galois connections in category theory, topology and logic

Galois Connections in Category Theory, Topology and Logic

W. Gahler

Abstract

The notion of a Galois connection is important in different branches of mathematics. It even is used for defining basic notions in several theories. In this paper the role of Galois connections is demonstrated in reference to known results and moreover in presenting new ones. AMS Mathematics Subject Classification: 06D35,18B35,03E12. Key words: Galois connection, Non-classical logic, MV-algebra, Partially ordered monad, Fuzzy filter.

1. Different kinds of Galois connections

Standard kind. G. Birkhoff ([1]) introduced a notion of Galois con-nection, related to power sets. Later on 0. Ore ([15]) and J. Schmidt ([18]) studied a generalization of this notion, which can be considered as the standard kind of a Galois connection. Its "covariant form" is defined as follows: A pair of isotone mappings f : (X, :S) -----t (Y, :S) and g : (Y, :S) -----t (X, :S) with (X, :S) and (Y, :S) partial ordered sets, is a Galois connection provided the equivalence f( x ) :S y -:::::::? x :S g(y) holds for all x E X and y E Y . (!,g) is a Galois connection if and only if for each y E Y the supremum U{x EX I j(x) :S y} exists and coincides with g(y). Therefore, g will be called the sup-inverse of f. Categorical notions of Galois connections. The following kind of Galois connections, pointed out by H. Herrlich and M. Husek ([10]), is more general. In this case instead of isotone mappings f : (X, :S ) -----t (Y, :S) and g : (Y, :S) -----t (X, :S) two concrete functors

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a category X. The pair ( r.p, 1/J ) is called a concrete Galois connection provided the following equivalence holds

r.pA ~ B E MorB ~ A ~ 'lj;B E MorA for A E ObA, B E ObB and f E MorX. We obtain the classical kind if we take (X,~) and (Y, ~) as the concrete categories A and B, respectively, with X and Y the sets of objects and the partial orderings of X and Y as the related sets of morphisms and interpreting X trivially as a category having only one object and only the identity morphism. A further kind of Galois connection, also discussed in [10], is that of an adjoint situation, which consists of a pair (

Galois Connections in Category Theory, Topology and Logic 427

(MO) rpX is empty in case X is empty. (Ml) For any set X and each pair of different elements x and y of X,

the infimum of rJx(x) and TJx(Y) does not exist. (M2) For all mappings f, g: Y ------t rpX from f ~ g it follows J-Lx o rpf ~

J-Lx o rpg, where ~ is defined argumentwise with respect to the partial ordering of rpX.

(M3) For each set X, J-Lx: (rprpX, ~) ------t (rpX, ~)preserves non-empty suprema.

The partial orderings ~ of the sets rpX are considered as finer rela-tions. For each set X, the elements of rpX are called rp- objects on X, and the minimal elements of rpX also ultra objects. For each non-empty subset A of rpX the supremum V M exists.

MEA Hence, whenever for a subset A of rpX a lower bound in rpX exists, the infimum A M of this subset exists.

MEA For each non-empty set X we denote the supremum V TJx(x) by

xEX TJx[X]. A rp-object M on X for which M ~ TJx[X] holds, is called stratified. A rp-object Mona set X for which M < TJx(x) holds for some x EX, is called a microobject at x. Because of condition (Ml), x is uniquely associated to M. Of course, microobjects are special stratified

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filter functor-, which assigns to each set X the set F X of all (proper) filters on X. ::; indicates that the sets F X are equipped with the finer relations of filters, that is , the inversion of the inclusion. ry and p, are natural transformations consisting of all mappings 'r/x : X --+ F X and fJ,x: FFX--+ FX respectively, where for each x EX, 'r!x (x) = {M ~ X I X E M} and for each filter .c on F X , Mx(.C) = u n M. In

AE.CMEA this classical case microobjects do not exist. In the general case of a partially ordered monad


(C2) M ---+ x and N::; M imply N---+ x, and

(C3) M---+ x implies M V TJx(x) ---+ x.

