Q-convergence of ideals in fuzzy lattices and its applications

ZZY sets and systems Fuzzy Sets and Systems92 (1997) 357-363

ELSEVIER

Q-convergence of ideals in fuzzy lattices and its applications Bai Shi-Zhong Department of Mathematics,Xiangtan University,Xiangten Hunan 411105, People'sRepublic of China ReceivedAugust 1995;revisedJune 1996

Abstract In this paper, we study Q-convergence and Q*-convergence of ideals in fuzzy lattices by the concept of Q-remoteneighborhood. Then we introduce the strong S-irresoluteness, S*-irresoluteness and S*-strong semicontinuity on fuzzy lattices. We also study some properties of the notions above and strong semicontinuity [2, 3] and S-irresoluteness [3, 4] with the aid of the Q-convergence(Q*-convergence) of ideals. © 1997 Elsevier Science B.V. Keywords: Fuzzy lattice; Strongly semiclosedelement;Ideal; Q-remote-neighborhood;Q-convergence;Order-homomorphism; Strong semicontinuity

1. Introduction and preliminaries Wang established the theory of topological molecular lattices by three basic concepts: molecule, remote-neighborhood and order-homomorphism in [7-9]. In it he established the complete Moore-Smith convergence theory by remoteneighborhood. Yang established the convergence theory of ideals in fuzzy lattices in [11]. In the author's paper [4], the concept of Q-remote-neighborhood is introduced and the Q-convergence theory of molecular nets in fuzzy lattices is established. Here, in Section 2 of this paper, we establish the Q-convergence (Q*-convergence) theory of ideals in fuzzy lattices by Q-remote-neighborhood. In Section 3 we discuss the relationship between Q*convergence of ideals and that of molecular nets in fuzzy lattices. The order-homomorphisms [10] in fuzzy lattices is a generalization of the concept of Zadeh's functions (or called fuzzy functions) [12]. It is one of the

most important tools of studying topology in fuzzy lattices. In [3], we first introduced and studied the strongly semiopen elements and strongly semicontinuous order-homomorphisms, which are extensions of corresponding concepts in the author's paper [2]. In Section 4 of this paper we introduce the strongly S-irresolute, S*-irresolute and S*strongly semicontinuous order-homomorphisms on fuzzy lattices. Then we study some properties of the notions above and strongly semicontinuous and S-irresolute order-homomorphisms with the aid of the Q-convergence (Q*-convergence) of ideals. In this paper, L denotes a fuzzy lattice, i.e., completely distributive lattice with order-reversing involutions ..... . M denotes the set consisting of all nonzero v-irreducible elements (or molecules, or points for short) in L. 0 and 1 denote the least and greatest elements in L, respectively. (L(M), b) will denote a topological molecular lattice (TML) with the topology b [7]. (L(M), 6) also briefly denote

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(L, 6) or L. Denote 6' = {A: A' e ~}. A °, A - and A' will denote the interior, closure and complement of A ~ L, respectively.

Definition 1.1 (Wang [7]). Let (L(M), 6) be a TML, e e M, P e 6' and e ~ P. Then P is called a remoteneighborhood (RN) of e, and the set of all RNs of e will be denoted by r/(e).

Definition 1.6 (Wang [10]). A mapping f : L1 ~ L2 is called an order-homomorphism if the following conditions hold: (1) f(O) = O. (2) f ( V A i ) = Vf(A,) for {Ai} ~ L1. (3) f - I(B') = ( f - I(B))' for each B 6 L2.

2. Q-convergence and Q*-convergence of ideals Definition 1.2 (Bai [3]). Let (L, 6) be a T M L and A e L. A is called a strongly semiopen element iff there is a B ~ 6 such that B ~< A ~< B-o. A is called a strongly semiclosed element iff there is a B ~ 6' such that B ° - ~< A ~< B. Q and Q' will denote the family of strongly semiopen elements and family of strongly semiclosed elements of (L, 6), respectively. Clearly, every open element is strongly semiopen and every strongly semiopen element is not only semiopen [3] but also preopen [3]. That none of the converses need be true is shown in [3].

