Finite fuzzy topological spaces

Finite fuzzy topological spaces

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Fuzzy Sets and Systems ••• (••••) •••–•••

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www.elsevier.com/locate/fss

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Finite fuzzy topological spaces

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Moussa Benoumhani, Ali Jaballah ∗

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Department of Mathematics, The University of Sharjah, Sharjah, P.O. Box: 27272, United Arab Emirates

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Received 6 February 2016; received in revised form 12 November 2016; accepted 15 November 2016

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Abstract

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We compute for the first time, the number of fuzzy topologies defined on a finite set and having a small number of open sets. Certain cases, where the number of open sets is large, are also considered. Several well known results are obtained as corollaries. The paper is ended by some questions for future investigations. © 2016 Elsevier B.V. All rights reserved.

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A fuzzy set as defined by Lotfi Zadeh is a function from a set X on [0, 1]. This definition has been extended to more general sets than the unit interval; for example to a complete lattice. Let X be a set, M be a totally ordered one, and let F = F(X, M) = M X be the collection of fuzzy subsets of X with membership values in M, or equivalently, the set F of functions from X to M. F is partially ordered by: μ  ν ⇐⇒ μ(x) ≤ ν(x) for every x ∈ X. This set is also a complete lattice with the same partial order. We also have μ ≺ ν, if and only if μ  ν and μ(x) < ν(x), for some x ∈ X.

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The fuzzy subsets 0F and 1F are the functions defined respectively by 0F (x) = 0 for every x ∈ X, and 1F (x) = 1 for every x ∈ X. For every fuzzy subset μ different from 0F and 1F , we have 0F ≺ μ ≺ 1F . A fuzzy topology τ on the set X, as defined in [8], isa collection of fuzzy subsets of X such that 0F and 1F are in τ , sup ui is in τ for every (ui )i∈I ∈ τ , and min ui , uj is in τ for every ui and uj in τ . Note that a fuzzy topology i∈I

on F is just a sublattice of F , containing the least and the greatest elements. Since its definition, fuzzy topology has

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1. Introduction

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Keywords: Fuzzy topology; Number of fuzzy topologies; Combinatorics; Topology

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* Corresponding author. Fax: +971 65050318.

E-mail address: [email protected] (A. Jaballah). http://dx.doi.org/10.1016/j.fss.2016.11.003 0165-0114/© 2016 Elsevier B.V. All rights reserved.

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attracted many researchers. See [10,24,26], and [33] for recent investigations in this field. On the other hand fuzzy topological spaces on finite sets have not been considered yet. The number of finite topologies in the classical case is an outstanding and open problem. There is no known explicit formula for the total number of topologies T (n), one can define on an n-element set. Yet, there are some results. In fact the bijective correspondence between topologies and quasiorders (relations which are reflexive and transitive) on a finite set is known since the thirties of the last century; it was discovered independently by Alexandroff [1] and Birkhoff [6]. The antisymmetric quasiorders are called orders, they correspond to T0 topologies; recall that a topology is T0 , if for every x = y, there is an open set containing exactly one of them. It is also known that there is a correspondence between transitive graphs (or 0–1 matrices) and finite topologies, [13] and [12]. This result was used by Evans et al. [13] to compute T (n) for 2 ≤ n ≤ 7. The sequence T (n) is known just for n ≤ 18, see [29]. Using the matrix representation of a finite topology, Krishnamurty [22] obtained the upper bound T (n) ≤ 2n(n−1) . n2 4

Asymptotically and up to homeomorphism, the total number of topologies on a set of cardinality n is 2 . This result is due to D. Kleitman and B. Rothschild [19]. Another approach of the subject is the enumeration according to the number of open sets. Let T (n, k) be the number of topologies on an n-element set having k (2 ≤ k ≤ 2n ) open sets. It is known that T (n, k) = 0, for 3 · 2n−2 < k < 2n and n ≥ 3. Stanley [30] computed T (n, k) for 2 ≤ k ≤ 5 and for k ≥ 7 · 2n−4 . In computing these values, Stanley used quasiorders (and orders instead of finite topologies), and hence the open sets are just the ideals of the order (or quasiorder). This approach was fruitful for some small values as well as for some large values of k, and the limits of this method were reached. Then, the only way to extended T (n, k) for k ≤ 17 and k ≥ 5 · 2n−4 was to move from the other equivalent concepts of a finite topology (quasiorder, matrix or a graph) and use the original definition of a topology; its very rich structure supplies tools that are not existing in the other notions and are easier to handle. This enabled the authors of [4,20,21] to push the calculations of T (n, k) for other values of k. On the other hand, fuzzy topological spaces satisfying some finiteness conditions have not been considered yet unlike the classical topology where this field is still active and attracting several researchers by its importance and by the numerous long-standing unsolved problems, [3,4,7]. Also, finite topology has several applications such as image processing, which is concerned with visual information; it creates, stores, manipulates and displays digital images. In all these operations finite topology is involved. For example, we would like to know to what degree an image (a photo, for example) is an exact reflexion of the real world image. Also, after a digital transformation of a picture, we need the topological aspects of the image intact. To answer these questions (and others), the plane Z × Z = Z2 is endowed with a topology on it called the digital topology. The points (k, l), where k and l are odd numbers, are the pixels and form the minimal open sets of this topology. The subspace of the digital plane consisting of all open points is called the visible screen. This corresponds to the pixels in a digital image display. Any finite number of points of the visible screen is a finite subspace. The investigations of these finite topologies supply information about the image display and the other questions related to image processing. Another application is in biology; where the mutation of RNA molecules is investigated. The model is a finite topological space based on a probability distribution of a mutation. For more details about the previous and other examples see [1]. Finite topology is also present in chemistry. An approach to molecular topology is based on the atomic adjacency, so any topology modeling a hydrocarbon molecule must contain a bond as an open set. In other words if atoms a and b are connected, then the set {a, b} is open. This topology is the smallest one generated by the subbasis of all bonds of 2-element set of atoms. This is the bond topology. Any space in the bond topology is a product of its components, s s so it has the form R r S s S S where r, s, s , s are the number of the components of each type. Once we have the form of the topology, we can restitute the general chemical formula of the molecule. For other topologies defined on the molecule see [23]. The enumeration of finite fuzzy structures, and the combinatorial aspect in general in the fuzzy realm is curiously rare, except some sporadic papers dealing with fuzzy subgroups of finite fuzzy groups and fuzzy ideals of finite rings such as [14,32] and the references therein. The main purpose of this work is to remedy this lack and initiate the corresponding fuzzy side of these problems. In [27] the digital topology, known in image processing is shown to be deficient, and another (somewhat mixed) model is presented.