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cp- t 2 of cpt2 exists. We introduce the closure operator cl : cpX ---+ cpX ofT ([3]) by means of the sup-inverse cp;t1 defining

cl is a hull operator, that is, M ::::; elM holds for all M E cpX . If M = elM, then M is called closed. Since cl is a hull operator, we have that the infimum of any set of of closed cp-objects on X is closed, as far as this infimum exists. As neighbourhood operator nb : cpX ---+ cpX of the -convergence structure T we mean the neighbourhood operator of the associated -pretopology p ofT, which is defined by p(x) = V M for all x E X.

M--tx nb is also a hull operator and can be introduced by means of the sup-inverse cp- t 2 ([6]) by

This shows that the notions of a closure operator and of a neighbour-hood operator are in some sense dual. More exactly, here we can speak of an associated pair of Galois connections. In general topol-ogy both notions of closure operator and of neighbourhood operator are used for different reasons. This differs from the classical case of a topological space and even of a pretopological space in which each of the notions of a closure operator and of a neighbourhood opera-tor determines uniquely the other one. This simple situation in the classical case depends on the fact that for the interior and the clo-sure of subsets of a space we have X\ intM = cl(X \ M), where intM = { x E X I M E p( x)}. Because of this reason, classical pre-topological spaces are sometimes called closure spaces. That in general such a one-to-one correspondence does not exist, will be demonstrated at the end of this paper by examples of different -pretopologies which have one and the same closure operator. In the following let T be a -convergence structure on a set X and let nb be its neighbourhood operator. A cp-object M on X is called open provided that M = nbM.

Proposition 2 ([6]) The set 0 of all open cp-objects on X fulfills the following condition


(0) 0 is closed with respect to all non-empty suprema and all infima, as far as these infima exist.

Each if>-topology p can be characterized by the set 0 of all open

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for each a E La Galois connection is given, consisting of the mappings fa : L ---+ L and ga : L---+ L , where fa(x) = a 1\ x and ga(Y) = a---+ y. Clearly, fa(x) :S y +-+ x :S ga(Y) holds for all x , y E L. Quantale case. A structure more general than that of a frame, is that of a commutative quantale (L, :::; , *) (cf. Rosenthal's paper [17]) . It is defined as a non-degenerate complete lattice which is equipped with a commutative and associative binary operation * such that the complete distributivity law

iEI iEI

holds. We always assume that the maximal element 1 of the underlying lattice is the unit element with respect to * Analogously as in case of a frame, in a commutative quantale impli-cations a---+ bare defined and Galois connections are introduced. The only difference is that 1\ is replaced by *, that is, here we have

a ---+ y = max { x E L I a * x :S y } and for the mappings fa : L ---+ L and ga : L ---+ L, given by fa ( x) = a*X and ga(Y) =a---+ y, we have fa(x) :S y +-+ x :S ga(Y) for all x , y E L. Clearly, a frame is a commutative quantale with 1\ : L2 ---+ L the related binary operation. Further examples of commutative quantales are given by means of the continuous t-norms * : 2 ---+ L. They are defined as follows:

(T1) L is the unit interval [0, 1] equipped with the usual ordering.

(T2) * is a continuous commutative and associative binary operation with unit element 1.

(T3) * is order preserving, that is, a :::; band c :::; dimply a* c :::; b *d.

Because of the continuity of*, for at-norm the complete distributivity law with respect to * holds. We note some special examples:

Godel t-norm a* b =min{ a, b} Product t-norm a * b = a b (product of reals) Lukasiewicz t-norm a* b = max{O, a+ b- 1}


Case of Girard monoids. A commutative quantale which fulfills the condition of double negation -.(-a) =a, is a complete Girard monoid. Note that the negation of a is given by a = a --t 0. For the Godel t-norm and the product t-norm we have

if a> 0

if a= 0.

Hence, for these t-norms the condition of double negation is not ful-filled and they therefore do not define Girard monoids.

Case of MY-algebras. By a complete MV-algebra we mean a com-plete Girard monoid (L , ::; , *) which is divisible, that is, for all a, b E L from a ::; b it follows a = b * c for some c E L. Examples of complete MY-algebras: (1) The Lukasiewicz t-norm *defines a complete MY-algebra ([0, 1],::; '* ). (2) All complete Boolean algebras are complete MY-algebras. They can be characterized as those complete MY-algebras (L, ::;, *)for which * is idempotent, that is, a * a = a for all a E L. In this case, then * equals 1\.

