CHARACTERIZATIONS OF L-CONVEX SPACES

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1 Iranian Journal of Fuzzy Systems Vol 13, No 4, (2016) pp CHARACTERIZATIONS OF L-CONVEX SPACES B PANG AND Y ZHAO Abstract In this paper, the concepts of L-concave structures, concave L- interior operators and concave L-neighborhood systems are introduced It is shown that the category of L-concave spaces and the category of concave L- interior spaces are isomorphic, and they are both isomorphic to the category of concave L-neighborhood systems whenever L is a completely distributive lattice Also, it is proved that these categories are all isomorphic to the category of L-convex spaces whenever L is a completely distributive lattice with an order-reversing involution operator 1 Introduction Abstract convexity theory [20, 23] is an important branch of mathematics As in the situation of topology theory, convexity theory usually deals with set-theoretic structures satisfying axioms similar to that usual convex sets fulfill Convexities exist in many different mathematical research areas, such as convexities in lattices and in Boolean algebras [22, 24], convexities in metric spaces and graphs [8, 11, 19] Especially, convexities appear naturally in topology and possess many topological properties, such as product spaces [4, 5], convex invariants [9, 16, 17, 18, 21] and separation [2, 3, 6, 7] Like closure operators in topology, hull operators (also called algebraic closure operators) can be used to characterize convex structures [23] With the development of fuzzy mathematics, convex structures have been endowed with fuzzy set theory In [10] and [14], the authors independently introduced the concept of L-convex structures Based on L-convex structures, Pang and Shi [12] proposed several types of L-convex structures and investigated their categorical relations Since algebraic closure operators cannot be directly generalized to L-convex spaces, there have not been algebraic L-closure operators That is to say, there have not been suitable axiomatic definition for algebraic L-closure operators in the framework of L-convex spaces Recently, Shi and Xiu [15] provided a new approach to fuzzification of convex structures and proposed the notion of M-fuzzifying convex structures In this environment, algebraic M-fuzzifying closure operators are introduced and they are shown to be equivalent to M-fuzzifying convex structures Inspired by this, Pang and Shi [13] introduced axiomatic L-hull operators to characterize L-convex structures In the theory of L-topology, there are several characterizations of L-topologies, including L-closure operators, L-interior operators and L-neighborhood systems Received: December 2015; Revised: February 2016; Accepted: April 2016 Key words and phrases: L-convex structure, L-concave structure, Convex L-closure operator, Concave L-interior operator, Concave L-neighborhood system

2 52 B Pang and Y Zhao As we all know, L-convex structures possess so many characters of L-topologies This motivates us to consider the corresponding characterizations of L-convex structures In this paper, we will firstly introduce the dual concept of L-convex structures, which is called L-concave structures And then we will consider the corresponding fuzzy interior operators and fuzzy neighborhood systems 2 Preliminaries Throughout this paper, let L denote a complete lattice The smallest element and the largest element in L are denoted by and, respectively For a, b L, we say that a is wedge below b in L, in symbols a b, if for every subset D L, D b implies d a for some d D An element a in L is called coprime if a b c implies a b or a c The set of nonzero coprime elements in L is denoted by J(L) Let β(a) = {b L b a} and β (a) = β(a) J(L) A complete lattice L is completely distributive if and only if a = β(a) (or a = β (a)) for each a L Also, we adopt the convention that = and = For a nonempty set X, L X denotes the set of all L-subsets on X L X is also a complete lattice when it inherits the structure of the lattice L in a natural way, by defining, and pointwisely The smallest element and the largest element in L X are denoted by and, respectively We say {A j } j J is a directed (resp co-directed) subset of L X dir, in symbols {A j } j J L X cdir (resp {A j } j J L X ), if for each A j1, A j2 {A j } j J, there exists A j3 {A j } j J such that A j1, A j2 A j3 (resp A j1, A j2 A j3 ) Let X, Y be two nonempty sets and ϕ : X Y be a mapping Define ϕ : L X L Y and ϕ : L Y L X by ϕ (A)(y) = ϕ(x)=y A(x) for A LX and y Y, and ϕ (B) = B ϕ for B L Y, respectively Definition 21 [10, 14] An L-convex structure E on X is a subset of L X which satisfies: (LC1), E; (LC2) {A k } k K E implies k K A k E; (LC3) If {A j } j J E is totally ordered, then j J A j E For an L-convex structure E on X, the pair (X, E) is called an L-convex space For partially ordered sets, in [1], the researchers present the following result Proposition 22 A partially ordered set D is closed for directed unions if and only if each chain in D has a supremum Obviously, for an L-convex structure E on X, since the order on L X is defined pointwisely, (E, ) is also a partially ordered set Hence, we obtain the following result Proposition 23 Let E be an L-closure system on X, that is, E L X satisfies (LC1) and (LC2) Then the following statements are equivalent (LC3) If {A j } j J E is totally ordered, then j J A j E (LC3) {A j } j J dir E implies j J A j E

