Fuzzy Convex Invariants and Product Spaces

Transcription

1 International Mathematical Forum, Vol. 6, 2011, no. 57, Fuzzy Convex Invariants and Product Spaces Lissy Jacob Department of Mathematics Bharata Mata College, Thrikkakara, Kerala, India, PIN M. V. Rosa Department of Mathematics Bharata Mata College, Thrikkakara,Kerala, India, PIN Abstract In abstract convexity theory, the classical convex invariants namely Helly number, Caratheodory number, Radon number and Exchange number play a central role. In [3], some basic relations between these invariants in a fuzzy convex structure are studied. The behaviour of these invariants under the formation of FCP images are also studied. In this paper, the fuzzy convex invariants of product spaces are discussed and certain relations are established. Subject Classification: 52A35, 03E72 Keywords: Fuzzy convex structure, Helly number, Caratheodory number, Exchange number 1 Introduction Caratheodory and Helly numbers of convex product structures are defined and studied by G. Sierksma in In the crisp case [6], it was shown that the Helly number of a convex product structure is the supremum of the Helly numbers of the factor spaces. In this paper, it is proved in the fuzzy context.the Caratheodory number of the convex product space with two factors is also determined in the fuzzy context.

2 2816 L. Jacob and M. V. Rosa 2 Basic definitions and properties Definition 2.1 [1] A fuzzy subset A on a nonempty set X is a function from X to I =[0, 1]. The set of all fuzzy subsets of X is denoted by I X. A fuzzy point a α is a fuzzy subset defined as a α (y) = { α, if y = a 0, otherwise Definition 2.2 [1] Support of a fuzzy subset A is Supp (A) ={x X A (x) > 0}. A fuzzy subset is said to be finite if its support is finite. Definition 2.3 [1] Given two fuzzy sets A, B I X,A B (ora B) if A (x) B (x), x X. Also, A\B = A B. Definition 2.4 [4] A family C of fuzzy subsets of X is called a fuzzy convexity if (i) 0, 1 C (ii)if F C, then F C (iii) If F C is non-empty and totally ordered by inclusion, then F C. The pair (X, C) is called a fuzzy convex structure or a fuzzy convexity space. Members of C are called fuzzy convex sets. Note. a denotes a constant function whose value at x is a, x X Definition 2.5 [3] If F is any fuzzy subset, then convex hull of F, is defined as Co(F )= {A C F A}. Definition 2.6 [2] Assume X is the universe of discourse, I is the unit interval [0, 1] and N is the set of nonnegative integers. Then a fuzzy bag A is a mapping from the product X I to N characterized by FC A (u/x) :X I N. A fuzzy bag is also denoted as A (u/x) =[FC A (u/x) / (u/x)] where FC A (u/x) is the count of x with u as its membership grade. Definition 2.7 [3] Let (X, C) be a fuzzy convex structure. A nonzero finite fuzzy subset F of X is Helly dependent (or H-dependent) if Co(F \a α ) 0 a α F where F \a α = F a α. Otherwise, it is H-independent. The Helly number of X is the smallest n such that for each nonzero finite fuzzy subset F of X, with cardinality of its support at least n+1, is Helly dependent. It is denoted by h(x) or by h.