Related logics. In the following there are listed basic structures together with related non-classical logics:

Frames intuitionistic logic Commutative quantales monoidallogic Complete Girard-monoids - linear logic Complete MY -algebras Lukasiewicz logic

The following is well-known in first order logic:

Proposition 4 ( [11]) A formula is valid with respect to all non-degenerate complete MV-algebras if and only if it is valid with respect to ([0, 1], ::;, *) with* the Lukasiewicz t-norm.

Proposition 5 ([16]) A formula is valid with respect to all non-degenerate complete Boolean algebras if and only if it is valid with respect to the Boolean algebra consisting only of 0 and 1.

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5. Fuzzy filters in the frame case There are different kinds of fuzzy filters. The following deals with one of these kinds. If (L, :S::) is a complete lattice, then for each set X the mappings f E Lx are called fuzzy subsets of X. For each a E L let a denote the constant fuzzy subset of X with value a. Let L = (L, :S::) be a non-degenerate frame. A fuzzy filter on X in the frame case, also called an L-filter, is a mapping M : Lx --* L such that

(Fl) M(O) = 0 and M(I) = 1, (F2) M(f 1\ g)= M(f) 1\ M(g) for all f,g E Lx. In the following, a definition of the related partially ordered fuzzy filter-monad (FL, :::;;, ry, J-t) , also called the partially ordered L-filter mo-nad, is given. For each set X, FLX consists of all L-filters on X. The partial ordering on each set FLX is given by

M :S:: N ~ M(f) :::=: N(f) for all f E Lx.

For each mapping f : X --* Y, the mapping FLf : FLX --* FLY assigns to each M E FLX the fuzzy filter FLf(M) on Y given by FLf(M)(g) = M(g of) for all g E Lx . For each set X, each x E X and f E Lx let 'TJx(x )(f) = f(x) , and for each E FLFLX and f E Lx let J-tx(.C)(f) = .C(ef) , where ef : FLX--* L is the mapping M r-+ M(f). We distinguish some types of fuzzy filters. An L-filter M on a set X is called

bounded if M(a) :::;; a holds for all a E L and homogeneous if M(a) =a holds for all a E L.

As a specialization of the general property of being stratified, here we have that an L-filter on a set X is

stratified if and only if M(a) :::=: a holds for all a E L. Each fuzzy filter coarser than a bounded fuzzy filter is also bounded. Each fuzzy filter finer than a stratified fuzzy filter is also stratified.


Proposition 6 There are three partially ordered submonads of (FL , ::; , TJ, f.L), defined by all bounded fuzzy filters , all homogeneous fuzzy filters and all stratified fuzzy filters , respectively.

The following is related to suprema and infima of fuzzy filters .

Proposition 7 Let X be a set. For the supremum of a non-empty subset A of FLX we have ( v M)(f) = A M(f) (2)

M EA M EA

for all f E Lx . If the infimum of a subset A of FLX exists, then (A M)(f) = MEA h 1\ 1\ fnS!

Ml , ... ,MnEA

for all f E LX.

6. Principal fuzzy filters in the frame case Let X be a non-empty set . For each mapping f E Lx and each a E L with a ::; Uf by

[/, a[(g) = { ; iff t g ' if f ::; g =!= I , if g =I

for all g E Lx, a bounded fuzzy filter [f, a] is defined, called a principal fuzzy filter on X (first defined in [4]).

Proposition 8 For each bounded fuzzy filter M in the frame case the infimum M 1\ TJx[X] exists, called the homogenization of M.

Since for each f E Lx the principal fuzzy filter [f, Uf] is bounded, its homogenization [ f , Uf) 1\ TJx[X] exists. For any f E Lx and all g E Lx we have [f](g) = V (Uf 1\a) vng . Clearly, [OJ= TJx[X].

f /\a5:c g

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In the following some representations of fuzzy filters by means of prin-cipal fuzzy filters are given.