3 Characterizations of L-convex Spaces 53 In the sequel, we usually adopt (LC3) instead of (LC3) Definition 24 [13] A mapping ϕ : (X, E X ) (Y, E Y ) between L-convex spaces is called L-convexity-preserving (L-CEP, in short) provided that B E Y implies ϕ (B) E X It is easy to check that all L-convex spaces as objects and all L-CEP mappings as morphisms form a category, denoted by L-CE In [13], Pang and Shi provided a characterization of L-convex structures by L- hull operators, which is called convex L-closure operators in this paper Definition 25 [13] A convex L-closure operator on X is a mapping Cl : L X L X which satisfies: (CLC1) Cl( ) = ; (CLC2) A Cl(A); (CLC3) Cl(Cl(A)) = Cl(A); (CLC4) Cl( j J A j) = j J Cl(A j) for {A j j J} dir L X For a convex L-closure operator Cl on X, the pair (X, Cl) is called a convex L- closure space Definition 26 [13] A mapping ϕ : (X, Cl X ) (Y, Cl Y ) between convex L- closure spaces is called L-closure-preserving (L-CLP, in short) provided that A L X, ϕ (Cl X (A)) Cl Y (ϕ (A)) It is easy to check that all convex L-closure spaces as objects and all L-CLP mappings as morphisms form a category, denoted by L-CLC Theorem 27 [13] L-CE is isomorphic to L-CLC 3 L-concave Spaces and Concave L-interior Spaces In this section, we will introduce the concept of L-concave spaces, which a dual concept of L-convex spaces Then we will propose the concept of concave L-interior spaces to characterize L-concave spaces Firstly, we give the definitions of L-concave structures and concave L-interior operators Definition 31 An L-concave structure A on X is a subset of L X which satisfies: (LA1), A; (LA2) {A k } k K A implies k K A k A; cdir (LA3) {A j } j J A implies j J A j A For an L-concave structure A on X, the pair (X, A) is called an L-concave space Definition 32 A mapping ϕ : (X, A X ) (Y, A Y ) is called L-concavity-preserving (L-CAP, in short) provided that B A Y implies ϕ (B) A X It is easy to check that all L-concave spaces as objects and all L-CAP mappings as morphisms form a category, denoted by L-CA

4 54 B Pang and Y Zhao Definition 33 A concave L-interior operator on X is a mapping Int : L X L X which satisfies: (CLI1) Int( ) = ; (CLI2) Int(A) A; (CLI3) Int(Int(A)) = Int(A); (CLI4) Int( j J A j) = j J Int(A j) for {A j j J} cdir L X For a concave L-interior operator Int on X, the pair (X, Int) is called a concave L-interior space Remark 34 By (CLI4), it is easy to check that Int is order preserving That is to say, for A, B L X, A B implies Int(A) Int(B) Definition 35 A mapping ϕ : (X, Int X ) (Y, Int Y ) between concave L- interior spaces is called L-interior-preserving (L-INP, in short) provided that B L Y, ϕ (Int Y (B)) Int X (ϕ (B)) It is easy to check that all concave L-interior spaces as objects and all L-INP mappings as morphisms form a category, denoted by L-CLI Next we will establish the relations between L-concave spaces and concave L- interior spaces Proposition 36 Let (X, Int) be a concave L-interior space and define A Int L X by A Int = {A L X A = Int(A)} Then A Int is an L-concave structure on X Proof It is enough to verify that A Int satisfies (LA1) (LA3) In fact, (LA1) Since = Int( ) and Int( ), we know, A Int (LA2) Take any {A k k K} A Int Then A k = Int(A k ) for each k K Since Int is order preserving, we have Int( k K A k) Int(A k ) for each k K Then it follows that ( ) Int A k Int(A k ) = A k k K k K By (CLI2), the inverse inequality Int( k K A k) k K A k holds obviously Thus, Int( k K A k) = k K A k This means k K A k A Int cdir (LA3) Take any {A j } j J A Int It follows that A j = Int(A j ) for each j J By (CLI4), we have k K Int A j = Int(A j ) = A j j J j J j J This means j J A j A Int, as desired Proposition 37 If ϕ : (X, Int X ) (Y, Int Y ) is L-INP, then ϕ : (X, A Int X ) (Y, A Int Y ) is L-CAP