3 Fuzzy convex invariants and product spaces 2817 Definition 2.8 [3] Let (X, C) be a fuzzy convex structure. A nonzero finite fuzzy subset F of X is Caratheodory dependent (or C-dependent) provided Co(F ) Co(F \a α ) Otherwise, it is C-independent. a α F The Caratheodory number of X is the smallest n such that for each nonzero finite fuzzy subset F of X, with cardinality of its support at least n+1, is Caratheodory dependent. It is denoted by c(x) or by c. Definition 2.9 [3] Let (X, C) be a fuzzy convex structure. A nonzero finite fuzzy subset F of X is Exchange dependent (or E-dependent) if for each p α F, Co (F \p α ) (Co(F \a β );a β F \p α ; a p). Otherwise, it is E-independent. The Exchange number of X is the smallest n such that for each nonzero finite fuzzy subset F of X, with cardinality of its support at least n+1, is Exchange dependent. It is denoted by e(x) or by e. Definition 2.10 [5] Let (X, C 1 ), (Y,C 2 ) be fuzzy convex structures. Let f : X Y. Then fis said to be a fuzzy convexity preserving function (FCP) if for each fuzzy convex set K in Y, f 1 (K) is a fuzzy convex set in X. Definition 2.11 [5] Let (X, C 1 ), (Y,C 2 ) be fuzzy convex structures. Let f : X Y. Then f is said to be a fuzzy convex to convex function (FCC) if for each fuzzy convex set K in X, f (K) is a fuzzy convex set in Y. Proposition 2.12 [5] Let {(X i, C i );i I} be a family of fuzzy convex structures. Let X = X i be the product and let π i : X X i be the projection map. Then X can be equipped with the fuzzy convexity C generated by the fuzzy convex sets of the form { π 1 i (C i ) /C i C i ; i I }. Then C is called the product fuzzy convexity on X and (X, C) is called the product fuzzy convexity space. Proposition 2.13 [5] In a product space, the polytopes are of the product type i.e., for each finite fuzzy subset F of the product,co(f )= Co(π i (F )). Proposition 2.14 [5] The projection map π i of a product to its factors is both FCP and FCC. Theorem 2.15 [3] Let (X, C) be a fuzzy convex structure and let n<. If each finite collection of fuzzy convex sets in X meeting n by n has a nonzero intersection then h n.

4 2818 L. Jacob and M. V. Rosa Theorem 2.16 [3] Let (X, C) be a fuzzy convex structure and let n<. Then c n iff for each nonzero fuzzy subset A X and p α Co(A) there is a fuzzy subset F of A with cardinality of its support at most n and having p α Co(F ). Theorem 2.17 [3] Let (X, C 1 ), (Y,C 2 ) be fuzzy convex structures. Let f : X Y be a surjective FCP function. Then h (X) h (Y ). 3 Relationships between fuzzy convex invariants In this section, we establish some relationships between fuzzy convex invariants in a fuzzy convex product space. Theorem 3.1 Let {X i,i I} be a family of nonzero fuzzy convex structures and let X = X i be the fuzzy convex product space. Then the Helly number h of X is the supremum of the Helly numbers of the factor spaces. i.e., h = Sup{h i ; i I} where h i s are the Helly numbers of the factor spaces. Let {(X i, C i ) i I} be a family of nonzero fuzzy convex structures and let X = X i be the product space. Let π i : X X i be the projection map which is a surjective FCP function. By Theorem 2.17, a surjective FCP function does not increase the Helly number. Hence h h i ; i I. So, h Sup{h i ; i I}. (1) Next suppose n = Sup{h i ; i I} is finite. Let F be a fuzzy bag on X I with support{x 1,x 2,..., x m }; m>n. Let F k = F \{x kα }.By the product formula for polytopes we have Co(F k )= Co(π i (F k )) = (π i Co(F k )) since the projection map π i of a product to its factors is both FCP and FCC. Now m>n m>h i, i. Hence π i (F ) is Helly dependent. Then by definition, m π i Co(F k ) 0. Consider k=1 { } m m Co(F k )= (π i Co(F k )) = { m } (π i Co(F k )) 0 k=1 k=1 k=1