Proposition 9 ([8]) Let X be a set of at least two elements. Then faT each fuzzy filter M on X we have: M is the infimum of all principal fuzzy filters [f, a] coaT"ser than M , that is, of all principal fuzzy filters [f, a] for which a ::; M (f) 1\ U.f holds:

M 1\ [f, a]. a ~M(f) 1\ Uf

Moreover then faT each stmtified fuzzy filter M on X we have: M is the infimum of all fuzzy filters [.f] which are coarser than M , that is, of all fuzzy filters [.f] for which U.f ::; M (f) holds:

M= 1\ [f]. Uf~M(f)

Proposition 10 ([8]) Let X be a set of only one elem ent, then we have: Each bounded fuzzy filter on X can be represented analogously as in Pmposition 9 by means of pTincipal fuzzy filters. Non of the stmtified, non-homogeneous fuzzy filters on X can be rep-resented in this way.

For any non-empty set X we have [ 0] 1\ [ .f, nf ]. / ELX

7. Fuzzy filters in the quantale case In the following let L = ( L , ::;, *) be a non-degenerate commutative quantale. A fuzzy filter on a set X in the quantale case, also called an L-filter, is a mapping M : Lx -+ L such that

(Ql) M(O) = 0 and M(l ) = 1. (Q2) f ::; g implies M(J) ::; M(g) for all f ,g E L x. (Q3) M(J) * M(g) ::; M(J *g) for all f ,g E L x .


If* is the infimum 1\ in (L , ::;), then the notion of a fuzzy filter in both the frame case and in the quantale case coincide. Hence, the quantale case is more general than the frame case. In the quantale case, the related partially ordered fuzzy filter monad (.ri, ::;, ry, fJ), also called the L-filter monad, is defined in the same way as in the frame case. In particular, the L-filters Tlx(x) : Lx ---* L are the same mappings as in the frame case. Moreover, the notions of bounded, homogeneous and stratified fuzzy filter are defined in the same way as in the frame case. There exist the partially ordered fuzzy filter submonads of (FL, ::;, ry, 11) defined by all bounded, homogeneous and stratified L-filters, respectively. For each set X the supremum of a non-empty subset A of FLX has the same representation as in the frame case given in Proposition 7 by equation (2). It follows that for each non-empty set X , T]x [X] = V Tfx(x) also is a fuzzy filter in the quantale case.

xEX For the infima of fuzzy filters in the quantale case we have:

Proposition 11 Let X be a set. If the infimum of a subset A of FLX exists, then

(A M)(J) = MEA

for all f E Lx.

v h fn S:: f M1 , ... ,MnEA

The following is an easy consequence of Proposition 11.

Proposition 12 Let X be a non-empty set and A a linearly ordered subset of (FLX, ::;). Then the infimum 1\ M exists and for each

MEA f E Lx we have

(A M)(J) v M(J) . MEA MEA

A bounded fuzzy filter on a set X will be called a bounded ultra fuzzy filter if there is no properly finer bounded fuzzy filter on X. Analo-gously the notions of homogeneous ultra fuzzy filter and of stratified ultra fuzzy filter are defined.

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It follows by means of Zorn's lemma from Proposition 12 that:

Proposition 13 Let X be a non-empty set. In the quantale case we have: Each bounded fuzzy filter has a finer bounded ultra fuzzy filter. Analogously, each homogeneous and each stratified fuzzy filter has a finer homogeneous and a finer stratified ultra fuzzy filter, respectively.

Remark. Clearly, this proposition also holds in the frame case. In this case because of Proposition 8 we even have that each bounded ultra fuzzy filter is a homogeneous fuzzy filter.

8. Principal fuzzy filters in the quantale case In the quantale case we introduce a notion of principal fuzzy filter in defining the mappings [f, a] : Lx ~ L exactly as in the frame case. However only some of these mappings appear as fuzzy filters in the quantale case. We have the following:

Proposition 14 For each set X , each f E Lx and each a :::; Uf the mapping [f, a] is a fuzzy filter in the quantale case if and only if f :::; f * f or a * a = 0 holds.

Clearly in the frame case we especially have f = f 1\ f for all f E Lx. The following counterexample is obtained as an immediate conse-quence of Proposition 14.

Proposition 15 Let a and b be elements of L such that b * b < b < a :::; a* a holds. Moreover, let X be a set of at least two elements and x 0 an element of X. Then [f, a] with f E Lx defined by

f(x) = { : if X= Xo

if X -::j::. Xo

for all x E X, is not a fuzzy filter in the quantale case.


Under some conditions, in the quantale case, homogeneous fuzzy fil-ters [f] exist.

