric, then (5)-(8) hold, where (5) U (c) = U (b); ε ε cb (6) U (b) ≤ B (c) ≤ B (b); ε ε ε cb (7) W (e) = W (a); λ λ ea (8) W (a) ≤ P (e) ≤ P (a). λ λ λ ea Lemma 2.5 [27] Letε,λ ∈ (0,+∞) and a, b ∈ M(L). If d is a Shi’s p. metric, then (1) P (z) = U (b); λ λ zb (2) U (z) ≤ P (a). ε ε za Fuzzy Inf. Eng. (2010) 2: 187-200 193 ∨ ∨ Lemma 2.6 If d is a Shi’s p. metric, then (M ) holds, and (M ) is equivalent to 4 4 ∨ ∗ (M ) , where (M ) d(b, x) = d(a, y); xa yb ∨ ∗ (M ) ∃x a ,d(b, x)>ε iﬀ ∃y b ,d(a, y)>ε. Proof Suppose that d(b, x) >ε. Then there exists x a such that d(b, x) >ε. xa By Lemma 2.2(6) and 2.5(1) we see that b U (x) = P (z) , i.e., b P (z) for ε ε ε zx each z x .By a x we get b P (a) , i.e., P (a) b . Thus, there exists y b ε ε such that y ≤ P (a). This implies d(a, y) ≥ ε. Hence d(a, y) ≥ ε. This shows yb d(b, x) ≤ d(a, y). Similarly we can prove d(b, x) ≥ d(a, y). (M )is xa yb xa yb proved. ∨ ∨ ∗ It is straightforward to prove that (M ) is equivalent to (M ) . 4 4 Lemma 2.7 [27] Let ε,λ ∈ (0,+∞) and a, b ∈ M(L). If d is a Shi’s p. metric, then (1) B (z) = W (a); ε ε za (2) W (z) ≤ B (b); λ λ zb (3) U (b) ≤ W (z) ≤ B (b); λ λ λ zb (4) W (a) ≤ U (z) ≤ P (a). ε ε ε za Definition 2.3 [27] Let F = { f |η ∈ (0,+∞)}⊂ N (L) and G = {g |β ∈ (0,+∞)}⊂ η β R(L) be families of mappings from M(L) into L. Then g is said to be conjugate to f if g (a) = f (z) for each a ∈ M(L). Similarly, f is said to be conjugate to g η η η β β za if f (a) = g (z) for each a ∈ M(L). F and G are called conjugate families of β β za mappings if either g is conjugate to f for eachη ∈ (0,+∞),or f is conjugate to g η η β β for eachβ ∈ (0,+∞). Remark 2.2 Let (L, d) be a PPMFL. From Lemma 2.5 and 2.7 it can be seen that {B |ε ∈ (0,+∞)} and{W |λ ∈ (0,+∞)} are conjugate families of mappings; and so ε λ are{U |ε ∈ (0,+∞)} and{P |λ ∈ (0,+∞)}. ε λ 3. Some Representation Results In this section, by virtue of the conjugate families of mappings, we shall eventually obtain the representations of pointwise metric and p. metric for fuzzy lattices. Theorem 3.1 Let d be a p. q. metric with (M ). For each ε ∈ (0,+∞) def ine F ∈ N (L) such that ∀b ∈ M(L),U (b) ≤ F (b) ≤ B (b). Then the following ε ε ε ε statements are valid: (WN ) ∀b ∈ M(L), F (b) = 1;a ≤ b implies ∀ε ∈ (0,+∞),a ≤ F (b). 1 ε ε ε>0 (N ) ∀ε,δ ∈ (0,+∞),F ◦ F ≤ B . 2 ε δ ε+δ (N ) ∀ε ∈ (0,+∞),U = F ≤ F = B . 