Michał Baczyński

(S, N)- and R-implications: A state-of-the-art survey

In this work we give a state-of-the-art review of two of the most established classes of fuzzy implications, viz., (S,N)- and R-implications. Firstly, we discuss their properties, characterizations and representations. Many new results concerning fuzzy negations and (S,N)-implications, notably their characterizations with respect to the identity principle and ordering property, are presented, which give rise to some representation results. Finally, using the presented facts, an almost complete characterization of the intersections that exist among some subfamilies of (S,N)- and R-implications are obtained.

Residual implications revisited. Notes on the Smets-Magrez Theorem

Abstract It is well known that continuous residual implications are conjugate with the Łukasiewicz implication. This fact was first obtained by Smets and Magrez (Internat. J. Approx. Reason. 1 (1987) 327). In this paper we show that the assumption of the monotonicity in this theorem can be omitted. We are also interested in the characterization of the other classes of fuzzy implications. As a result, the characterization of fuzzy implications, which are conjugate with the Goguen implication, is obtained. Besides these main results we present some new facts concerning the properties of fuzzy implications.

On the characterizations of (S,N)-implications

The characterization of S-implications generated from strong negations presented firstly by Trillas and Valverde in 1985 is well-known in the literature. In this paper we show that some assumptions are needless and present two characterizations of S-implications with mutually independent requirements. We also present characterizations of (S,N)-implications obtained from continuous fuzzy negations or strict negations. Besides these main results some new facts concerning fuzzy implications, fuzzy negations and laws of contraposition are proved.

(U,N)-implications and their characterizations

Recently, we have presented characterizations of (S,N)-implications generated from t-conorms and continuous (strict, strong) negations. Uninorms were introduced by Yager and Rybalov in 1996 as a generalization of t-norms and t-conorms, thus they are another fertile source based on which one can define fuzzy implications. (U,N)-implications are a generalization of (S,N)-implications, where a t-conorm S is replaced by a (disjunctive) uninorm U. In this work we present characterizations of (U,N)-implications obtained from disjunctive uninorms and continuous negations as well as (U,N)-operations defined from uninorms and continuous negations.

On the Distributivity of Fuzzy Implications Over Nilpotent or Strict Triangular Conorms

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.

QL-implications: Some properties and intersections

In this paper, we attempt a systematic study of QL-implications. Towards this end, firstly, we investigate the conditions under which a QL-operation becomes a fuzzy implication without imposing any conditions on the underlying operations. Following this, we discuss the conditions under which this family satisfies some desirable algebraic properties. Based on the obtained results and existing characterization results, the intersections between QL-implications and the two most established families of fuzzy implications, viz., (S,N)- and R-implications are determined. It is shown that QL-implications contain the set of all R-implications obtained from left-continuous t-norms that are also (S,N)-implications. Finally, the overlaps between QL-implications and the recently proposed f- and g-implications are also studied.

Conjugacy Classes of Fuzzy Implications

We discuss the conjugacy problem in the family of fuzzy implications. Particularly we examine a compatibility of conjugacy classes with induced order and induced convergence in the family of fuzzy implications. Conjugacy classes can be indexed by elements of adequate groups.

Fuzzy Implications: Past, Present, and Future

Fuzzy implications are a generalization of the classical two-valued implication to the multi-valued setting. They play a very important role both in the theory and applications, as can be seen from their use in, among others, multivalued mathematical logic, approximate reasoning, fuzzy control, image processing, and data analysis. The goal of this chapter is to present the evolution of fuzzy implications from their beginnings to the current days. From the theoretical point of view, we present the basic facts, as well as the main topics and lines of research around fuzzy implications. We also devote a specific section to state and recall a list of main application fields where fuzzy implications are employed, as well as another one to the main open problems on the topic.

Advances in Fuzzy Implication Functions

Fuzzy implication functions are one of the main operations in fuzzy logic. They generalize the classical implication, which takes values in the set {0,1}, to fuzzy logic, where the truth values belong to the unit interval [0,1]. These functions are not only fundamental for fuzzy logic systems, fuzzy control, approximate reasoning and expert systems, but they also play a significant role in mathematical fuzzy logic, in fuzzy mathematical morphology and image processing, in defining fuzzy subsethood measures and in solving fuzzy relational equations. This volume collects 8 research papers on fuzzy implication functions. Three articles focus on the construction methods, on different ways of generating new classes and on the common properties of implications and their dependencies. Two articles discuss implications defined on lattices, in particular implication functions in interval-valued fuzzy set theories. One paper summarizes the sufficient and necessary conditions of solutions for one distributivity equation of implication. The following paper analyzes compositions based on a binary operation * and discusses the dependencies between the algebraic properties of this operation and the induced sup-* composition. The last article discusses some open problems related to fuzzy implications, which have either been completely solved or those for which partial answers are known. These papers aim to present todays state-of-the-art in this area.

Distributive Equations of Implications Based on Continuous Triangular Norms (I)

In order to avoid combinatorial rule explosion in fuzzy reasoning, in this paper, we explore the distributive equations of implications. In detail, by means of the sections of <i>I</i>, we give out the sufficient and necessary conditions of solutions for the distributive equation of implication <i>I</i>(<i>x</i>,<i>T</i>₁(<i>y</i>,<i>z</i>))=<i>T</i>₂(<i>I</i>(<i>x</i>,<i>y</i>),<i>I</i>(<i>x</i>,<i>z</i>)), when <i>T</i>₁ is a continuous but not Archimedean triangular norm, <i>T</i>₂ is a continuous and Archimedean triangular norm, and <i>I</i> is an unknown function. This obtained characterizations indicate that there are no continuous solutions for the previous functional equation, satisfying the boundary conditions of implications. However, under the assumptions that <i>I</i> is continuous except for the point (0,0), we get its complete characterizations. Here, it should be pointed out that these results make differences with recent results that are obtained by Baczyński and Qin. Moreover, our method can still apply to the three other functional equations that are related closely to the distributive equation of implication.