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We hope that the present paper may be helpful to improve the existing models of image processing. The Stone representation theorem states that every boolean algebra is isomorphic to a power set, its analogous in multi-valued logic is the fact that a Łukasiewicz algebra is isomorphic to a fuzzy algebra [9]. Here too, it is expectable to use this work to find topological representations of multi-valued algebras. For a set X of cardinality n, and a complete lattice M of cardinality m, let TF (n, m) be the total number of fuzzy topologies on X, with membership values in M, and let TF (n, m, k) be the number of fuzzy topologies on X, with membership values in M, having k open sets. We have trivially TF (n, m, 2) = 1, TF (n, m, 3) = mn − 2. For k ≥ 4, calculations are not as immediate as for k = 2, or 3. In the next section we give conditions under which the number of fuzzy topologies on F is finite. In section 3, we compute TF (n, m, 4). Section 4 is devoted to the determination of TF (n, m, 5). Then non-discrete topologies of maximal cardinalities are investigated in section 5, where the number and the cardinality of such topologies is established in Theorem 21. Several known results for finite classical topologies are obtained as corollaries of the established results in this work for the fuzzy setting, see Corollaries 6, 10, 13, 18, and Remark 22. The conclusion contains some open questions and directions for other investigations. Throughout this paper, X is a nonempty set of cardinality n, M is a totally ordered set with cardinality m ≥ 2 so that we can define fuzzy subsets of X.

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Proposition 2. The number of fuzzy topologies on X, with membership values in M, is finite if and only if X and M are both finite.

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The existence of a topology τ in the collection of fuzzy subsets of F with membership values in M implies necessarily that 0F and 1F are open sets in τ . Since M is finite, it has a minimum usually denoted by 0 and a maximum usually denoted by 1, so that the empty set 0F assigning the value 0, and the whole set 1F assigning the value 1 for each element of X, are part of each fuzzy topology defined on X. Concerning the finiteness of the number of such topologies we have the following result.

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(1) The fuzzy subsets 0F (also called the empty set) and 1F (also called the whole set) are in τ . (2) The union ∨i∈I ui of any collection {ui : i ∈ I } of elements of τ is also in τ . (3) The intersection u1 ∩ u2 of any two elements u1 and u2 of τ is also in τ .

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Definition 1. A fuzzy topology τ on a set X consists of a collection of fuzzy subsets of X called open sets, satisfying the following three axioms:

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Many notions and facts needed in the sequel are gathered in this section. The intersection and union of two fuzzy subsets μ and ν defined on a set X are defined for every x ∈ X by: μ ∩ ν(x) = min(μ(x), ν(x)), and μ ∪ ν(x) = max(μ(x), ν(x)). The union of the elements of a collection C = {μi : i ∈ I } of fuzzy subsets is defined by (∨i∈I μi ) (x) = sup{μi (x) : i ∈ I }. Recall that 0F and 1F are the functions defined respectively by 0F (x) = 0 for every x ∈ X, and 1F (x) = 1 for every x ∈ X.

