Nearly Frank T-norms and the Characterization of T-measures

We introduce nearly Frank t-norms as t-norms which are generated from Frank t-norms by means of a negation-preserving automorphism. We state basic properties of nearly Frank t-norms and we show their speci c role in the characterization of T -measures. 1 Basic facts about T -norms A t-norm (fuzzy conjunction) is a binary operation T : [0; 1]2 ! [0; 1] which is commutative, associative, nondecreasing, and satis es the boundary condition T (a; 1) = a for all a 2 [0; 1] (see [16]). We restrict our attention to continuous t-norms. A continuous t-norm T is called strict if a < b; 0 < c =) T (a; c) < T (b; c): A continuous t-norm T is called nilpotent if it is not strict and T (a; a) < a for all a 2 (0; 1). For the other necessary fuzzy logical operations, we take the standard fuzzy negation 0: [0; 1]! [0; 1] de ned by a0 := 1 a, and the t-conorm S: [0; 1]2 ! [0; 1] dual to T , i.e., S(a; b) := T (a0; b0)0. The Frank family of t-norms Ts, s 2 [0;1], was de ned in [5]. For s 2 (0;1) n f1g, the Frank t-norms are de ned by the formula Ts(a; b) := logs 1 + (sa 1)(sb 1) s 1 : The limit cases are de ned as follows: T0(a; b) := min(a; b); T1(a; b) := max(a+ b 1; 0); T1(a; b) := a b: The t-norms T0; T1; T1 are called the minimum, Lukasiewicz and product t-norm, resp. (In [4], Frank t-norms are called fundamental t-norms because of their exceptional role.) A Frank t-norm Ts is strict i s 2 (0;1) and it is nilpotent i s =1. As a main result of [5], the following property is characteristic for Frank t-norms. Theorem 1.1 : Let T be a strict t-norm and S its dual t-conorm. The equality T (a; b) + S(a; b) = a+ b is satis ed for all a; b 2 [0; 1] i T is a strict Frank t-norm, i.e., T = Ts for some s 2 (0;1). Remark 1.2 : All non-strict continuous solutions of the equality from Th. 1.1 can be expressed as ordinal sums of Frank t-norms (see [5]). Theorem 1.3 [6, 9, 15]: Let g be an automorphism of [0; 1]. Then the formula (G) T (a; b) := g 1(g(a) g(b)) de nes a strict t-norm T . Conversely, every strict t-norm T is of the form (G) for some automorphism g called the multiplicative generator of T . A multiplicative generator is unique up to raising to a positive power, i.e., each multiplicative generator gr of T is of the form gr(a) = (g(a))r, r > 0. Corollary 1.4 : Every strict t-norm has a unique multiplicative generator g such that g(1=2) = 1=2. The product (=product t-norm) in Th. 1.3 can be replaced by any other strict t-norm: Theorem 1.5 : For every automorphism h of [0; 1] and for every strict t-norm T , the formula (NF) T (a; b) := h 1(T (h(a); h(b))) de nes a strict t-norm T . Conversely, for all strict t-norms T; T there is a unique automorphism h of [0; 1] satisfying (NF) and h(1=2) = 1=2. 2 2 Nearly Frank t-norms In this section we de ne nearly Frank t-norms and we state their basic properties. De nition 2.1 : An automorphism h of [0; 1] is called a negation-preserving automorphism if it commutes with the standard fuzzy negation, i.e., h(a0) = h(a)0 for all a 2 [0; 1]. All negation-preserving automorphisms are characterized by the following proposition: Proposition 2.2 : If i is an automorphism of [0; 1=2], then the mapping h: [0; 1]! [0; 1] de ned by h(a) = i(a) if a 1=2; i(a0)0 if a > 1=2; is a negation-preserving automorphism. All negation-preserving automorphisms are of this form. The inverse of a negation-preserving automorphism is also a negation-preserving automorphism. De nition 2.3 : A t-norm T is called nearly Frank if there is a Frank t-norm T and a negation-preserving automorphism h satisfying the equation (NF). Obviously, every Frank t-norm is nearly Frank. It is obtained from (NF) by taking the identity, id : [0; 1]! [0; 1], for h. Proposition 2.4 : The equation (NF) is satis ed for a negation-preserving automorphism h and Frank t-norms T; T i T = T and h = id. Corollary 2.5 : Let h be a negation-preserving automorphism di erent from id and let T be a Frank strict t-norm. The nearly Frank strict t-norm T de ned by (NF) is not Frank. As a direct consequence of Th. 1.5, Prop. 2.4 and Cor. 2.5 we obtain: Corollary 2.6 : For every nearly Frank t-norm T there is a unique Frank t-norm T and a unique negationpreserving automorphism h satisfying (NF). There are strict t-norms which are not nearly Frank. However, we still lack for an algorithm allowing to decide whether a given strict t-norm is nearly Frank or not. Taking the minimum t-norm T0 in (NF) gives again T0. Taking the Lukasiewicz t-norm T1 in (NF), we obtain nilpotent nearly Frank t-norms. They are all of the form T (a; b) := h 1(T1(h(a); h(b))): 3 T -measures Let X be a set and B a -algebra of subsets of X . The B-generated tribe is the collection T of all functions A:X ! [0; 1] (fuzzy subsets of X) which are B-measurable. We extend the operations T , 0, S to operations T, c, S (fuzzy intersection, fuzzy complement and fuzzy union) on T pointwise: T(A;B)(x) := T (A(x); B(x)); Ac(x) := A(x)0; S(A;B)(x) := S(A(x); B(x)): A function m: T ! R+ (where R+ denotes the set of all nonnegative real numbers) is a monotone T -measure if it satis es the following axioms: (M1) m(0) = 0; (M2) m(T(A;B)) +m(S(A;B)) = m(A) +m(B); (M3) An % A =) m(An)% m(A); where the symbol % denotes monotone increasing convergence. If, moreover, m(1) = 1, then m is called a probability T -measure. 