Proposition 16 For any f E Lx for which f :::; f * f or Uf * Uf = 0 holds and there is an x EX with f(x) = Uj, the fuzzy filter infimum in the quantale case

[f] = [f,Uf] !\1Jx[X] exists and for each g E Lx we have [ f ](g) V (uf *a) v ng.

f*a~g

9. Distinguished and adjoint fuzzy filters, the frame case

Let L = (L, :::;) be a non-degenerate frame. An L-filter M on a set X is said to be distinguished ([9]) provided that

M(f v g) = M(f) v M(g) holds for all f, g E Lx. In the filter case distinguished means that this filter F is prime, that is for non-empty subsets M and N of X from M UN E Fit follows M E F or N E F. Hence, in this case distinguished means that F is an ultra filter. In the general frame case the property of a fuzzy filter to be distin-guished is more complex. As will be shown in the example in Section 12, there are distinguished fuzzy filters which are neither homoge-neous ultra fuzzy filters nor stratified ultra fuzzy filters and on the other hand there are homogeneous ultra fuzzy filters which fail to be distinguished. In the following the zero-condition (Z) of L is important. (Z) means that

a > 0 and b > 0 imply a!\ b > 0

for all a, b E L. This condition is needed for proving some types of compactifications ( cf. [9]). Clearly this condition is fulfilled in case

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(L, ~) is a complete chain.

Proposition 17 ([9]) If L fulfills the condition (Z), then each strati-fied ultra fuzzy filter is distinguished. If L is a non-degenerate complete chain, then each distinguished ho-mogeneous fuzzy filter is a homogeneous ultra fuzzy filter.

Remark. ([9]) Propositions 13 and 17 imply that under the condition (Z) each fuzzy filter has a finer distinguished fuzzy filter (which may not be homogeneous). In the following let an antitone involution c : L --t L be fixed. Clearly, for all a, bEL we have a V b = c(c(a) 1\ c(b)). By means of c each distinguished fuzzy filter M on X can be assigned a further fuzzy filter Me defined by

Me(!) = c(M(c o !)) for all f E Lx. Me is called the c-adjoint of M.

Proposition 18 The c-adjoint of each distinguished fuzzy filter M is also a distinguished fuzzy filter and we have (Me)e = M . If M and N are distinguished fuzzy filters, then M ~ N implies Me 2: Ne.

From the example in Section 12 we will obtain, that there are distin-guished fuzzy filters which differ from their c-adjoints and also some which do not. If M and N are distinguished fuzzy filters and M = Ne , hence also Me = N, then {M ,N} will be called a c-adjoint pair. In case M = Me, the fuzzy filter M will be called c-selfadjoint.

Proposition 19 Let M be a distinguished fuzzy filter. If one of the fuzzy filter-s M and Me is bounded the other- one is stratified and vice ver-sa. M is homogeneous if and only if Me is homogeneous.

Clearly, if we restrict ourselves to the partially ordered submonad of all bounded fuzzy filters or of the partially ordered submonad of all


stratified fuzzy filters, then for each non-homogeneous distinguished fuzzy filter, its c-adjoint does not appear.

10. Distinguished and adjoint fuzzy filters, the quantale case In the following let L = (L , ~' *) be a non-degenerate commutative quantale. If * differs from /\ , for defining the notion of distinguished fuzzy filter in the quantale case always an antitone involution c : L -t L is to be fixed. If not specified we always assume in the following that such a mapping c is given. Let E9 denote the binary operation on L defined for all a, bEL by

a E9 b = c(c(a) * c(b)).

If L is a Girard monoid, then as c we can take the negation -, : L -t L

If L = [0, 1], cis given by c(a) = 1- a and *is a continuous t-norm, then E9 is the t-conorm associated to * Both notions oft-norm and t-conorm, introduced by Schweitzer and Sklar ([19]) , play an important role in fuzzy logic. For a fixed antitone involution c of L, an L-filter on a set X is called c-distinguished provided that

M(f E9 g) ~ M(f) E9 M(g)

holds for all f, g E Lx. Notice that in this definition only an inequality appears, which is in accordance with the inequality in condition (Q3) of the definition of a fuzzy filter in the quantale case. Analogously as in the frame case, to each c-distinguished fuzzy filter M can be associated a further fuzzy filter Me defined by M e(!) = c(M(c of)) for all f E Lx. Me is called the c-adjoint of M. Propo-sitions 18 and 19 hold analogously in the quantale case. The notions c-adjoint pair and c-selfadjoint fuzzy filter are defined analogously as in the frame case. In the example which will be presented in Section 13 all c-distinguished fuzzy filters M are c-selfadjoint.