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2. Preliminaries

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Proof. If M and X are both finite of cardinalities m and n respectively, then a topology on X is a collection of elements n of F . Therefore the number of topologies cannot exceed 2m . On the other hand if M is infinite, let t1 , t2 , ..., tn , ... be an infinite sequence of elements of M, and let ut be the fuzzy subset defined by ut (s) = t for every s ∈ X. Let τi = {0F , uti , 1F }. Then τi , i ≥ 1, is an infinite collection of fuzzy topologies of F . Finally if X is infinite, let s1 , s2 , ..., sn , ... be an infinite sequence of elements of X, and let u1s be the fuzzy subset defined by u1s (s) = 1 and u1s (x) = 0 for every x = s. Let σi = {0F , μ1si , 1F }. Then σi , i ≥ 1, is an infinite collection of fuzzy topologies of F . This proves the result. We assume throughout this paper that X and M are both finite, with X = {s1 , s2 ...., sn }, and M = {t0 , t1 , ..., tm−1 } is a totally ordered set such that: 0 = t0 < t1 < ... < tm−1 = 1. F is partially ordered by: μ  ν, if and only if μ(si ) ≤ ν(si ), for each i ∈ {1, 2, ..., n}.

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We also have

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μ ≺ ν, if and only if μ  ν and μ(si ) < ν(si ), for some i ∈ {1, 2, ..., n}. Let us denote each μ in F by (μ(s1 ), μ(s2 ), ..., μ(sn )). So, under this notation, the usual empty set will be represented by 0F = (0, 0, ..., 0); and the whole set X will be represented by 1F = (1, 1, ..., 1). For every μ ∈ F , we have 0F  μ  1F . Let us classify the elements of F in levels, where in each level the fuzzy subsets are incomparable. Since each of the component of (μ(s1 ), μ(s2 ), ..., μ(sn )) can be increased from t0 to tm−1 , we have exactly n(m − 1) + 1 levels in the ordered set F . The lowest level (level 0) contains only the element 0F . The level 1 consists of elements obtained from (μ(s1 ), μ(s2 ), ..., μ(sn )) = (0, 0, ..., 0) by exactly one-step increase. That is μ(si ) = t0 = 0 for n − 1 components and is equal to t1 for only one component. The level 2 consists of elements obtained from (μ(s1 ), μ(s2 ), ..., μ(sn )) = (0, 0, ..., 0) by exactly two one-step increases. That is either μ(si ) = t0 = 0 for n − 1 components and is equal to t2 for only one component, or μ(si ) = t0 = 0 for n − 2 components and is equal to t1 for the remaining two components, etc. The highest level (level n(m − 1)) contains only the element 1F , where all components are equal to tm−1 = 1. A fuzzy topology consisting of exactly two open sets, is the trivial topology consisting of 0F and 1F . On the other hand, a topology with exactly three open sets consists necessarily of a chain containing any fuzzy subset of X in addition to 0F and 1F . Therefore, there are exactly mn − 2 fuzzy topologies consisting of three open sets in F . The case of topologies consisting of more than three open sets is not as trivial as the last two cases. In order to compute the cardinalities of such topologies, we need the following result.

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(i1 .i2 ,...,in

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Now assume that



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yi1 yi2 ...yik+1

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This proves the lemma.

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At several stages of this paper, we need to compute the number of chains of a certain length in a finite ordered set. The following well known algorithm enables us to compute these numbers.

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Proof. Let us proceed by induction on n. If n = 1, we have  m    yi 1 = yi .

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Lemma 3. Let m and n be positive integers, then for any numbers y1 , y2 , ..., ym , we have:  m n   yi1 yi2 ...yin = yi .

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Algorithm 1. Let P be a finite ordered set and let us denote by ck (P ) or ck the number of chains with k elements in the ordered set P . Also, for each u ∈ P , we let ck (u) be the number of chains with k elements from P and with maximal element u. The numbers c1 , c2 , ..., cn are obtained recursively as follows:

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Fig. 1. Topologies with four open sets.

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3. Fuzzy topologies with four open sets

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A topology with four open sets consists, in addition to the empty and whole sets, either of a two elements chain as in Fig. 1(a), or of a two elements antichain having the empty set as intersection and the whole set as union, as in Fig. 1(b). To compute the number of topologies with exactly four open sets, we need to compute the number of chains and the  := F \ {0F , 1F }. number of antichains of size two, and not containing both the empty and whole sets. Let us define F  consists of levels 1, 2, ... , and n(m − 1) − 1 from F . Before computing the number of such topologies in F , we F first need to compute the number of such chains and antichains. The next two lemmata will be helpful. We start with chain topologies as in Fig. 1(a). To compute the number of such topologies it is enough to find the number of chains . of length 2 in F

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 is given by: Theorem 4. The number of chains of length 2 in F  n  = m+1 c2 (F) − 3mn + 3. 2

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c2 (ti1 , ti2 , ..., tin ) = (i1 + 1)(i2 + 1)...(in + 1) − 2.

