The Experts below are selected from a list of 102 Experts worldwide ranked by ideXlab platform
Michal Baczynski - One of the best experts on this subject based on the ideXlab platform.
-
some functional equations connected to the distributivity laws for fuzzy implications and Triangular Conorms
IEEE International Conference on Fuzzy Systems, 2015Co-Authors: Wanda Niemyska, Michal BaczynskiAbstract:Recently in some considerations connected with the distributivity laws of fuzzy implications over Triangular norms and Conorms, the following functional equation appeared ƒ(min(x + y, a)) = min(ƒ(x) + ƒ(y), b), (1) where a; b are finite or infinite nonnegative constants (see [1]). In [2] we considered a generalized version of this equation in the case when both a and b are finite, namely the equation ƒ(m 1 (x + y)) = m 2 (ƒ(x) + ƒ(y)), where m 1 , m 2 are functions defined on some finite intervals of ℝ satisfying additional assumptions. In this article we enhance the results from [2], [3] and consider generalized versions of the equation (1) in the cases when a or b is infinite. We show that some well known solutions of several functional equations, that we presented earlier in [1], [4], can be obtained as corollaries of these new facts.
-
distributivity equations of implications based on continuous Triangular Conorms ii
Fuzzy Sets and Systems, 2014Co-Authors: Feng Qin, Michal BaczynskiAbstract:In order to avoid combinatorial rule explosion in fuzzy reasoning, Qin and Baczynski, in [16], investigated the distributivity equation of implication I(x,T"1(y,z))=T"2(I(x,y),I(x,z)), when T"1 is a continuous but not Archimedean Triangular norm, T"2 is a continuous and Archimedean Triangular norm and I is an unknown function. In fact, it partially answered the open problem suggested by Baczynski and Jayaram in [5]. In this work we continue to explore the distributivity equation of implication I(x,S"1(y,z))=S"2(I(x,y),I(x,z)), when both S"1 and S"2 are continuous but not Archimedean Triangular Conorms, and I is an unknown function. Here it should be pointed out that these results make difference with recent ones obtained in [16]. Moreover, our method can still apply to the three other functional equations related closely to this equation. It is in this sense that we have completely solved the open problem commented above.
-
on the distributivity of fuzzy implications over continuous and archimedean Triangular Conorms
Fuzzy Sets and Systems, 2010Co-Authors: Michal BaczynskiAbstract:Recently, we have examined the solutions of the following distributive functional equation I(x,S"1(y,z))=S"2(I(x,y),I(x,z)), when S"1, S"2 are either both strict or nilpotent t-Conorms and I is an unknown function. In particular, between these solutions, we have presented functions which are fuzzy implications. In this paper we continue these investigations for the situation when S"1, S"2 are continuous and Archimedean t-Conorms, i.e., we consider in detail the situation when S"1 is a strict t-conorm and S"2 is a nilpotent t-conorm and vice versa. Towards this end, we firstly present solutions of two functional equations related to the additive Cauchy functional equation. Using obtained results we show that the above distributive equation does not hold when S"1, S"2 are continuous and Archimedean t-Conorms and I is a continuous fuzzy implication. Further, we present the solutions I which are non-continuous fuzzy implications. Obtained results are not only theoretical but also useful for the practical problems, since such equations have an important role to play in efficient inferencing in approximate reasoning, especially in fuzzy control systems.
-
on the distributivity of fuzzy implications over nilpotent or strict Triangular Conorms
IEEE Transactions on Fuzzy Systems, 2009Co-Authors: Michal Baczynski, Balasubramaniam JayaramAbstract:Recently, many works have appeared in this very journal dealing with the distributivity of fuzzy implications over t-norms and t-Conorms. These equations have a very important role to play in efficient inferencing in approximate reasoning, especially fuzzy control systems. Of all the four equations considered, the equation I(x, S1 (y,z)) = S2(I(x,y),I(x,z)), when S1,S2 are both t-Conorms and I is an R-implication obtained from a strict t-norm, was not solved. In this paper, we characterize functions I that satisfy the previous functional equation when S1,S2 are either both strict or nilpotent t-Conorms. Using the obtained characterizations, we show that the previous equation does not hold when S1,S2 are either both strict or nilpotent t-Conorms, and I is a continuous fuzzy implication. Moreover, the previous equation does not hold when I is an R -implication obtained from a strict t-norm, and S1,S2 are both strict t-Conorms, while it holds for an R-implication I obtained from a strict t-norm T if and only if the t-Conorms S1 = S2 are Phi-conjugate to the Lukasiewicz t-conorm for some increasing bijection phi of the unit interval, which is also a multiplicative generator of T.