5 Characterizations of L-convex Spaces 55 Proof Since ϕ : (X, Int X ) (Y, Int Y ) is L-INP, it follows that B L Y, ϕ (Int Y (B)) Int X (ϕ (B)) Then for each C A Int Y, that is, C = Int Y (C), we have Int X (ϕ (C)) ϕ (Int Y (C)) = ϕ (C) By (LCI2), we have Int X (ϕ (C)) = ϕ (C) This implies ϕ (C) A Int X Hence, ϕ : (X, A Int X ) (Y, A Int Y ) is L-CAP By Propositions 36 and 37, we obtain a functor A I : L-CLI L-CA by L-CLI L-CA, A I : (X, Int) (X, A Int ), ϕ ϕ Proposition 38 Let (X, A) be an L-concave space and define Int A : L X L X by A L X, Int A (A) = {B L X B A, B A} Then Int A is a concave L-interior operator on X Proof It is enough to show that Int A satisfies (CLI1) (CLI4) (CLI1) and (CLI2) are obvious We need only verify (CLI3) and (CLI4) (CLI3) By (LA2), we know Int A (A) = {B L X B A, B A} A This implies Int A (A) Int A (Int A (A)) Since the inverse inequality Int A (A) Int A (Int A (A)) holds obviously, we obtain Int A (A) = Int A (Int A (A)) (CLI4) By the definition of Int A, we know Int A is order preserving ( That ) is to say, A B implies Int A (A) Int A (B) This means Int A j J A j j J IntA (A j ) Since {A j } j J is co-directed, it follows that {Int A (A j )} j J is codirected Further, by (LA2), it follows that Int A (A) = {B L X B A, B A} A This implies j J IntA (A j ) A As j J IntA (A j ) j J A j, we have ( j J IntA (A j ) Int A j J A j as desired ) This proves j J IntA (A j ) = Int A ( j J A j Proposition 39 If ϕ : (X, A X ) (Y, A Y ) is L-CAP, then ϕ : (X, Int A X ) (Y, Int A Y ) is L-INP Proof Since ϕ : (X, A X ) (Y, A Y ) is L-CAP, it follows that Then we have B L Y, B A Y implies ϕ (B) A X ϕ (Int A Y (B)) = ϕ ( {C L Y C B, C A Y }) {ϕ (C) L X ϕ (C) ϕ (B), ϕ (C) A X } {A L X A ϕ (B), A A X } = Int A X (ϕ (B)) ),

6 56 B Pang and Y Zhao This proves that ϕ : (X, Int A X ) (Y, Int A Y ) is L-INP By Propositions 38 and 39, we obtain a functor I A : L-CA L-CLI by L-CA L-CLI, I A : (X, A) (X, Int A ), ϕ ϕ Theorem 310 L-CLI is isomorphic to L-CA Proof It suffices to show that I A A I = I L CLI and A I I A = I L CA That is to say, we need only verify (1) Int AInt = Int and (2) A IntA = A For (1), take each A L X Then Int AInt (A) = {B L X B A, B A Int } = {B L X B A, B = Int(B)} On one hand, take each B L X such that B A and B = Int(B), we have B = Int(B) Int(A) This means Int AInt (A) Int(A) On the other hand, let B = Int(A) Then B A and Int(B) = Int(Int(A)) = Int(A) = B This means that Int(A) Int AInt (A), as desired (2) Take each A L X Then A A IntA A = Int A (A) = {B L X B A, B A} A = {B L X B A, B A} A A This means A IntA = A Theorem 311 Suppose that L is equipped with an order-reversing involution operator Then L-CE is isomorphic to L-CA Proof Given an L-convex space (X, E), define A E L X by A E = {A L X A E} Then (X, A E ) is an L-concave space Similarly, given an L-concave space (X, A), define E A L X by E A = {A L X A A} Then (X, E A ) is an L-convex space Since is an order-reversing involution operator on L, it can be easily checked that L-CE is isomorphic to L-CA By Theorems 27, 310 and 311, we obtain Corollary 312 Suppose that L is equipped with an order-reversing involution operator Then L-CE, L-CA, L-CLC and L-CLI are isomorphic