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Taking the sum over all elements of F lying on levels 2 to n(m − 1) − 1, we obtain that the number c2 of chains of  is given by: length 2 in F   = c2 (F) c2 (ti1 , ti2 , ..., tin )

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 is first subject to the condition that there are enough levels to contain Proof. The existence of chains of length 2 in F such chains. That is we assume first that n(m − 1) − 1 ≥ 2, or equivalently n(m − 1) ≥ 3. To compute the number of  we use Algorithm 1. We have c1 (ti1 , ti2 , ..., tin ) = 1, for each (ti1 , ti2 , ..., tin ) ∈ F . chains of length 2 in F For each (tj1 , tj2 , ..., tjn ) satisfying (tj1 , tj2 , ..., tjn )  (ti1 , ti2 , ..., tin ), we have 0 ≤ jk ≤ ik for each k in {1.2...., n}. That is there are exactly (i1 + 1)(i2 + 1)...(in + 1) such (tj1 , tj2 , ..., tjn )’s. Removing (i1 , i2 , ..., in ) and (0, 0, ...0) from  and ending with (ti1 , ti2 , ..., tin ) is: the list we obtain that the number of chains containing two elements in F

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(a) either m = 2 and n = 1, (b) or m = 2 and n = 2, (c) or m = 3 and n = 1.

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On one hand, these values, substituted in the above formula, give the value 0. On the other hand these values result in the ordered sets shown in Fig. 2. It is easy to check that we have no chain topology with 4 open sets in all cases, (a), (b) and (c), which is consistent with the obtained formula. This finishes the proof of the theorem.

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Fig. 3. Topologies with n(m − 1) ≤ 1.

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Corollary 5. The number of chains of length 4 in F , and containing both 0F and 1F is given by:  n  = m(m + 1) − 3mn + 3. c2 (F) 2

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Lemma 7. The number of antichains of size 2 in F consisting of two elements having 0F as intersection and 1F as union is given by: 2

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Proof. Let (ti1 , ti2 , ..., tin ) and (tj1 , tj2 , ..., tjn ) form such an antichain. The two fuzzy subsets are different from 0F and 1F , and satisfy min(tik , tjk ) = 0 and max(tik , tjk ) = 1 for each 1 ≤ k ≤ n. That is tik = 0 if and only if tjk = 1, and tik  = 1if and only if tjk = 0. Thus (tj1 , tj2 , ..., tjn ) is automatically   n n such (ti1 , ti2 , ..., tin )’s containing exactly one 0, such determined by (ti1 , ti2 , ..., tin ). There are exactly 1 2   n containing (n − 1) 0’s. That is; in total we have (ti1 , ti2 , ..., tin )’s containing exactly two 0’s, ..., n−1       n n n + + ... + = 2n − 2 = 2(2n−1 − 1) 1 2 n−1 different such (ti1 , ti2 , ..., tin )’s. Since each pair of (ti1 , ti2 , ..., tin ) and corresponding (tj1 , tj2 , ..., tjn ) is repeated twice by this process, we have exactly 2n−1 − 1 different antichains. If m and n are positive integers such that n(m − 1) + 1 ≤ 2, or equivalently n(m − 1) ≤ 1, and since we assumed that n ≥ 1 and m ≥ 2, we obtain m = 2 and n = 1. This leads to the ordered set shown in Fig. 3. It is easy to check that we have no topologies with 4 open sets in this case, which is consistent with the obtained formula.This completes the proof of the lemma.

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Having finished with chain topologies, we compute in the following results the number of non-chain topologies with four open sets in F .

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Proof. This is just obtained by letting m = 2 in the previous Corollary 5.

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Corollary 6. Let X be a nonempty finite set of cardinality n. The number of chains of subsets of X of length 4 and containing both X and the empty set is given by:

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Corollary 8. The number of non-chain topologies of F and consisting of 4 open sets is given by: 2

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Theorem 9. The number of fuzzy topologies in F consisting of four open sets is given by:   m(m + 1) n TF (n, m, 4) = − 3mn + 2n−1 + 2. 2

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2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

Fig. 4. Topologies with five open sets.

13 14 15 16 17 18 19 20 21 22

14

Proof. The number of fuzzy topologies with four open sets in F is obtained by adding the number of chain topologies to non-chain topologies, that is:   m(m + 1) n TF (n, m, 4) = 2n−1 − 1 − 2(mn − 2 − n) + − mn − 2n − 1 2   m(m + 1) n − 3mn + 2. = 2n−1 + 2 This proves the theorem.

23 24

27 28

Taking m = 2, we obtain the following result for classical topologies.

35 36 37 38

41 42 43 44 45 46 47 48

51 52

20 21 22

24

26 27 28

30

32

34 35 36 37 38 39

Let us call the topologies of Fig. 4(a) chain topologies and denote their number by TF ,C (n, m, 5). Let us also call the topologies of Figs. 4(b) and 4(c) kite topologies and denote their number by TF ,K (n, m, 5). Existence of chain topologies is subject to the condition that there are enough levels to contain such chains. That is we must have n(m − 1) + 1 ≥ 5. To compute the number TF ,C (n, m, 5) we use again Algorithm 1. The following result gives the number of such topologies.  is given by: Theorem 11. The number of chains of length 3 in F n n n n n  = m (m + 1) (m + 2) − 4m (m + 1) + 6mn − 4. c3 (F) 6n 2n

49 50

19

33

(a) either a chain of length 3 as in Fig. 4(a), (b) or two two-element chains having a common minimum different from 0F , and different maximums whose union is 1F as in Fig. 4(b), (c) or two two-element chains having a common maximum different from 1F , and different minimums whose intersection is 0F as in Fig. 4(c).