3 ε δ δ ε δ<ε ε<δ 194 Xing-hua Zhu · Jian-zhong Xiao (2010) Proof For all ε ∈ (0,+∞), by U ≤ F ≤ B , it is clear that d(a, b) <ε implies ε ε ε a ≤ F (b), and a ≤ F (b) implies d(a, b) ≤ ε. ε ε (WN ) For each a ∈ M(L), by d(a, b) ∈ [0,+∞), there isδ> 0 such that d(a, b) < δ. Hence, a ≤ F (b) ≤ F (b). This shows that F (b) = 1. If a ≤ b, then for δ ε ε ε>0 ε>0 eachε ∈ (0,+∞), d(a, b)<ε, and hence a ≤ F (b). (N ) Suppose that a F ◦ F (b), where a, b ∈ M(L). Then there is c ∈ M(L) 2 ε δ such that a ≤ F (c) and c ≤ F (b). Thus, d(a, c) ≤ ε and d(c, b) ≤ δ.By (M )we ε δ 2 have d(a, b) ≤ ε+δ, and so a ≤ B (b). This shows F ◦ F ≤ B . ε+δ ε δ ε+δ (N ) By Lemma 2.3(1),(2),(5) and (6), we obtain that U (b) = U (b) ≤ F (b) ≤ B (b) = U (b), ε δ δ δ ε δ<ε δ<ε δ<ε B (b) = U (b) ≤ F (b) ≤ B (b) = B (b). ε δ δ δ ε ε<δ ε<δ ε<δ Hence, U = F , B = F . ε δ ε δ δ<ε ε<δ Theorem 3.2 Let {F | F : M(L) → L,ε ∈ (0,+∞)} be a family of mappings ε ε satisfying (WN ), (N ) and (N ), where U = F ,B = F . Def ine d from 1 2 3 ε δ ε δ δ<ε ε<δ M(L)× M(L) into [0,+∞) by d(a, b) = {ε| a F (b)}. Then d is a p. q. metric with (M ),{F | ε ∈ (0,+∞)}⊂ N (L) and d(a, b) = {ε | a ≤ F (b)}. ε ε Proof We first prove the following: (1) d(a, b)<ε implies a ≤ F (b), and a ≤ F (b) implies d(a, b) ≤ ε. ε ε (2) d(a, b)<ε implies a ≤ U (b), and d(a, b) ≤ ε iﬀ a ≤ B (b). ε ε In fact, by a F (b)wehave d(a, b) = {δ | a F (b)}≥ ε. This shows that ε δ d(a, b) <ε implies a ≤ F (b). Suppose that d(a, b) = {δ | a F (b)}>ε. Then ε δ there isδ>ε such that a F (b). By (N )wehave F (b) ≥ F (b), and hence δ 3 δ ε a F (b). This shows that a ≤ F (b) implies d(a, b) ≤ ε. Therefore the assertion ε ε (1) is valid. Let d(a, b) <ε. Then there exists δ such that d(a, b) <δ<ε. By (1) we have a ≤ F (b) ≤ F (b) = U (b). If d(a, b) ≤ ε, then for eachδ>ε we have δ δ ε δ<ε d(a, b)<δ. By (1), a ≤ F (b). Hence a ≤ F (b)= B (b). Conversely, if a ≤ B (b) δ δ ε ε ε<δ = F (b), then for eachδ>ε, a ≤ F (b). By (1), d(a, b) ≤ δ. Since δ is arbitrary, δ δ ε<δ we have d(a, b) ≤ ε. Therefore the assertion (2) holds. Without loss of generality we suppose that{ε| a F (b)} Ø, by the appointment Ø = 0. From (WN )wehave a ≤ F (a) for each ε ∈ (0,+∞). This shows that {F | ε ∈ 1 ε ε (0,+∞)}⊂ N (L). (WM ) Let a, b ∈ M(L). From F (b) = 1 it is clear that d(a, b) ∈ [0,+∞). 1 ε ε>0 Since a ≤ b implies for allε> 0, a ≤ F (b), by (1) we have d(a, b) ≤ ε for each ε> 0. This shows that d(a, b) = 0. (M ) Suppose that d(a, b) = δ and d(b, c) = ε. Then for eachσ> 0wehave d(a, b)<δ+σ, d(b, c)<ε+σ. Condition (1) implies that a ≤ F (b), b ≤ F (c). δ+σ ε+σ Thus, a ≤ F ◦ F (c). From (N ), a ≤ B (c). By (2), it means d(a, c) ≤ δ+σ ε+σ 2 δ+ε+2σ δ+ε+ 2σ. Letting σ −→ 0 we obtain d(a, c) ≤ δ+ε. Fuzzy Inf. Eng. (2010) 2: 187-200 195 (M ) Suppose that d(a, b) =ε> 0. Then, for each δ ∈ (0,ε)wehave d(a, b)>δ. By (1), it means a F (b). Thus, there exists e a such that e F (b), i.e., δ δ d(e, b) ≥ δ. Hence, d(e, b) ≥ δ. Since δ is arbitrary, we have d(e, b) ≥ ε. ea ea Now we further claim that d(e, b) = ε. Otherwise, if d(e, b) >δ>ε, then ea ea there