Theorem 1.3 (Bai [3]). Any union (intersection) of strongly semiopen (semiclosed) elements is strongly semiopen (semiclosed). The intersection (union) of any two strongly semiopen (semiclosed) elements need not be strongly semiopen (semiclosed) [3]. Definition 1.4 (Bai [3]). Let L be a T M L and A e L. Then A zx= V{B: B ~< A, B e Q}, A- =A{B:A~ A1 v A 2. (2) ICB.

Definition 2.1 (Bai I-4]). Let (L(M), (5) be a TML, e e M, P e Q' and e ~ P. Then P is called a Qremote-neighborhood of e. The set of all Q-remoteneighborhoods (briefly, Q-R-neighborhoods or Q-RNs) of e will be denoted by ¢(e). Remark 2.2. Since the union of any two strongly semiclosed elements need not be a strongly semiclosed element, P e ~(e) and R ~ ~(e) do not necessarily lead to P v R e ~(e). Let (*(e) = { P c ¢(e): for each R e ~(e), P v R e ~(e)}, it is easy to prove that ~*(e) is an ideal base on L. Clearly, r/(e) c ~.*(e) c ((e). Definition 2.3. Let I be an ideal in (L(M), 6) and e e M. Then (1) e is said to be a Q-limit point of I (or I Qconverges to e; in symbols, I ~ e(Q)), if ~(e) c I. (2) e is said to be a Q-cluster point of 1 (or I Q-accumulates to e; in symbols, I ~ e(Q)), if for each A e I and each P ~ ~(e), A v P ~ 1. (3) e is said to be a Q*-limit point of I (or I Q*converges to e; in symbols, I ~ e(Q*)) if ~*(e) ~ I. (4) e is said to be a Q*-cluster point of I (or I Q*-accumulates to e, in symbols, I ~ e(Q*)), if for each A e I and each P ~ ~*(e), A v P # 1. Examples 1. (1) Let e be a Q-limit (Q*-limit) point of I. Then e is a limit point [11] of I. (2) Let e be a Q-cluster (Q*-cluster) point of I. Then e is a cluster point E11] of I. The union of all Q-limit points, all Q-cluster points, all Q*-limit points and all Q*-cluster points of I will be denoted by Q-lim I, Q-ad I, Q*-lim I and Q*-adI, respectively. Obviously, Q-lim I ~< Q-ad I. Q*-limI ~< Q*-adl.

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Q-limI ~< Q*-lim I ~< limI [113. Q-adI <<,Q*-adI ~< a d I [11].

But not vice versa. One can readily check the following proposition. Proposition 2.4. Let I and J be both ideals in (L(M), ~), I c J and e • M. Then (1) I f l ~ e ( Q ) , then J -~e(Q). (2) I f l -~ e(Q*), then J -~e(Q*). (3) I f J ~ e(Q), then I ~ e ( Q ) . (4) I f J ~ e(Q*), then I ~ e(Q*). (5) I f I ~ e(Q) and d <<.e, then I ~ d ( Q ) . (6) I f l ~ e ( Q * ) and d <~ e, then I ~ d ( Q * ) . (7) I f I ~ e(Q) and d <. e, then I ~ d(Q). (8) I f I ~ e ( Q * ) and d <<.e, then I ~ d ( Q * ) . Definition 2.5. Let fi be an ideal base in L and l(fi) = {A • L: there exists B • fl such that B >~ A};

then 1([t) is an ideal in L and is called the ideal generated by fl [11]. Define Q-limfl = Q-limI(fl) and Q*-lim fl = Q*-lim I(fl). Theorem 2.6. Let I be an ideal in (L(M), 6) and e • M. I f I oo e(Q*), then there exists an ideal J ~ I in L such that J --* e(Q*). Proof. If I o0 e(Q*), then for each P • ~*(e) and each B • I we have B v P ~ 1. Put fi = {B v P: B • I, P • ~*(e)}.