39 40

18

31

A topology with five open sets consists, in addition to the empty and whole sets, of three elements forming

33 34

17

29

4. Fuzzy topologies with five open sets

31 32

16

25

Corollary 10. The number of topologies consisting of four open sets in a set of cardinality n is given by:   T (n, 4) = 3n − 5 2n−1 + 2.

29 30

15

23

25 26

13

40 41 42 43 44 45 46 47 48 49

 we use Algorithm 1. Proof. We first assume that n(m − 1) + 1 ≥ 5. To compute the number of chains of length 3 in F   and with maximal For each (ti1 , ti2 , ..., tin ) in F , let ck (ti1 , ti2 , ..., tin ) be the number of chains with k elements from F element (ti1 , ti2 , ..., tin ). We have then:

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c1 (ti1 , ti2 , ..., tin ) = 1,

1

4

c2 (ti1 , ti2 , ..., tin ) = (i1 + 1)(i2 + 1)...(in + 1) − 2, by Eq. (1), and  c3 (ti1 , ti2 , ..., tin ) = c2 (tj1 , tj2 , ..., tjn )

5

0F ≺(tj1 ,tj2 ,...,tjn )≺(ti1 ,ti2 ,...,tin )

3

6

10 11 12

15 16 17

20

=



=1+ =1+

n  k=1 n  k=1

45 46

49 50 51 52

n 

24

(ik + 1)

25 26

k=1

(ik + 2)(ik + 1) −2 2



n 

27

(ik + 1)

28

n 

29 30

(ik + 1)

31 32

k=1

34 35



n n   (ik + 2)(ik + 1) − 2 (ik + 1) 1+ 2 k=1

k=1

37 38 39 40 41

(m + 1)n (m)n c 3 + c2 + 1 + − 2mn 2n   n n    (ik + 2)(ik + 1) = − 2 (ik + 1) 1+ 2 0F (ti1 ,ti2 ,...,tin )1F

= mn +

33

36



Therefore

47 48

20

k=1

0F ≺(ti1 ,ti2 ,...,tin )≺1F

40

44

y1 y2 ...yn ⎠ − 2

(1 + 2 + ... + (ik + 1)) − 2

=

39

43

19

23



 (ti1 ,ti2 ,...,tin )∈F

38

42

18

 we obtain: Taking the sum over all elements of F  c 3 + c2 = c3 (ti1 , ti2 , ..., tin ) + c2 (ti1 , ti2 , ..., tin )

37

17

22

1≤yk ≤ik +1,1≤k≤n

36

41

16

((j1 + 1)(j2 + 1)...(jn + 1) − 2)

21



=1+⎝

32

35

15

(y1 y2 ...yn − 2)

0≤yk −1≤ik ,1≤k≤n

30

34



((j1 + 1)(j2 + 1)...(jn + 1) − 2)



=1+

29

33

14

0≤jk ≤ik ,1≤k≤n

27

31

12

0F (j1 ,j2 ,...,jn )(i1 ,i2 ,...,in )



26

28

11

((j1 + 1)(j2 + 1)...(jn + 1) − 2)

= − ((0 + 1)(0 + 1)...(0 + 1) − 2) +

24 25

10

13

0F ≺(j1 ,j2 ,...,jn )(i1 ,i2 ,...,in )

23

7

9



=1+

6

8

0F ≺(tj1 ,tj2 ,...,tjn )(ti1 ,ti2 ,...,tin )

21 22

c2 (tj1 , tj2 , ..., tjn )

c3 (ti1 , ti2 , ..., tin ) + c2 (ti1 , ti2 , ..., tin )  = c2 (tj1 , tj2 , ..., tjn )

18 19

4

Hence we have:

13 14

3

0F ≺(tj1 ,tj2 ,...,tjn )(ti1 ,ti2 ,...,tin )

8 9

2

5



= −c2 (ti1 , ti2 , ..., tin ) +

7

9

m−1 

1 ( 2n

ik =0

k=1

k=1

m−1 

(ik + 2)(ik + 1))n − 2(

ik =0

43 44 45 46 47

(ik + 1))n

ik =0

1  2mn (m + 1)n = mn + n ( (ik + 2)(ik + 1))n − 2 2n m−1

42

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10

1

Since



2

c2 =

3 4 5 6 7 8

m(m + 1) 2

1

n

2

− 3mn + 3,

3 4

and

5

m−1 

6

m  (ik + 2)(ik + 1)) = ( (j + 1)j )n

(

n

ik =0

9

7 8

j =1

9

m  =( j 2 + j )n

10 11

10 11

j =1



12

m(m + 1)(2m + 1) 3m(m + 1) + 6 6  n m(m + 1)(2m + 4) = 6   m(m + 1)(m + 2) n = , 3

n

12

=

13 14 15 16 17 18

13 14 15 16 17 18

19 20 21 22 23 24 25 26

19

we have:

 m(m + 1) n (m + 1)n (m)n c3 + − 3mn + 3 + 1 + − 2mn 2 2n   1 m(m + 1)(m + 2) n 2mn (m + 1)n = mn + n − 2 3 2n

That is

30 31

37 38 39 40

43 44 45 46 47 48 49 50 51 52

23 24 25

28 29 30 31

c3 = −4 + 6mn −

2n

+

mn (m + 1)n (m + 2)n 6n

33

.