5 Fuzzy convex invariants and product spaces 2819 Then by definition, F is Helly dependent. So h n. From (1) and (2), i.e., h Sup{h i ; i I}. (2) h = Sup{h i ; i I} Theorem 3.2 Let X 1,X 2 be two fuzzy convex structures and X = X 1 X 2 be the fuzzy convex product space. If c 1,c 2 and c are the Caratheodory numbers of X 1,X 2 and X respectively, then c c 1 + c 2. Let A be a fuzzy subset of X and p α Co(A). Let π i : X X i ; i =1, 2 denote the i th projection. Then π 1 (A) and π 2 (A) are the projections of A on X 1 and X 2 respectively. Given that c i is the Caratheodory number of X i for i =1, 2. Hence by Theorem 2.16, for each π i (A) X i and π i (p α ) Co(π i (A)), there is a fuzzy subset F i of A with cardinality of its support at most c i such that π i (p α ) Co(π i (F i )) for i =1, 2. But p α Co(A) p α Co(π 1 F 1 ) Co(π 2 F 2 ) where Thus Clearly, Co(π 1 F 1 ) Co(π 2 F 2 ) Co(π 1 (F 1 F 2 )) Co(π 2 (F 1 F 2 )) = Co(F 1 F 2 ) p α Co(F 1 F 2 ). #Supp (F 1 F 2 ) c 1 + c 2 i.e., for each fuzzy subset A X and p α Co(A), there is a fuzzy subset F 1 F 2 of A with cardinality of its support at most c 1 + c 2 and having p α Co(F 1 F 2 ). Then by Theorem 2.16, c c 1 + c 2. Theorem 3.3 Let X 1,X 2 be two fuzzy convex structures and X = X 1 X 2 be the fuzzy convex product space. Let c i and e i be the Caratheodory number and Exchange number of X i for i =1, 2 and c be the Caratheodory number of X.Ifc i e i for at least one of i =1, 2 then c c 1 + c 2 1. Suppose c 2 e 2. Let A be fuzzy subset of Xand p α Co(A). Then as above, there are finite fuzzy subsets F 1,F 2 of A with cardinality of its support at most

6 2820 L. Jacob and M. V. Rosa c i and having π i (p α ) Co(π i (F i )) ; i =1, 2. Also p α Co(F 1 F 2 ). If #Supp (F i ) <c i for some i or if F 1 F 2 0then #Supp (F 1 F 2 ) <c 1 +c 2 c 1 +c 2 1. If #Supp (F i )=c i for some i and if F 1 F 2 =0then take a fuzzy point a β F 1. Since #Supp (F 2 a β ) >c 2 e 2, the fuzzy subset F 2 a β is E-dependent. Then by definition, Co(F 2 ) {Co[(F 2 a β ) \b γ ];b γ F 2 } i.e., Co(F 2 ) Co(F 2 ) where F 2 =(F 2 a β ) \b γ ; b γ F 2. Since π i (p α ) Co(π i (F i )) ; i =1, 2 π 2 (p α ) Co(π 2 (F 2 )) π 2 (p α ) Co(π 2 (F 2 )) Hence, p α Co(F 1 F2 ) where cardinality of support of F 1 F2 is at most c 1 + c 2 1. Thus, c c 1 + c 2 1. Theorem 3.4 Let X 1,X 2 be two fuzzy convex structures and X = X 1 X 2, the fuzzy convex product space. Suppose F i X i be a fuzzy set with #Supp(F i ) > 1 for i =1, 2. Let p i Supp(F i ) such that Co(F i ) {Co(F i \q α ); q α F i ; q p i }for i =1, 2. Then the fuzzy subset F =[{p 1 } (F 2 \ p 2 )] [(F 1 \ p 1 ) {p 2 }] of X is C- independent. For i =1, 2, consider a fuzzy point x iλi Co(F i \q α ); q α F i ; q p i. in Co(F i ) but not in the sets i.e., x iλi Co(F i ) \ {Co(F i \q α ); q α F i ; q p i }for i = 1, 2. (3) Let x λ =(x 1λ1,x 2λ2 ). Since π i (F )=F i,i=1, 2 we get x λ Co(F ). Let a β be a fuzzy point in F. Then either a β =(p 1,q β ) with q p 2 or a β =(q β,p 2 ) with q p 1.Ifa β =(p 1,q β ) then π 2 (F \ a β )=F 2 \ q β. But by (3), x 2λ2 / Co(F 2 \q α ); q α F 2 ; q p 2.i.e., x 2λ2 / Co(π 2 (F \a β )). Similarly, if a β =(q β,p 2 ) with q p 1, we can show that x 1λ1 / Co(π 1 (F \a β )). Hence in either case, x λ / {Co(F \a β );a β F } ; where x λ Co(F ). Therefore, Co(F ) {Co(F\a β );a β F }. So by definition, F is C- independent.