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much exist a Q-limit point d of I such that d ~ P, i.e., P • ~(d). Hence, P • I, i.e., ~(e)~ I. Thus, I ~ e(Q). Theorem 2.8. Let I be an ideal in (L(M), 6) and e • M. Then Q-lim I and Q-ad I are strongly semiclosed. Proof. Let e ~< (Q-limI)-. Then Q-limI 4; P for each P • ~(e). Hence there exists d • M such that d ~< Q-lim I and d 4; P- Then P • ~(d). By Definition 2.3 we have P • I. Thus, e ~< Q-liml. This implies that Q-limI is strongly semiclosed. Similarly, Q-ad I is strongly semiclosed. Theorem 2.9. Let (L(M), 6) be a T M L , A • L and e • M. I f e <~ A - , then there exists an ideal I in L such that Aq~I and I-~e(Q*). Proof. Let e ~< A -. Then A 4; P for each P • ~*(e). Clearly, ~*(e) is an ideal base in L. Put I = {B • L: there exists P • ~*(e) such that P/> B}. Then I is the ideal generated by ~*(e). Thus, A(EI and 1 --* e(Q*). Theorem 2.10. Let I be an ideal in (L(M), 5), A • L and e • M. I r A q i and I-~e(Q), then e <<.A - . Proof. Let I --* e(Q) and A(~I. Then P • I for each P • ~(e), and A 4; P. By Definition 2.2 in [4], e is a Q-adherence point of A. From Theorem 2.3 in [4], e ~ A - .

Clearly, fi is an ideal base in L. Take J = {A • L: there exists C • fl such that C ~> A}, from Definition 2.5, J is the ideal generated by ft. One can easily prove that J = I and J --* e(Q*). Theorem 2.7. Let I be an ideal in (L(M), 5) and e • M. Then (1) I--*e(Q) iff e <<.Q-limI. (2) I ~ e ( Q * ) iff e <<.Q*-limI. (3) l ~ e ( Q ) iff e ~ Q-adI. (4) l ~ e ( Q * ) !fie <-GQ*-adI. ProoL We only prove the sufficiency of (1). Suppose that e ~< Q-limI and P • ~(e). Then e 4; P, so Q-lira 1 4; P. By the definition of Q-lim I, there very

3. Relationships of ideals and molecular nets Definition 3.1 (Bai [4]). Let S be a molecular net in (L(M), 6) and e • M. Then e is said to be a Q*-limit (Q*-cluster) point of S, if for each P • ~*(e), S is eventually (frequently) not in P, in symbols S ~ e(Q*) (S ~ e(Q*)). The union of all Q*-limit points and all Q*-cluster points of S will be denoted by Q*-lim S and Q*-ad S, respectively. Theorem 3.2 (Bai [4]). Let S be a molecular net in (L(M), 6) and e c M. Then (1) S--*e(Q*) iff e <<,Q*-limS. (2) Sc~e(Q*) iff e ~ Q*-ad S.

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Definition 3.3 (Yang [11]). (1) Let I be an ideal in (L(M), 6) and

O(l) = {(e,A): e ~ M, A e I and e ~ A}. For every pair of elements (el, A1) and (e2, A2) in

D(I), we define that (el, A1) ~< (e2, A2) iffA1 ~< A2. Then D(I) forms a directed set with the relation. Clearly,

S(I) = {S(I)(e, A) = e: (e, A) e D(I)} is a molecular net in L and is called the molecular net induced by I. (2) Let S be a molecular net in (L(M), 6). Then

S oo e(Q*) implies I(S) oo e(Q*). Let S oo e(Q*). Then S is not frequently in P for each P ~ ¢*(e). On the other hand, S is not eventually in A for each A ~ I(S). Hence, S is not frequently in P v A for each P ~ ~*(e) and each A ~ 1(S). This means that P v A ~ 1. Thus, I(S) oo e(Q*).