This proves the lemma. Corollary 12. The number of chains of length 5 in F , and containing both 1F and 0F is given by: n n n n n  = m (m + 1) (m + 2) − 4m (m + 1) + 6mn − 4. c3 (F) 6n 2n

41 42

22

32

4mn (m + 1)n

35 36

21

27

Hence

33 34

20

26

n

c3 + 2

29

32



m(m + 1) − 5mn + 4 2   1 m(m + 1)(m + 2) n 2mn (m + 1)n n − =m + n 2 3 2n

27 28



34 35 36 37 38 39 40 41

Corollary 13. The number of chains of length 5, of subsets of a finite set X of cardinality n, and containing both X and the empty set is given by: 4 − 4(3 ) + 6(2 ) − 4. n

n

n

Having finished with chain topologies, we compute in the following the number of non-chain topologies with five open sets in F . We start with the following result.  and having 0F as intersection is given by: Lemma 14. The number of antichains consisting of two elements, in F, (2m − 1)n − mn − (m − 1)n − 2n + 2 . 2

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M. Benoumhani, A. Jaballah / Fuzzy Sets and Systems ••• (••••) •••–••• 1 2 3 4 5 6 7 8 9 10 11

Proof. Assume that (ti1 , ti2 , ..., tin ) and (tj1 , tj2 , ..., tjn ) form such an antichain. Thus we must have min(tik , tjk ) = 0, for every k, 1 ≤ k ≤ n. Assume that ti1 , ti2 , ..., tip are the only zeros of (ti1 , ti2 , ..., tin ), then tjp+1 , tjp+2 , ..., tjn are necessarily zeros, while tip+1 , tip+2 , ..., tin are free to take any value in {t1 , ..., tm−1 }; and tj1 , tj2 , ..., tjp are free to take any value in  {t0 , t1 , ..., tm−1 }. That is there are (m − 1)n−p mp such pairs. Nowsince  (tj1 , tj2 , ..., tjn ) = 0F , and n  n (m − 1)n−p mp − 1 pairs there are exactly such (ti1 , ti2 , ..., tin )’s containing exactly p zeros, we have p p of (ti1 , ti2 , ..., tin ) and (tj1 , tj2 , ..., tjn ) with each (ti1 , ti2 , ..., tin ) containing exactly p zeros, 1 ≤ p ≤ n − 1. That is in total we have n−1      n (m − 1)n−k mk − 1 k k=1

12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

11

=

n−1    n

k

(m − 1)

n−k

m − k

n−1    n

k k=1 k=1   = (2m − 1)n − (m − 1)n + mn − 2n − 2

33 34 35 36 37 38 39 40 41 42

45 46 47 48 49

52

4 5 6 7 8 9 10 11 12 13 14 15 16

 consisting of two elements having 0F as intersection and a union Corollary 15. The number of antichains in F, different from 1F is given by:

21

(2m − 1) + 4 − (m − 1) 2 n

n

− mn

− 2n+1

18 19 20

22 23 24

.

 and having 0F as intersection and a union Proof. Let a1 be the number of antichains consisting of two elements, in F,  and having 0F as intersection. different from 1F . Let a2 be the number of antichains consisting of two elements, in F,  Let a3 be the number of antichains consisting of two elements, in F, and having 0F as intersection and 1F as union. Then

25 26 27 28 29 30 31

a 1 = a 2 − a3   (2m − 1)n − (m − 1)n + mn − (2n − 2)  n−1 = −1 − 2 2  (2m − 1)n − (m − 1)n + mn − 2 (2n − 2) = 2 (2m − 1)n + 4 − (m − 1)n − mn − 2n+1 = . 2 Corollary 16. The number of kite topologies in F , and consisting of five open sets is given by: TF ,K (n, m, 5) = (2m − 1)n − mn − (m − 1)n − 2n+1 + 4.

32 33 34 35 36 37 38 39 40 41 42 43

Proof. Recall that we adopted the notation M = {t0 , t1 , ..., tm−1 } with 0 = t0 < t1 < ... < tm−1 = 1. Define θ : F → F by θ (ti1 , ti2 , ..., tin ) = (tm−1−i1 , tm−1−i2 , ..., tm−1−in ). It is clear that θ ◦ θ = idF ; and μ1  μ2 if and only if θ (μ2 )  θ (μ1 ). Let 0F ≺ μ1 , μ2 ≺ μ3 ≺ 1F be a topology of F , as in Fig. 4(b). Then θ (1F ) = 0F ≺ θ (μ3 ) ≺ θ (μ2 ), θ (μ1 ) ≺ θ (0F ) = 1F is a topology of F , as in Fig. 4(c). Also θ transforms a topology of F as in Fig. 4(c) to a topology of F as in Fig. 4(b). That is θ is a one-to-one correspondence between topologies as in Fig. 4(b) and topologies as in Fig. 4(c). This means that TF ,K (n, m, 5) is twice as much as the result in Corollary 15, which finishes the proof.