7 Fuzzy convex invariants and product spaces 2821 Theorem 3.5 Let X 1,X 2 be two fuzzy convex structures and X = X 1 X 2 be the fuzzy convex product space. Let c i and e i be the Caratheodory number and Exchange number of X i for i =1, 2. Then the Caratheodory number c of the product space is determined as follows. i) If c i <e i for i =1, 2 then c = c 1 + c 2. ii) If c i <e i for exactly one of i =1, 2 then c = c 1 + c 2 1. i) In Theorem 3.2, it is proved that c c 1 + c 2. (4) Since e i >c i for i =1, 2 for each i, there is a nonexchangeable fuzzy set F i X i with cardinality of its support >c i. In other words, F i ; fori =1, 2is E - independent. Then, by definition, for each i, there is a point p i Supp(F i ) such that Co(F i \ p i ) {Co(F i \q α ); q α F i ; q p i }. Hence Co(F i ) {Co(F i \q α ); q α F i ; q p i } for i =1, 2. Then by Theorem 3.4, the fuzzy subset F =[{p 1 } (F 2 \ p 2 )] [(F 1 \ p 1 ) {p 2 }] of X is C- independent. Since the cardinality of support of F is at least c 1 +c 2, we have c c 1 + c 2. (5) Combining (4) and (5), c = c 1 + c 2. ii) Suppose c 1 <e 1 and c 2 e 2. As c 2 e 2, by Theorem 3.3, c c 1 + c 2 1. (6) Since e 1 >c 1, there is a nonexchangeable fuzzy set F 1 X 1 with cardinality of its support at least c i.e., F 1 is E - independent. So, there is a point p 1 Supp(F 1 ) such that Co(F 1 \ p 1 ) {Co(F 1 \q α ); q α F 1 ; q p 1 }. Hence Co(F 1 ) {Co(F 1 \q α ); q α F 1 ; q p 1 }. Since c 2 e 2, there is a C- independent fuzzy set F 2 X 2, with cardinality of its support as c 2 and having a point p 2 Supp(F 2 ) such that Co(F 2 ) {Co(F 2 \q α ); q α F 2 ; q p 2 }.

8 2822 L. Jacob and M. V. Rosa Then by Theorem 3.4, the fuzzy subset F =[{p 1 } (F 2 \ p 2 )] [(F 1 \ p 1 ) {p 2 }] of X is C- independent. As the cardinality of support of F is at least c 1 +c 2 1, Combining (6) and (7), c c 1 + c 2 1. (7) c = c 1 + c 2 1. ACKNOWLEDGEMENT. The authors wish to thank Prof.T.Thrivikraman for his guidance during the preparation of this paper. References [1] George J. Klir and Bo Yuan, Fuzzy sets and Fuzzy logic -Theory and Applications, Prentice-Hall of India Pvt. Ltd,2002. [2] LI Baowen,Fuzzy bags and applications, Fuzzy sets and systems, 34 (1990), [3] Lissy Jacob, M.V. Rosa, Shiny Philip, On some fuzzy convex invariants, International Mathematical Forum, 6(20) (2011) [4] Lissy Jacob, M.V.Rosa, On fuzzy join operators, Applied Mathematical Sciences, 5(35) (2011) [5] M.V.Rosa, Separation axioms in fuzzy topology fuzzy convexity spaces, Jl. of Fuzzy Mathematics, 2(3) (1994), [6] M. van de Vel, Theory of Convex Structures, North Holland, N.Y., Received: May, 2011