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11. "Basic example" In this section a special example of fuzzy filters in the frame case will be considered. For different reasons, this example has been already considered in [9] in the general case and at first , restricted to homo-geneous fuzzy filters also in [2] . We are here especially interested in the distinguished fuzzy filters in this example. Let X = {0, 1} and L = {0, ~ ' 1} . There are 9 fuzzy subsets on X , and among them there are 6 fuzzy subsets which are non-constant. They are presented in Fig. 2.

L J 0 f s

Fig. 2. All nonconstant mappings of X= {0, 1} into L = {0, ~ ' 1}.

There are 25 fuzzy filters in this example. They are shown in Fig. 3, in which finer filters are situated more downwards. In particular there are

9 bounded, non-homogeneous fuzzy filters , indicated by small circles, 11 homogeneous fuzzy filt ers, indicated by boldfaced dots, and 5 stratified, non-homogeneous fuzzy filt ers, indicated by small dots. Whereas in this example all the principal fuzzy filters are the bounded, non- homogeneous fuzzy filters , their homogenizations as well as H1s = [!I] A [fs], H16 = [!I] A [!6], Hs2 = [is] A [!2], H52 = [!6]/\ [!2] are the homogeneous fuzzy filters . Moreover, Mo = [h] A [!6], M1 = [!4]/\ [is], Mo1 = [fs] 1\ [!6] and 50 = [!I] A [fs]A [f6], S1 = [!2] A [fs] A [!6] are the stratified, non-homogeneous fuzzy filters.

Proposition 20 Th e following hold: (1) M0 and M1 are the only micro fuzzy filters and the only stratified ultra fuzzy filters . (2) TJx(O), 7Jx(1), H16 and Hs2 are the only homogeneous ultra fuzzy filters.


(3) In this example [h, 1], [!4 , 1], 7Jx(O), 7Jx(1), M0 and M1 are the distinguished fuzzy filters, indicated in Fig. 3 additionally by overlines. (4) {M 0 , [h, 1]} and {M 1 , [!4 , 1]} are adjoint pairs. (5) Both fuzzy filter-s 1Jx(O) and 7Jx(1) are selfadjoint.

TJx( =[h]

(0, 0]

Fig. 3. Graph of all fuzzy filters on X = {0, 1} with L = {0, ~' 1} in the frame case.

Notice the following:

7Jx(O) and 7Jx(1) fail to be stratified ultra fuzzy filters. H16 and H52 are homogeneous ultra fuzzy filters which are not distin-guished. The distinguished fuzzy filters [h, 1] and [f4 , 1] are neither homoge-neous ultra fuzzy filters nor stratified ultra fuzzy filters.

12. Analogous lukasiewicz case As in the preceding section let X = { 0, 1} and let L be the set { 0, ~, 1}. We equip L with the usual ordering and with the binary operation *

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defined by

* 0 1 1 2

0 0 0 0 1 0 0 1 2 2 1 0 1 1 2

By a fuzzy filter here we mean an L-filter with L the quantale (L, :::; , * ). There are 22 fuzzy filters in this example. They are shown in Figure 4, where the finer fuzzy filters are situated more downwards. In par-ticular, there are 11 bounded, non-homogeneous fuzzy filters, indicated by small circles, and 11 homogeneous fuzzy filters , indicated by bold faced dots. Stratified fuzzy filters which are non-homogeneous, in particular micro-fuzzy filters, do not exist. From Proposition 14 it follows:

Proposition 21 The mappings [!5, 1], [!6, 1], [f5] and [f6] are ex-cluded as fuzzy filters with respect to (L , :::; , *).

There are four bounded non-homogeneous fuzzy filters which are not single principal fuzzy filters. They are:

834 [h, ~], 1\ [!4, ~] B36 = [/s, ~] 1\ [!6, ~], 854 [f5, ~] 1\ [f4, ~], B56 = [f5, ~] 1\ [!6, ~].