50 51

3

17

43 44

2

Finally the number is:  (2m − 1)n − (m − 1)n + mn − (2n − 2) . 2

31 32

1

44 45 46 47 48 49 50

Adding together the results of Corollary 12 and Corollary 16, we obtain the total number of fuzzy topologies with five open sets in the next result.

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Corollary 17. The number TF (n, m, 5) of fuzzy topologies in F consisting of five open sets, is given by:  n  n m+2 m+1 TF (n, m, 5) = −4 + (2m − 1)n + 5mn − (m − 1)n − 2n+1 . 3 2 Taking m = 2, we obtain the following result for classical finite topologies, see [30] and [3] (actually, there is a misprint in Stanley paper [30] for the case T (n, 5)).

8 9 10 11 12 13 14 15 16 17 18 19 20 21

T (n, 5) = 4n − 3n+1 + 3(2n ) − 1 Example 19. Let us calculate the number of fuzzy topologies on X where n = 2 and m = 3. The numbers TF (n, m, k) are calculated by hand for 6 ≤ k ≤ 8, and using formulae for 2 ≤ k ≤ 5. TF (2, 3, 2) = T (2, 3, 32 ) = 1, TF (2, 3, 3) = 7, TF (2, 3, 4) = 13, TF (2, 3, 5) = 14, TF (2, 3, 6) = 6, TF (2, 3, 7) = 5, TF (2, 3, 8) = 2. So, the total number is TF (2, 3) = 49. Remark 20. In the classical case, we have T (n, k) = 0 for 3(2n−2 ) < k < 2n . In the fuzzy case, for n = 2 and m = 3, TF (n, m, k) = 0, for every k such that 2 ≤ k ≤ 32 . In general, for n = 2, and m ≥ 2, TF (n, m, k) = 0 for every 2 ≤ k ≤ m2 , and in particular TF (n, m, m2 − 1) = 2. It seems to be true that for n < m, TF (n, m, k) = 0 for every 2 ≤ k ≤ mn . But, for n ≥ m, we establish in the next section a result analogous to the classical case.

28

31

34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52

6 7

9 10 11 12 13 14 15 16 17 18 19 20 21

23

25

27 28 29

(1) TF (n, m, k) = 0 for mn − mn−2 < k < mn , and (2) TF (n, m, mn − mn−2 ) = n(n − 1).

32 33

5

26

Theorem 21. For n ≥ m ≥ 2, there are exactly n(n − 1) nondiscrete topologies of maximal cardinality. Every such fuzzy topology has exactly mn − mn−2 open sets. That is for n ≥ m ≥ 2, we have:

29 30

4

24

The next theorem gives the number of nondiscrete fuzzy topologies having maximal number of open sets.

26 27

3

22

5. Fuzzy topologies with large number of open sets

24 25

2

8

Corollary 18. The number of topologies on a finite set of cardinality n and consisting of five open sets is given by:

22 23

1

30 31 32

Proof. Let bp,q := (0, 0, ...., tq , ..., 0) be the fuzzy subset of X with tq in position p and 0 in all other remaining positions, and B = {bp,q |1 ≤ p ≤ n, 1 ≤ q ≤ m − 1}. It is clear that every fuzzy subset of X is either a union or an intersection of a subfamily of B. Let τ be a maximal non-discrete topology of F (i.e.; having the maximum number of open sets < mn ). Then bp0 q0 is not an open set of τ for some p0 ∈ {1, 2, ..., n} and q0 ∈ {1, 2, ..., m − 1}. If b1,m−1 is not an open set of τ , let A1 be the collection of elements (ti1 , ti2 , ..., tin ) of τ such that ti1 = tm−1 = 1, and A2 be the collection of elements of τ such that ti1 = 1. A1 cannot contain a family of fuzzy subsets μ2 , μ3 , ..., μn with μi having 0 in position i, as this leads to b1,m−1 being an open set of τ . Therefore tip = 0 for all elements of A1 for some p ∈ {2, ..., n}. There is no loss of generality if we assume p = 2. Then A1 would consists of all fuzzy subsets (ti1 , ti2 , ..., tin ) such that ti1 = 1, ti2 = 0, and ti3 , ..., tin ∈ {t0 , t1 , ..., tm−1 }. This indicates that there are exactly (m − 1)mn−2 different open sets in A1 . On the other hand every element (ti1 , ti2 , ..., tin ) of A2 satisfy ti1 = 1 and ti2 , ..., tin ∈ {t0 , t1 , ..., tm−1 }, then A2 has exactly (m − 1)mn−1 different open sets. Since every element of τ is either in A1 or in A2 , τ has exactly (m − 1)mn−2 + (m − 1)mn−1 = (m2 − 1)mn−2 different open sets. Now let μ1 and μ2 be two elements of τ . It is easy to check that μ1 ∩ μ2 and μ1 ∪ μ2 are in A1 if both are in A1 , and in A2 if both are in A2 . On the other hand if μ1 is in A1 and μ2 is in A2 , then it is easy tot check that μ1 ∩ μ2 is in A2 and μ1 ∪ μ2 is in A1 . That proves that τ is a fuzzy topology. Now let an additional fuzzy subset (ti1 , ti2 , ..., tin ) of X be added to τ . We necessarily have ti1 = 1, and ti2 = 0. This will necessarily imply that b1,m−1 , and therefore every fuzzy subset of X is in the topology τ . Contradicting the fact that τ is a non-discrete topology. This shows that τ is a maximal topology of F . Similarly, for each p ∈ {1, 2, ..., n},