Definition 3.6 (Yang [11]). An ideal I in L is called a maximal ideal, if I c J for each ideal J in L we have J = 1. Definition 3.7 (Yang [11]). A molecular net S in L is called a universal net, if there exists a maximal ideal I in L such that S is a subnet of S(I).

I(S) = {A ~ L: S is not eventually in A} is an ideal in L and is called the ideal induced by S.

Theorem 3.4. Let I be an ideal in L. Then (1) Q*-lim I = Q*-lim S(I). (2) Q*-ad I = Q*-ad S(I). Proof. (1) Let e ~< Q*-limI. Then I ~ e ( Q * ) by Theorem 2.7, i.e., P e I for each P e ~*(e). Hence, (e, P) ~ D(I). If (a, A) ~ D(1) and (a, A) ~> (a, P), we have

S(I)(a, A) = a ~ A >~e. Therefore S(1)(a, A ) ~ P. This means that S(1) is not eventually in P for each P e ~*(e), i.e., S(I) ~ e(Q*). F r o m Theorem 3.2 we have e ~< Q*lira S(I). Conversely, let e ~ Q*-limS(1). Then S ( 1 ) ~ e(Q*) by Theorem 3.2. Therefore for each P ~ ~*(e) there exists (a, A ) ~ D(1) such that S(1)(b, B ) = b ~ P whenever (b, B)/> (a, A) and (b, B) ~ D(I). This shows that b ~ A implies b ~ P, equivalently e ~< P implies b ~< A. Hence, P ~< A follows from Proposition 2.17 in [9]. Note that I is a lower set and A ~ I, so P e L Hence 1 --, e(Q*). From Theorem 2.7 we have e ~< Q*-limI. Thus (1) holds. (2) This is analogous to the proof of (1).

Theorem 3.5. Let S be a molecular net in L. Then (1) Q*-lim S -- Q*-lim I(S). (2) Q*-ad S ~< Q*-ad I(S). Proof. We prove only (2). In accordance with Theorems 2.7 and 3.2, we need only to prove that

Theorem 3.8. Let (L(M), 6) be a TML. Then following are equivalent: (1) Every ideal in (L(M),6) has a Q*-cluster point. (2) Every maximal ideal in (L(M), 6) has a Q*limit point. (3) Every universal net in (L(M), 6) has a Q*-limit point. Proof. ( 1 ) ~ (2): This follows directly from Theorem 2.6. (2) ~ (1): Let I be an ideal in (L(M), 6) and e e M. Then there exists a maximal ideal J in (L(M),6) such that J ~ I. By hypothesis, J ~ e(Q*), i.e., ~*(e) c J. Hence for each P E ~*(e) and each A e I we have P v A e J, i.e., P v A ¢ 1. This implies that I oo e(Q*). (2) ~ (3): Let S be a universal net in (L(M), 6) and e e M. Then there exists a maximal ideal J in (L(M), 6) such that S is a subnet of S(J). Since J ~ e ( Q * ) , S(J)--*e(Q*) by Theorem 3.4. Thus S ~ e(Q*). (3) ~ (2): Let J be a maximal ideal in (L(M), 6). Then S(J) is a universal net in (L(M), 6) by Definition 3.7. By hypothesis, S(J)-*e(Q*). Hence J --, e(Q*). 4. Applications Definition 4.1. Let f : (LI(M1), 61) ~ (L2(M2), 62) he an order-homomorphism, f is called: (1) Strongly semicontinuous if f - l ( B ) ~ Q1 for each B e 62 [3].