exists e a such that d(e, b) >δ. Condition (1) implies that e F (b). So a F (b), i.e.,ε = d(a, b) ≥ δ, a contradiction. Set d(a, b) = 0. By (1), it is clear that d(e, b) = 0. Therefore, d(e, b) = d(a, b). ea ea Letβ = {ε| a ≤ F (b)}. Now we prove thatβ = d(a, b). Ifβ< d(a, b), then there is δ such thatβ<δ< d(a, b). By {ε | a ≤ F (b)}<δ, there existsε<δ such that a ≤ F (b), i.e., d(a, b) ≤ ε, which contradictsδ< d(a, b). Ifβ> d(a, b), then there is ρ such thatβ>ρ> d(a, b). By {ε | a ≤ F (b)} >ρ we have a F (b), i.e., ε ρ d(a, b) ≥ ρ, which contradicts d(a, b)<ρ. Thereforeβ = d(a, b). Remark 3.1 From Theorem 3.1 and 3.2 we see that the family of neighborhood mappings {F } can represent the p. q. metric with (M ), if it satisf ies (WN ), (N ) ε 1 2 and (N ). There are some continuity and monotonicity in (N ). (N ) is a description 3 3 2 of the axiom (M ). Theorem 3.3 Let d be a Shi’s p. q. metric. For each λ ∈ (0,+∞) and b ∈ M(L), def ine G ∈ R(L) by W (b) ≤ G (b) ≤ P (b). Then the following statements are λ λ λ λ valid: (WR ) ∀a ∈ M(L), G (a) = 0;a ≤ b implies ∀λ ∈ (0,+∞),b G (a). 1 λ λ λ>0 (R ) ∀λ,μ ∈ (0,+∞),G G ≥ W . 2 μ λ λ+μ (R ) ∀λ ∈ (0,+∞), W = G ≤ G = P . 3 λ μ μ λ λ<μ μ<λ Proof For all λ ∈ (0,+∞), by W ≤ G ≤ P , it is clear that d(a, b) >λ implies λ λ λ b ≤ G (a), and b ≤ G (a) implies d(a, b) ≥ λ. λ λ (WR )If G (a) 0, then there is b ∈ M(L) such that b ≤ G (a). Thus, 1 λ λ λ>0 λ>0 for all λ ∈ (0,+∞), we have b ≤ G (a), i.e., d(a, b) ≥ λ. It yields d(a, b) =+∞,a contradiction. Hence, G (a) = 0. Suppose that a ≤ b. Then for each λ ∈ (0,+∞), λ>0 we have d(a, b) = 0<λ, and so b G (a). (R ) Suppose that c W (a). From Lemma 2.2(4) we obtain d(a, c) >μ + λ. 2 λ+μ Now we prove c ≤ G G (a). If c G G (a), where a, c ∈ M(L), then there is μ λ μ λ b ∈ M(L) such that c G (b) and b G (a), i.e., d(b, c) ≤ μ and d(a, b) ≤ λ.By μ λ (M ) we derive d(a, c) ≤ μ+λ, a contradiction. Hence, G G ≥ W . 2 μ λ λ+μ (R ) By Lemma 2.3 (3),(4),(7) and (8) we get W (a) = W (a) ≤ G (a) ≤ P (a) = W (a). λ μ μ μ λ λ<μ λ<μ λ<μ P (a) = W (a) ≤ G (a) ≤ P (a) = P (a). λ μ μ μ λ μ<λ μ<λ μ<λ Hence, W = G , P = G . λ μ λ μ λ<μ μ<λ 196 Xing-hua Zhu · Jian-zhong Xiao (2010) Theorem 3.4 Let {G | G : M(L) −→ L,λ ∈ (0,+∞)} be a family of mappings λ λ satisfying (WR ), (R ) and (R ), where W = G ,P = G . Def ine d from 1 2 3 λ μ λ μ λ<μ μ<λ M(L) × M(L) into [0,+∞) by d(a, b) = {λ | b G (a)}. Then d is a Shi’s p. q. metric,{G |λ ∈ (0,+∞)}⊂ R(L) and d(a, b) = {λ| b ≤ G (a)}. λ λ Proof We first prove the following: (1) d(a, b)>λ implies b ≤ G (a), and b ≤ G (a) implies d(a, b) ≥ λ. λ λ (2) d(a, b)>λ implies b ≤ W (a), and d(a, b) ≥ λ iﬀ b ≤ P (a). λ λ In fact, by b G (a)wehave d(a, b) = {μ | b G (a)}≤ λ. This shows λ μ that d(a, b) >λ implies b ≤ G (a). Suppose that d(a, b) = {μ | b G (a)}<λ. λ μ Then there isμ<λ such that b G (a). By (R )wehave G (a) ≤ G (a), and so μ 3 λ μ b G (a). This shows that b ≤ G (a) implies d(a, b) ≥ λ. Therefore the assertion λ λ (1) is valid. Let d(a, b)>λ. Then there exists μ such that d(a, b)>μ>λ. By (1) we have b ≤ G (a) ≤ G (a) = W (a). From (1) and G (a) = P (a) we also have μ μ λ μ λ λ<μ λ<μ d(a, b) ≥ λ iﬀ b ≤ P (a). Therefore the assertion (2) is valid. Without loss of generality, we suppose that {λ | b ≤ G (a)} Ø, by the appoint- ment Ø = 0. From (WR )wehave ∀λ ∈ (0,+∞), a G (a). This shows that {G | λ ∈ 1 λ λ (0,+∞)}⊂ R(L). (WM ) Let a, b ∈ M(L). By G (a) = 0 we get b G (a). Then there exists 1 λ λ λ>0 λ>0 λ> 0 such that b G (a), i.e., d(a, b) ≤ λ, which shows that d(a, b) ∈ [0,+∞). If a ≤ b, then for each λ ∈ (0,+∞), b G (a), i.e., d(a, b) ≤ λ, which shows that d(a, b) = 0. (M ) Suppose that d(a, b) = λ, d(b, c) = μ. Then for eachρ> 0wehave d(a, b) < λ+ρ, d(b, c)<μ+ρ. By (1) we have b G (a), c G (b). Thus, c G λ+ρ μ+ρ μ+ρ G (a). From (R )wehave c W (a). By (2) we have d(a, c) ≤ λ + μ + 2ρ. λ+ρ 2 λ+μ+2ρ Sinceρ is arbitrary, we obtain d(a, c) ≤ λ+μ. (M ) Without loss of generality, we suppose that d(a, b) =λ> 0 and d(a, c) = cb μ> 0. For eachν<λ we have d(a, b) >ν, i.e., b ≤ G (a). Thus, for each c b, we have c ≤ G (a), i.e., d(a, c) ≥ ν, and hence μ = d(a, c) ≥ ν. This shows that cb μ ≥ λ. Conversely, for each c b,we have d(a, c) ≥ μ, i.e., c ≤ P (a). Hence, b ≤ P (a). This shows d(a, b) = λ ≥ μ. Therefore λ = μ. Let β = {λ | b ≤ G (a)}. Now we prove that β = d(a, b). Ifβ< d(a, b), then there is δ such thatβ<δ< d(a, b). By {λ | b ≤ G (a)}<δ we have b G (a), i.e., λ δ d(a, b) ≤ δ, which contradictsδ< d(a, b). Ifβ> d(a, b), then there exists ρ such that β>ρ> d(a, b). By {λ | b ≤ G (a)} >ρ, there isλ>ρ such that b ≤ G (a), i.e., λ λ d(a, b) ≥ λ, which contradicts withρ> d(a, b). Therefore,β = d(a, b). Remark 3.2 From Theorem 3.3 and 3.4, we see that the family of remote-neighborhood mappings {G } can represent Shi’s p. q. metric, if it satisf ies (WR ), (R ) and (R ). λ 1 2 3 There are some continuity and monotonicity in (R ). (R ) is a description of the axiom 3 2 (M ). Theorem 3.5 Let (L, d) be a PPMFL. Let F = {F | ε ∈ (0,+∞)}⊂ N (L) and G ε Fuzzy Inf. Eng. (2010) 2: 187-200 197 = {G | λ ∈ (0,+∞)}⊂ R(L) be conjugate families of mappings. Suppose that one of the follow conditions is satisf ied: (i) U ≤ F ≤ B ,if G is conjugate to F , ∀ε ∈ (0,+∞). ε ε ε ε ε (ii) W ≤ G ≤ P ,if F is conjugate to G ,∀λ ∈ (0,+∞). λ λ λ λ λ Then the following statements are valid: (WN ) ∀b ∈ M(L), F (b) = 1;a ≤ b implies ∀ε> 0,a ≤ F (b). 1 ε ε ε>0 (WR ) ∀a ∈ M(L), G (a) = 0;a ≤ b implies ∀λ> 0,b G (a). 