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M. Benoumhani, A. Jaballah / Fuzzy Sets and Systems ••• (••••) •••–••• 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

13

B\{bp,m−1 } generates (n − 1) topologies, each having exactly (m2 − 1)mn−2 different open sets. That is, in total we have n(n − 1) maximal fuzzy topologies each consisting of (m2 − 1)mn−2 different open sets. Now if b1,q is not an open set of τ , for q = m − 1, let B1 be the collection of elements (ti1 , ti2 , ..., tin ) of τ such that ti1 ≥ tq , B2 be the collection of elements of τ such that ti1 < tq . B1 cannot contain a family of fuzzy subsets μ2 , μ3 , ..., μn with μi having 0 in position i, as this leads that b1,q is an open set of τ . Therefore tir = 0 for all elements of B1 for some r ∈ {2, ..., n}. There is no loss of generality if we assume r = 2. Then B1 would contain all fuzzy subsets (ti1 , ti2 , ..., tin ) such that ti1 ≥ tq , ti2 = 0, and ti3 , ..., tin ∈ {0, 1, ..., m − 1}. This indicates that there are exactly (m − q)(m − 1)mn−2 different open sets in B1 . On the other hand every element (ti1 , ti2 , ..., tin ) of B2 satisfy ti1 < tq and ti1 , ti2 , ..., tin ∈ {t0 , t1 , ..., tm−1 }, then B2 has exactly qmn−1 different open sets. It is clear that every element of τ is either in B1 or in B2 . Therefore τ has exactly (m − q)(m − 1)mn−2 + qmn−1 = (m2 − m + q)mn−2 different open sets. Now let μ1 and μ2 be two elements of τ . It is easy to check that μ1 ∩ μ2 and μ1 ∪ μ2 are in τ whether μ1 is in B1 or B2 , and μ2 is in B1 or B2 . That is τ is a fuzzy topology. Now let an additional fuzzy subset (ti1 , ti2 , ..., tin ) of X be added to τ . We necessarily have ti1 ≥ tq , and ti2 = 0. This will necessarily lead that b1,q , and therefore every fuzzy subset of X is in the topology. This shows that τ is a maximal topology of F consisting of exactly (m2 − m + q)mn−2 different open sets. Since q ∈ {1, 2, ..., m − 1}, we can see that the maximum value for (m2 − m + q)mn−2 is obtained for q = m − 1. That is the maximum is (m2 − 1)mn−2 , obtained when bp,m−1 , 1 ≤ p ≤ n, is not an open set of the maximal nondiscrete topology. This finishes the proof of the theorem.

21 22 23

Remark 22. The case m = 2 gives the number TF classical case, see [3,28,31].

This agrees with the

28 29

32 33 34 35 36 37 38 39

42 43

At the best knowledge of the authors, this is the first time that the subject of finite fuzzy topologies is considered in the mathematical literature, at least under this aspect. We believe, that this is a new and rich area which has never been explored. More results will be considered in a forthcoming paper, see [5]. We also pose the following problems.

48

Problem 23. Determine TF (n, m, k) for 6 ≤ k

9 10 11 12 13 14 15 16 17 18 19 20

22

27 28 29

32

Problem 24. In the classical case, it is known that T (n, k) = 0 for 2 ≤ k ≤ k0 and that k0 ≥ 2 , see [25]. Find the corresponding k0 for the sequence TF (n, m, k). The TF (n, m, k) for 2 ≤ k ≤ 5 are the numbers of labeled fuzzy topologies with k open sets. Problem 25. Determine the numbers TF ,h (n, m, k), 2 ≤ k ≤ 5, of unlabeled (up to homeomorphism) fuzzy topologies with k open sets.

33 34 35 36 37 38 39 40

Problem 26. Is it possible to represent a finite fuzzy topology by a matrix? a graph? Problem 27. Is there any interesting link with the trivalent logic of Łukasiewicz, when m = 3?

41 42 43 44

Acknowledgements

45 46

The authors would like to thank the editors and the referees for their valuable comments and suggestions. The authors would like to thank the University of Sharjah for its support.

47 48 49

Uncited references

50 51

51 52

8

31

− mn−2 . 3 4n

49 50

7

30

< mn

46 47

6

26

44 45

5

25

6. Conclusion

40 41

4

24

30 31

3

23

26 27

2

21

(n, 2, 3(2n−2 )) = T (n, 3(2n−2 )) = n(n − 1).

24 25

1

[2] [11] [15] [16] [17] [18]

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References

2 3 4 5 6 7 8 9 10

2

[1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

11 12

[11]

13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

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[12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33]

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