7 Characterizations of L-convex Spaces 57 4 Concave L-neighborhood Systems In this section, we introduce the concept of concave L-neighborhood systems and study its relations with L-concave structures In this section, L is assumed to be a completely distributive lattice Definition 41 A concave L-neighborhood system on X is a set N = {N xλ x λ J(L X )}, where N xλ L X satisfies: (CLN1) X L N x λ, X L N x λ ; (CLN2) A N xλ, x λ A; (CLN3) A N xλ, A B implies B N xλ ; cdir (CLN4) {A j } j J N xλ, j J A j N xλ ; (CLN5) A N xλ if and only if B L X st x λ B A and y µ B, B N yµ For a concave L-neighborhood system N on X, the pair (X, N ) is called a concave L-neighborhood space Definition 42 A mapping ϕ : (X, N X ) (Y, N Y ) between concave L-neighborhood spaces is called L-neighborhood-preserving (L-NEP, in short) provided that x λ J(L X ), B L Y, B N Y ϕ(x) λ implies ϕ (B) N X x λ It is easy to check that all concave L-neighborhood spaces as objects and all L-NEP mappings as morphisms form a category, denoted by L-CN Next we will investigate the relations between L-CN and L-CA Proposition 43 Let (X, N ) be a concave L-neighborhood space and define A N L X as follows: A N = {A L X x λ A, A N xλ } Then A N is an L-concave structure on X Proof It suffices to verify that A N satisfies (LA1), (LA2) and (LA3) In fact, (LA1) Obviously (LA2) Take each {A k } k K A N Then for each k K, it follows that A k N xλ for each x λ A k For each x λ J(L X ) satisfying x λ k K A k, there exists k 0 K such that x λ A k0 This implies that A k0 N xλ By (CLN3), we have k K A k N xλ This shows for each x λ k K A k, k K A k N xλ Thus, we get k K A k A N cdir (LA3) Take each {A j } j J A N Then for each j J, it follows that A j N xλ for each x λ A j For each x λ J(L X ) satisfying x λ j J A j, we have x λ A j for each j J This means A j N xλ for each j J By (CLN4), we obtain j J A j N xλ This proves that for each x λ j J A j, j J A j N xλ Thus, we obtain j J A j A N Proposition 44 If ϕ : (X, N X ) (Y, N Y ) is L-NEP, then ϕ : (X, A N X ) (Y, A N Y ) is L-CAP

8 58 B Pang and Y Zhao Proof Since ϕ : (X, N X ) (Y, N X ) is L-NEP, it follows that x λ J(L X ), B L Y, B Nϕ(x) Y λ implies ϕ (B) Nx X λ Then for each C L Y, we obtain C A N Y y µ C, C Ny Y µ = ϕ(x) λ C, C Nϕ(x) Y λ = x λ ϕ (C), ϕ (C) N X x λ ϕ (C) A N X This proves that ϕ : (X, A N X ) (Y, A N Y ) is L-CAP By Propositions 43 and 44, we obtain a functor A N : L-CN L-CA by L-CN L-CA, A N : (X, N ) (X, A N ), ϕ ϕ Proposition 45 Let (X, A) be an L-concave space and define Nx A λ L X by N A x λ = {A L X B A st x λ B A} Then N A = {N A x λ x λ J(L X )} is a concave L-neighborhood system on X Proof It is enough to show that N A satisfies (CLN1) (CLN5) (CLN1) (CLN3) are obvious cdir (CLN4) Take {A j } j J Nx A λ Then for each j J, there exists B j A such that x λ B j A j By (LA2), we know Int A (A j ) A and x λ B j Int A (A j ) A j Since Int A is order preserving, we obtain {Int A (A j ) j J} cdir A Put B = j J IntA (A j ) By (LA3), it follows that B A Further, x λ B j J A j Thus, we get j J A j Nx A λ (CLN5) Necessity Take each A Nx A λ Then there exists B A such that x λ B A Since B A, for each y µ B, there exists C(= B) A such that y µ C B This means B Ny A µ for each y µ B, as desired Sufficiency We first prove the following result (CLN0) A N A x λ µ β (λ), A N A x µ On one hand, if A Nx A λ, then there exists B A such that x λ B A Take each µ β (λ) Then x µ x λ B A This implies A Nx A µ for µ β (λ) On the other hand, since A Nx A µ for µ β (λ), it follows that for each µ β (λ), there exists B µ A such that x µ B µ A Let B = µ β (λ) B µ Then by (LA2), we have B A Further, x λ = x µ B µ = B A This implies that A N A x λ µ β (λ) µ β (λ)

9 Characterizations of L-convex Spaces 59 Now we know there exists B L X such that x λ B A and for each y µ B, B Ny A µ Thus, for each µ β (λ), it follows that x µ x λ B This means B Nx A µ By (CLN3), we know A Nx A µ By (CLN0), we get A Nx A λ This proves the sufficiency of (CLN5) Proposition 46 If ϕ : (X, A X ) (Y, A Y ) is L-CAP, then ϕ : (X, N A X ) (Y, N A Y ) is L-NEP Proof Since ϕ : (X, A X ) (Y, A Y ) is L-CAP, it follows that B L Y, B A Y implies ϕ (B) A X Then for each x λ J(L X ) and B L Y, we have B N A Y ϕ(x) λ C A Y st ϕ(x) λ C B = ϕ (C) A X st x λ ϕ (C) ϕ (B) = A A X st x λ A ϕ (B) ϕ (B) N A X x λ This shows that ϕ : (X, N A X ) (Y, N A Y ) is L-NEP By Propositions 45 and 46, we obtain a functor N A : L-CA L-CN by L-CA L-CN, N A : (X, A) (X, N A ), ϕ ϕ Next we present the main result in this section Theorem 47 L-CN is isomorphic to L-CA Proof It suffices to show that N A A N = I L CLI and A N N A = I L CA That is to say, we need only verify (1) N AN = N and (2) A N A = A For (1), take each A L X Then A N AN x λ B A N st x λ B A B L X st x λ B A and y µ B, B N yµ This shows N AN = N (2) Take each A L X Then A N xλ (by (CLN5)) A A N A x λ A, A Nx A λ x λ A, B A st x λ B A A = B A A x λ A x λ B A

10 60 B Pang and Y Zhao This means A N A = A Corollary 48 Suppose that L is a completely distributive lattice with an orderreversing involution operator Then L-CE, L-CA, L-CLC, L-CLI and L-CN are isomorphic 5 Conclusions In this paper, we provided several characterizations of L-convex structures from a categorical viewpoint, including L-concave structures, concave L-interior operators and concave L-neighborhood systems As in the situation of L-topology, all these resulting categories are isomorphic whenever L is a completely distributive lattice with an order-reversing involution operator Based on these new concepts, we will consider topological properties of L-convex spaces, such as compactness and separation property In the classical case, convex structures can be characterized by algebraic closure operators (also called domain finite closure operators) in [23] Convex L-closure operators in Definition 25 are not direct generalization of algebraic closure operators Since algebraic closure operators cannot be generalized to the fuzzy case directly, we firstly found an equivalent form of L-convex spaces in Proposition 23 and then used this equivalent form to propose Definition 25 Consequently, the concept of concave L-interior operators can also be defined dually and the concept of concave L-neighborhood systems can be proposed In the future, we will consider how to define algebraic L-closure operators in the framework of L-convex spaces and study its relations with convex L-closure operators Acknowledgements The authors would like to express their sincere thanks to the anonymous reviewers for their careful reading and constructive comments This work is supported by National Nature Science Foundation Committee (NSFC) of China (No ), China Postdoctoral Science Foundation (No 2015M581434), Fundamental Research Project of Shenzhen (No JCYJ and No JCYJ ) References [1] S Abramsky and A Jung, Domain theory, S Abramsky, D Gabbay, TSE Mailbaum (Eds), Handbook of Logic in Computer Science, Oxford University Press, Oxford (1994), [2] V Chepoi, Separation of two convex sets in convexity structures, J Geom, 50 (1994), [3] E Ellis, A general set-separation theorem, Duke Math J, 19 (1952), [4] J Eckhoff, Radon s theorem in convex product structures I, Monatsh Math, 72 (1968), [5] J Eckhoff, Radon s theorem in convex product structures II, Monatsh Math, 73 (1969), [6] R E Jamison, A general theory of convexity, Dissertation, University of Washington, Seattle, Washington, 1974 [7] D C Kay and E W Womble, Axiomatic convexity theory and the relationship between the Carathéodory, Helly and Radon numbers, Pacific J Math, 38 (1971), [8] M Lassak, On metric B-convexity for which diameters of any set and its hull are equal, Bull Acad Polon Sci, 25 (1977),

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