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(2) Strongly semicontinuous at a point e • MI if ( f - l ( p ) ) ~ • ix(e) for each P • r/2(f(e)) [3]. (3) S-irresolute if f - I ( B ) • Q1 for each B • Q2 [4-]. (4) S-irresolute at a point e • M1 if ( f - l ( p ) ) ~ • ~l(e) for each P • ~2(f(e)) [4]. Theorem 4.2 (Bai [3]). An order-homomorphism f : (LI(Mx), 61) ~ (L2(M2), 62) is strongly semicontinuous iff f is strongly semicontinuous for each point e • Mx. Theorem 4.3 (Bai [4]). An order-homomorphism f : (LI(M1), 61) ~ (L2(M2), fi2) is S-irresolute ifffis Sirresolute for each point e • Mx.

Definition 4.4. Let f : (LI(M1), 61) ~ (L2(M2),62) be an order-homomorphism and e • Mx. f is called: (1) S*-strongly semicontinuous at e i f ( f - l(p))~ • ~*(e) for each P • r/2(f(e)). (2) Strongly S-irresolute at e if ( f - l ( p ) ) ~ • (*(e) for each P • ¢2(f(e)). (3) S*-irresolute at e if ( f - l ( p ) ) ~ • ~'(e) for each P • ~*(f(e)). Definition 4.5. Let f : (LI(M1), 61) --* (L2(M2), 62) be an order-homomorphism, f is called: (1) S*-strongly semicontinuous if f is S*strongly semicontinuous for each point e • M1. (2) Strongly S-irresolute if f is strongly Sirresolute for each point e • M1. (3) S*-irresolute if f is S*-irresolute for each point e e M1. Example 2. Let f : L X ~ L r be a S*-strongly semicontinuous (strongly S-irresolute, S*-irresolute) function of Zadeh's type. Then f is a S*strongly semicontinuous (strongly S-irresolute, S*-irresolute) order-homomorphism. One can easily verify the following relational graph by Definitions 4.1, 4.4, 4.5 and Theorems 4.2, 4.3. strong S-irresoluteness ~ S-irresoluteness

L

strong semicontinuity

S*-irresoluteness =:- S*-strong semicontinuity

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Lemma 4.6 (Yang [11]). Let f : ( L l ( M x ) , 6 1 ) ~

(L2(M2),62) be an order-homomorphism and I an ideal in LI. Then (1) f * ( I ) = {B • L2: there exists A • I such that for each e ~ M1, e .~ A implies f ( e ) ~ B} is an ideal in L 2. (2) (f(I'))' is an ideal base in L2. Theorem 4.7. Let f : (LI(M1), 61) ~ (L2(M2), 62) be an order-homomorphism and f S*-strongly semicontinuous at e • M1. I f I ~ e(Q*) for each ideal I in L1, then (1) f * ( I ) ~ f ( e ) . (2) (f(I'))' ~ f ( e ) . Proof. (1) Let f be S*-strongly semicontinuous at e and I an ideal in L1 which I ~ e(Q*). Then for each P • r/2(f(e)) we have ( f - l ( p ) ) ~ • ~*(e) ~ I. So f - l ( p ) • I. Since e ~ f - l ( p ) implies f(e) .~ P, P • f * ( I ) follows from Lemma 4.6. Hence f * ( I ) ~ f ( e ) by Definition 1.1 in [11]. (2) Suppose that conditions are satisfied. Then for each P • q2(f(e)) we have ( f - l ( p ) ) ~ • ~*(e) = I. So f - l ( p ) • I. Since f ( f - l ( p , ) ) <~p,, p <~( f ( f - X ( p , ) ) ) , = ( f ( ( f - l ( p ) ) , ) ) , .

Put A = f - l ( p ) . Then A • I and P ~ (f(A'))' • (f(I'))'. Thus (f(I'))'-* f(e). Corollary 4.8. Let f : L 1 - ~ L 2 be a S*-strongly semicontinuous order-homomorphism. Then for each ideal I in LI we have (1) f(Q*-limI) <<.limf*(I). (2) f(Q*-lim I) ~< lim(f(l'))'. Proof. This follows directly from Definition 4.5 and Theorem 4.7.