Funda Karaçal - One of the best experts on this subject based on the ideXlab platform.
-
Idempotent nullnorms on bounded lattices
Information Sciences, 2018Co-Authors: Gül Deniz Çaylı, Funda KaraçalAbstract:Abstract Nullnorms are generalizations of Triangular norms and Triangular Conorms with a zero element to be an arbitrary point from a bounded lattice. In this paper, we study and discuss the existence of idempotent nullnorms on bounded lattices. Considering an arbitrary distributive bounded lattice L , we show that there exists a unique idempotent nullnorm on L . We prove that an idempotent nullnorm may not always exist on an arbitrary bounded lattice. Furthermore, we propose a construction method to obtain idempotent nullnorms on a bounded lattice L with an additional constraint on a for the given zero element a ∈ L \{0, 1}.
Gül Deniz Çaylı - One of the best experts on this subject based on the ideXlab platform.
-
Idempotent nullnorms on bounded lattices
Information Sciences, 2018Co-Authors: Gül Deniz Çaylı, Funda KaraçalAbstract:Abstract Nullnorms are generalizations of Triangular norms and Triangular Conorms with a zero element to be an arbitrary point from a bounded lattice. In this paper, we study and discuss the existence of idempotent nullnorms on bounded lattices. Considering an arbitrary distributive bounded lattice L , we show that there exists a unique idempotent nullnorm on L . We prove that an idempotent nullnorm may not always exist on an arbitrary bounded lattice. Furthermore, we propose a construction method to obtain idempotent nullnorms on a bounded lattice L with an additional constraint on a for the given zero element a ∈ L \{0, 1}.
Wanda Niemyska - One of the best experts on this subject based on the ideXlab platform.
-
some functional equations connected to the distributivity laws for fuzzy implications and Triangular Conorms
IEEE International Conference on Fuzzy Systems, 2015Co-Authors: Wanda Niemyska, Michal BaczynskiAbstract:Recently in some considerations connected with the distributivity laws of fuzzy implications over Triangular norms and Conorms, the following functional equation appeared ƒ(min(x + y, a)) = min(ƒ(x) + ƒ(y), b), (1) where a; b are finite or infinite nonnegative constants (see [1]). In [2] we considered a generalized version of this equation in the case when both a and b are finite, namely the equation ƒ(m 1 (x + y)) = m 2 (ƒ(x) + ƒ(y)), where m 1 , m 2 are functions defined on some finite intervals of ℝ satisfying additional assumptions. In this article we enhance the results from [2], [3] and consider generalized versions of the equation (1) in the cases when a or b is infinite. We show that some well known solutions of several functional equations, that we presented earlier in [1], [4], can be obtained as corollaries of these new facts.
Ewa Rak - One of the best experts on this subject based on the ideXlab platform.
-
the distributivity property of increasing binary operations
Fuzzy Sets and Systems, 2013Co-Authors: Ewa RakAbstract:Abstract This paper is mainly devoted to solving the functional equations of distributivity and conditional distributivity of increasing binary operations with the unit. Our investigations are motivated by distributive logical connectives and their generalizations used in fuzzy set theory. In particular, some assumptions (namely associativity and commutativity) and results about conditional distributivity of uninorms and Triangular norms and Triangular Conorms were simplified. Moreover, some other properties e.g. componentwise convexity (concavity) for special operations from the class of quasi-copulas is considered.