Notice that for the principle fuzzy filters [!,a] appearing in this ex-ample in some cases only the condition f :::; f * f (e.g. for [h, 1]) and in some other cases only the condition a* a = 0 (e.g. for [h , ~]) is fulfilled.


'TJX (0) =[h]

[0,0]

Fig. 4. Graph of all fuzzy filters on X = {0, 1} with L = {0, ~' 1} in the Lukasiewicz case.

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Among the principal fuzzy fil ters there are three which are homoge-neous, that is [ 0] = [J, ~] , [!I] = [/I ,~ ] and [h] = [h , ~ ]. The further homogeneous fuzzy filters are [h] = T,lx(O), [!4] = T,lx(1) and

H12 = [!1] 1\ [h] , H14 = [!I] 1\ [j4 , ~] , H32 = [h, ~] 1\ [h] , H3 = [h, ~] 1\ [0], H4 = [j4 , ~ ] 1\ [OJ , H34 = [h, ~] 1\ [!4, n

The property of a fuzzy filter to be distinguished here we understand with respect to the associated binary operation EB of * defined by means of the antitone involution c with c(a) = 1 - a , that is we have

EB 0 1 1 2 0 0 2 1 1 1 1 1 2 2 1 1 1 1

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Proposition 22 In this example the ultra fuzzy filters 1Jx(O), 7Jx(l) and H12are the only distinguished fuzzy filters (indicated in Fig. 4 ad-ditionally by overlines). They are selfadjoint.

13. Balanced fuzzy filters

Let L = (L, :::;) be a non-degenerate frame. An L-filter will be called balanced ([9]) whenever a finer distinguished homogeneous fuzzy filter exists. All balanced fuzzy filters are bounded. In the "basic example" H16 and H52 are homogeneous fuzzy filters which are not balanced.

Proposition 23 ([9]) Let L be a non-degenerate chain. Then we have:

(1) All balanced fuzzy filters define a partially ordered submonad of the partially ordered fuzzy filter monad.

(2) All homogeneous balanced fuzzy filters define a partially ordered submonad of the partially ordered fuzzy filter monad.

14. Fuzzy convergence structures

In the following let L = (L, :::;, *) be a non-degenerate commutative quantale and let = ( cp, :::; , 1], J.L) be the partially ordered L-filter monad or one of its partially ordered submonads. The special cases we will consider are listed in the following table. We describe these cases only by noting the related fuzzy filters.


Case (A) Case (S) Case (B) Case (H) Case (B')

Case (H')

Fuzzy filters of

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In the frame case conditions (2) and (3) are equivalent to the condition

(2)' intf 1\ intg = int(f 1\ g) for all j, g E Lx. In a -pretopological space a mapping f is called open provided that intf =f.

Proposition 25 The set T of all open fuzzy sets f E Lx of a -pretopology fulfills the condition

( 0') T is closed with respect to all suprema and all finite products !1 * * fn of fuzzy sets.

Each -topology p can be characterized as a subset T of Lx which ful-fills condition (0') by taking intf = V g for all f E Lx.

g"S_ f,gET

Remark. In the frame case, (0') is a well-known condition for char-acterizing a -topology. Notice that (0') and condition (0) in Propo-sition 3 look completely different.

Proposition 26 The set r of all open fuzzy sets of a fuzzy topology equipped with the argumentwise defined partial ordering and the argu-mentwise defined product *, is a quantale.

Remark. Notice that for each subset M of r the supremum of M in the lattice ( r, :::;) is the argumentwise defined supremum x H V f ( x) ( x E X) and the infimum of M in ( r , :::;) is the interior of the

! EM argumentwise defined infimum x H 1\ f(x) (x E X).

! EM

16. Closure operator of a fuzzy convergence structure Let L = (L, :=:;;, *) be a non-degenerate commutative quantale and let = (rp, :::;, ry, p,) be specified as in Section 15. Moreover, let T be a -convergence structure on a set X and let as usual t 1 : T ----+


Proposition 27 The sup-inverse 'P;t1 : (/)X ---+ (/JT of J.Lx o'Pt1 : (/JT ---+ 'PX is given by

v (4) for all M E 'PX and h E LT, wheTe joT each f E Lx, ef : 'PX ---+ L is defined by ef (N) = N (f).