Theorem 4.9. Let f : (L 1(MI), 61 ) ~ (L2 (M2), 62) be an order-homomorphism and e • M1. (1) If I ~ e ( Q * ) implies f * ( I ) ~ f ( e ) for each ideal I in L1, then f is strongly semicontinuous at e. (2) If I ~ e(Q*) implies (f(I'))' ~ f ( e ) for each ideal I in L1, then f is strongly semicontinuous at e.

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Proof. (1) Suppose that f is not strongly semicontinuous at e. Then there exists P • t/2(f(e)) such that (f-l(P))~q~l(e), i.e., e ~< (f-X(p))~. Hence there exists an ideal I in L1 such that f - l(P)q~I and I--*e(Q*) by Theorem 2.9. N o w we prove that P~f*(I). First, we assert that

f*(I) c {B • L2: P ~ B}. In fact, if there exists B • f * ( I ) with P ~< B, then there is A • I such that e ~ A implies f(e) 4~ B according to the definition of f*(I), and so f ( e ) ~ P . This shows that e~
Corollary 4.12. Let f : L I ~ L 2 be a strongly semicontinuous order-homomorphism. Then for each

ideal I in L1 we have (1) f(Q-limI)<<, limf*(I). (2) f(Q-lim 1) ~< lim(f*(I'))'. Theorem 4.13. Let f : (LI(M1), 61) --* (LE(Mz), 62) be an order-homomorphism and f S-irresolute at e • M1. I f I -* e(Q)for each ideal I in L1, then (1) f*(I)~f(e)(Q). (2) (f(I'))' ~ f(e)(Q).

Corollary 4.14. Let f :L1--+L2 be a S-irresolute order-homomorphism. Then for each ideal I in L1 we have (1) f(Q-limI) <~Q-lim f*(I). (2) f(Q-lim I) <~Q-lim(f(I'))'. Theorem 4.15. Let f : (LI(M1), 61) ~ (L2(M2), 62) be an order-homomorphism and f S*-irresolute at e • M 1. l f I --* e(Q*)for each ideal I in L1, then (l) f*(1)---,f(e)(Q*). (2) (f(I'))'-* f(e)(Q*).

Corollary 4.16. Let f : L1---rL2 be a S*-irresolute order-homomorphism. Then for each ideal I in L1 we have (1) f(Q*-limI) <~Q*-lim f *(I). (2) f(Q*-lim 1) <~Q*-lim(f(I'))'. Theorem 4.17. Let f : (LI(M1), 61) -~ (L2(M2), 62) be an order-homomorphism and f strongly S-irresolute at e • M1. I f I -~ e(Q*) for each ideal I in El, then (1) f*(I)-*f(e)(Q). (2) (f(I'))'-~f(e)(Q).

Corollary 4.18. Let f : L1 ~ L 2 be a strongly S-irresolute order-homomorphism. Then for each ideal I in L1 we have (1) f(Q*-limI) ~ Q-lim f*(1). (2) f(Q*-limI) ~ Q-lim(f(I'))'. Theorem 4.19. Let f : (LI(M1), 61) --4 (L2(M2) , 62) be an order-homomorphism and e • M1. (1) If I -~ e(Q*) implies f*(l) -*f(e)(Q) for each ideal I in L1, then f is S-irresolute at e. (2) I f I -~ e(Q* ) implies ( f (I') )' -~f (e)(Q )for each ideal I in L1, then f is S-irresolute at e.

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Corollary

4.20. Let f : L a ~ L 2 be an orderhomomorphism. (1) 1f f ( Q * - l i m I ) <<,Q-lim f * ( I ) for each ideal I in L1, then f is S-irresolute. (2) l f f ( Q * - l i m I ) <~ Q - l i m ( f ( I ' ) ) ' for each ideal I in L I , then f is S-irresolute.

Question:

W h e t h e r the c o n v e r s e s of T h e o r e m s 4.7, 4.15 a n d 4.17 are true?

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