3 Remark 3.1 : Condition (M3) implies monotony of m. In [3, 4], monotony is not required, but only monotoneT -measures are studied. In this paper, we work only with monotone nite T -measures (i.e., satisfying (M1){(M3)).Remark 3.2 : For the de nition of a T -measure on T , it is su cient to have the collection T [0; 1]X suchthat 0 2 T ;A 2 T =) Ac 2 T ;fAngn2N T =) Tn2N An 2 T ;fAngn2N T ; An % A =) A 2 T :Such a collection is called a T -tribe [4, 7]. A B-generated tribe is a T -tribe for any measurable t-norm T .Following [3, 4], we call a T -tribe generated if it is a B-generated tribe for some -algebra B.Assumption 3.3 : Unless stated otherwise, we assume in this paper that B is a -algebra of subsets of X andT is the B-generated tribe.The notion of T -measure is not only a natural generalization of a classical measure. It is also the base ofsuccessful applications in game theory. Many deep mathematical results, including a generalization of LiapunoTheorem, were proved in [1, 2, 4].The support of a fuzzy set A 2 T is the crisp set SuppA := fx 2 X : A(x) > 0g. The main result is thefollowing:Theorem 3.4 [14]: Let T be a strict t-norm which is not nearly Frank. Then every monotone T -measure mon T is of the formm(A) = (SuppA);where is a (classical) measure on B.The measure (SuppA) (understood as a function of A 2 T ) is called a support measure. It is a T -measurefor all strict t-norms T . Th. 3.4 says that a t-norm T which is not nearly Frank allows only support T -measures.A more complex characterization is obtained for nearly Frank t-norms:Theorem 3.5 [14]: Let T be a strict nearly Frank t-norm. Then every monotone T -measure m on T is of theformm(A) = (SuppA) + Z h A d ;where ; are (classical) measures on B and h is the (unique) negation-preserving automorphism such thatT (a; b) := h(T (h 1(a); h 1(b)))is a Frank t-norm.The second summand in Th. 3.5, R h A d (understood as a function of A 2 T ), is called a generalizedintegral measure. Th. 3.5 says that, for a nearly Frank t-norm T , every monotone T -measure is a sum of asupport measure and a generalized integral measure (which has a form dependent on h and hence on T ). If T isa Frank t-norm, then, according to Prop. 2.4, the generalized integral measure reduces to R A d , called a linearintegral measure, because it depends linearly on A. Thus we obtain the characterization for Frank t-norms (see[4]) as a special case of Th. 3.5:Corollary 3.6 [3, 4]: Let T be a strict Frank t-norm. Then every monotone T -measure on T is a sum of asupport measure and a linear integral measure.Nilpotent nearly Frank t-norms allow only generalized integral measures:Theorem 3.7 : Let T be a nilpotent nearly Frank t-norm. Then every monotone T -measure m on T is of theformm(A) = Z h A d ;where is a (classical) measure on B and h is the (unique) negation-preserving automorphism such thath(T (h 1(a); h 1(b))) = T1(a; b):4 4 Open problemsOur characterization of T -measures was done for generated tribes. For Frank t-norms, the characterization ofTs-tribes given in [10, 12] implies that Cor. 3.6 remains valid also for all Ts-tribes [11], s 2 (0;1).Problem 4.1 : Does Th. 3.5 hold also for T -tribes which are not generated?The situation for nilpotent t-norms is not completely known.Problem 4.2 : Characterize T -measures for nilpotent t-norms T which are not nearly Frank.All preceding works dealt with monotone T -measures.Problem 4.3 : Characterize nonmonotone T -measures. Do they exist for T strict, resp. nilpotent?The study of the new class of nearly Frank t-norms brings also some questions. The following problem seemsto be open:Problem 4.4 : Find an algorithm which, for a t-norm given by a formula, decides whether it is nearly Frankor not.References[1] A. Avallone, G. Barbieri: Range of nitely additive fuzzy measures. Fuzzy Sets Syst. 89 (1997), 231{241.[2] G. Barbieri, M.A. Lepellere, H. Weber: The Hahn decomposition theorem for fuzzy measures and appli-cations. To appear.[3] D. Butnariu, E.P. Klement: Triangular norm-based measures and their Markov kernel representation. J.Math. Anal. Appl. 162 (1991), 111-143.[4] D. Butnariu, E.P. Klement: Triangular Norm-Based Measures and Games with Fuzzy Coalitions. Kluwer,Dordrecht, 1993.[5] M.J. Frank: On the simultaneous associativity of F (x; y) and x + y F (x; y). Aequationes Math. 19(1979), 194{226.[6] M. Gehrke, C. Walker, E.A. Walker: DeMorgan systems on the unit interval. Int. J. Intelligent Syst. 11(1996), 733{750.[7] E.P. Klement: Construction of fuzzy -algebras using triangular norms. J. Math. Anal. 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