1 λ λ λ>0 (N ) ∀ε,δ ∈ (0,+∞),F ◦ F ≤ B . 2 ε δ ε+δ (R ) ∀λ,μ ∈ (0,+∞),G G ≥ W . 2 λ μ λ+μ (N ) ∀ε ∈ (0,+∞),U = F ≤ F = B . 3 ε δ δ ε δ<ε ε<δ (R ) ∀λ ∈ (0,+∞),W = G ≤ G = P . 3 λ μ μ λ λ<μ μ<λ (NR) ∀a, b ∈ M(L),a U (b) implies b P (a);b P (a) implies a ≤ U (b); λ λ λ λ b W (a) implies a B (b) and a B (b) implies b ≤ W (a). ε ε ε ε Proof If the condition (i) is satisfied, then W (a) = B (z) ≤ F (z) = G (a) ≤ U (z) ≤ P (a), λ λ λ λ λ λ za za za i.e., W ≤ G ≤ P ,∀λ ∈ (0,+∞). λ λ λ If the condition (ii) is satisfied, then U (b) = P (z) ≤ G (z) = F (b) ≤ W (z) ≤ B (b), ε ε ε ε ε ε zb zb zb i.e., U ≤ F ≤ B ,∀ε ∈ (0,+∞). ε ε ε By Theorem 3.1 and 3.3, we obtain (WN ), (N ), (N ), (WR ), (R ) and (R ). 1 2 3 1 2 3 (NR) is obviously true. Theorem 3.6 Let F = {F | ε ∈ (0,+∞)}⊂ N (L) and G = {G | λ ∈ (0,+∞)}⊂ ε λ R(L) be conjugate families of mappings from M(L) into L. Let F ≤ F whenδ<ε, δ ε or G ≥ G whenμ<λ. Let U = F ,B = F ,P = G ,W = G . μ λ ε δ ε δ λ μ λ μ δ<ε ε<δ μ<λ λ<μ Def ine d from M(L) × M(L) into [0,+∞) by d(a, b) = {ε | a F (b)} and ρ from M(L)× M(L) into [0,+∞) by ρ(a, b) = {λ | b G (a)}. Suppose that the statement (NR) is valid and so is one of the following: (i) F satisf ies (WN ) and (N ). 1 2 (ii) G satisf ies (WR ) and (R ). 1 2 Then (L, d) is a PPMFL, and d(a, b) = ρ(a, b)= {λ| b ≤ G (a)}= {ε| a ≤ F (b)}. λ ε Proof It is evident that F satisfies (N ) and G satisfies (R ). Duplicating the proof 3 3 of Theorem 3.2, from condition (N ) we have the following: (1) d(a, b)<ε implies a ≤ F (b), and a ≤ F (b) implies d(a, b) ≤ ε. ε ε (2) d(a, b)<ε implies a ≤ U (b), and d(a, b) ≤ ε iﬀ a ≤ B (b). ε ε Duplicating the proof of Theorem 3.4, from (R ) we also have the following: (3) ρ(a, b)>λ implies b ≤ G (a), and b ≤ G (a) impliesρ(a, b) ≥ λ. λ λ (4) ρ(a, b)>λ implies b ≤ W (a), and ρ(a, b) ≥ λ iﬀ b ≤ P (a). λ λ 198 Xing-hua Zhu · Jian-zhong Xiao (2010) Then, by (3)-(4) and (R )wehave(M ) holds with respect to ρ. We shall prove that 3 3 ρ(a, b) = d(a, b). Set ρ(a, b) = λ, d(a, b) = ε. Then by (4) and (2) we have a ≤ B (b) and b P (a) ε μ for eachμ>λ. From (NR) we have b W (a) and a ≤ U (b) ≤ B (b). By (2) and ε μ μ (4) we derive that ε = d(a, b) ≤ μ, i.e., ε ≤ λ, and λ = ρ(a, b) ≤ ε. This shows that ε = λ. By Theorem 3.2 and 3.4, we have d(a, b) = {ε| a ≤ F (b)}= ρ(a, b) = {λ| b ≤ G (a)}. If the condition (i) is satisfied, then, by duplicating the proof of Theorem 3.2, from (1)-(2), (WN ) and (N ), we can obtain that (WM ) and (M ) hold with respect to d. 1 2 1 2 If the condition (ii) is satisfied, then, by duplicating the proof of Theorem 3.4, from (3)-(4), (WR ) and (R ), we can also obtain that (WM ) and (M ) hold with 1 2 1 2 respect to ρ. Since ρ(a, b) = d(a, b), we conclude that (WM ), (M ) and (M ) hold. 1 2 3 To prove (M ), we shall check (M ) by Lemma 2.1(2). 