Because of this proposition the closure operator cl 'PX ---+ (/)X is given as follows.

Proposition 28 For each M E (/)X and f E LX we have

(elM)(!) = v N(h)NCJn) ~ f (x)

for all N -tx

(5)

In the following we assign to each finite subset { JI, ... , fn} of Lx as its fuzzy set closuTe a fuzzy set cl (!1, ... , fn) E LX defined for all X EX by

cl (h, , fn)(x) = V (N(h) * * N(.fn)) N-tx

Some properties of this fuzzy set closure are the following. (1) h * * fn = 0 implies cl(JI,, ... , fn) = 0, (2) h :::; 91, ... , fn:::; 9n implies cl(h, ... , fn) :::; cl(91, ... , 9n) and (3) h * ... * fn :::; cl (!1, ... 'fn) and cl (!1, ... 'fn, 91, ... '9m) :::;

(6)

cl (h, .. . , fn) * cl (91, ... , 9m) holds for all JI, ... , fn, 91, ... , 9m E Lx. By means ofthe notion offuzzy set closure, equation ( 4) can be written as follows:

(ciM)(.f) v M(h* *fn) (7) cl (JI , .. . .Jn) :S; J

450 W. Gahler

Because of the result on finer ultra


by p(O) = [!I], q(O) = [h, ~] and p(l) = q(l) = 7Jx(l) are different -pretopologies which have one and the same closure operator.

References

[1] G. Birkhoff, Lattice theory, Amer. Math. Soc. Colloq. Publ. 25, 1940.

[2] P. Eklund, W. Gahler, Completions and compactifications by means of monads, in: Fuzzy Logic, State of Art, Kluwer (1993) 39-56

[3] W. Gahler, Monadic topology- a new concept of generalized topology, in: Recent Developments of General Topology and its Applications, Akademie Verlag 1992, 136-149.

[4] W. Gahler, The general fuzzy filter approach to fuzzy topology, I, Fuzzy Sets and Systems 76 (1995) 205-224.

[5] W. Gahler, The general fuzzy filter approach to fuzzy topology, II, Fuzzy Sets and Systems 76 (1995) 225-246.

[6] W. Gahler, Monadic convergence structures, Seminarberichte aus dem Fachbereich Mathematik, Fernuniversitat Hagen 67 (1999) 111-130.

[7] W. Gahler, General topology - the monadic case, examples, applica-tions, Acta Math. Hungar. 88 (4) (2000) , 279-290.

[8] W. Gahler, Extension structures and completions in topology and algebra, Seminarberichte aus dem Fachbereich Mathematik, Fernuni-versitat Hagen (2001).

[9] W. Gahler, P. Eklund, Extension structures and compactifications, Universitat Bremen, Mathematik-Arbeitspapiere Nr. 54 (2000) 181-205.

452 W . Gahler

[10] H. Herrlich, M. Husek, Galois connections categorically, J. Pure Appl. Algebra 68 (1990) 165-180.

[11] U. Hohle, Monoidal logic, in: Fuzzy Systems in Computer Science, Vieweg Verlag 1994, 223-234.

[12] U. Hohle, Locales and L-topologies, Universitat Bremen, Mathematik-Arbeitspapiere Nr. 48 (1997) 223-250.

[13] D. C. Kent, On convergence groups and convergence uniformities, Fundamenta Math. 60 (1967) 213-222.

[14] H.-J. Kowalsky, Limesraume und Komplettierung, Math. Nachr. 12 (1954) 301-340.

[15] 0. Ore, Galois connections, Trans. Amer. Math. Soc. 55 (1944) 493-513.

[16] H. Rasiowa, R. Sikorski, The mathematics of metamathematics, Paristwowe Wydawnictwo Naukowe 1963.

[17] H. I. Rosenthal, Quantales and their applications, Pitman Research Notes in mathematics 234, Longman, Burnt Mill, Harlow, 1990.

[18] J. Schmidt, Beitrage zur Filtertheorie II, Math. Nachr. 10 (1953) 197-232.

[19] B. Schweitzer, A. Sklar, Probabilistic metric spaces, North-Holland, 1983.

Author's address: Werner Gahler, Scheibenbergstr. 37, 12685 Berlin

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