4 4 Case A. G is conjugate to F for each ε ∈ (0,+∞). ε ε Set ∃x a , d(x, b)<λ. Then there exists μ such that d(x, b)<μ<λ. By (4) we have b P (x) ≥ G (x) = F (z) , μ μ μ zx i.e., b F (z) , for each z x . Since a x ,we have b F (a) , i.e., F (a) b . μ μ μ Hence, there exists y b such that y F (a) ≤ U (a). From (NR) we have μ λ a P (y). By (4), this shows that ∃y b , d(y, a)<λ. The contrary is true, too. So (M ) holds. Case B. F is conjugate to G for eachλ ∈ (0,+∞). λ λ Set ∃x a , d(x, b)<λ. Then there exists μ such that d(x, b)<μ<λ. By (4) we have b P (x). Thus, b P (x) ≥ G (x) = F (a) ≥ U (a) , μ μ μ λ xa xa i.e., U (a) b . Hence, there exists y b such that y U (a). From (NR) we have λ λ a P (y). By (4), this shows that ∃y b , d(y, a)<λ. The contrary is true, too. So (M ) holds. Remark 3.3 From Theorem 3.5 and 3.6, we see that the conjugate families F and G can represent Shi’s p. metric, if they satisfy the conditions of Theorem 3.6. (NR) is a description of the axiom (M ). Theorem 3.7 Let (L, d) be a PMFL. Let F , G be as Theorem 3.5. Then the state- ments (N ), (N ), (R ), (R ) and (NR) to be as Theorem 3.5 are valid, and so are the 2 3 2 3 following: (N ) ∀b ∈ M(L), F (b) = 1; a ≤ biﬀ ∀ε> 0, a ≤ F (b). 1 ε ε ε>0 (R ) ∀a ∈ M(L), G (a) = 0; a ≤ biﬀ ∀λ> 0, b G (a). 1 λ λ λ>0 Proof It follows from (M ) and Theorem 3.5. 1 Fuzzy Inf. Eng. (2010) 2: 187-200 199 Theorem 3.8 Let F , G ,U ,B ,W ,P , d and ρ be as Theorem 3.6. Suppose that ε ε ε ε the statement (NR) is valid and so is one of the following: (i) F satisf ies (N ) and (N ). 1 2 (ii) G satisf ies (R ) and (R ). 1 2 Then (L, d) is a PMFL, and d(a, b) = ρ(a, b) = {λ| b ≤ G (a)} = {ε| a ≤ F (b)}. λ ε Proof It follows from (N ) (or (R )) and Theorem 3.6. 1 1 Example 3.1 An important example of PPMFL is the L-fuzzy real line (see [26]). By means of four gauge functionals introduced in [26], we can check that the L-fuzzy real line satisfies the conditions and conclusions of Theorem 3.1-3.6. Acknowledgments The authors are grateful to the referees for their suggestions. This work is supported by the Natural Science Foundation of Nanjing University of Information Science and Technology of China (No.20080286). References 1. Chen P, Shi F G (2008) A note on Erceg’s pseudo-metric and pointwise pseudo-metric. J. Math. Res. Expo. 28: 439-443 2. Erceg M A (1979) Metric in fuzzy set theory. J. Math. Anal. Appl. 69: 205-230 3. 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Journal
Fuzzy Information and Engineering
– Taylor & Francis
Published: Jun 1, 2010
Keywords: Fuzzy lattice; Pointwise metric; Conjugate mapping