In this article, we define the <inline-formula><tex-math notation="LaTeX">k</tex-math></inline-formula>-generated implications by continuous multiplicative generators <inline-formula><tex-math ...notation="LaTeX">k</tex-math></inline-formula> of <inline-formula><tex-math notation="LaTeX">t</tex-math></inline-formula>-norms, along with the research lines of Yager's <inline-formula><tex-math notation="LaTeX">f</tex-math></inline-formula>-generated implications by continuous additive generators <inline-formula><tex-math notation="LaTeX">f</tex-math></inline-formula> of <inline-formula><tex-math notation="LaTeX">t</tex-math></inline-formula>-norms and <inline-formula><tex-math notation="LaTeX">g</tex-math></inline-formula>-generated implications by continuous additive generators <inline-formula><tex-math notation="LaTeX">g</tex-math></inline-formula> of <inline-formula><tex-math notation="LaTeX">t</tex-math></inline-formula>-conorms and of Balasubramaniam's <inline-formula><tex-math notation="LaTeX">h</tex-math></inline-formula>-generated implications by continuous multiplicative generators <inline-formula><tex-math notation="LaTeX">h</tex-math></inline-formula> of <inline-formula><tex-math notation="LaTeX">t</tex-math></inline-formula>-conorms. This article is mainly motivated by many desirable properties of such generator generating implications for their flexible ordering property, unique determinations by induced natural negations, close relations to nilpotent <inline-formula><tex-math notation="LaTeX">t</tex-math></inline-formula>-norms, <inline-formula><tex-math notation="LaTeX">T</tex-math></inline-formula>-conditionality, the law of importation with more choices of <inline-formula><tex-math notation="LaTeX">t</tex-math></inline-formula>-norms <inline-formula><tex-math notation="LaTeX">T</tex-math></inline-formula> other than <inline-formula><tex-math notation="LaTeX">T_{P}</tex-math></inline-formula>, and good distributivity over <inline-formula><tex-math notation="LaTeX">t</tex-math></inline-formula>-norms and <inline-formula><tex-math notation="LaTeX">t</tex-math></inline-formula>-conorms. The relationships of <inline-formula><tex-math notation="LaTeX">k</tex-math></inline-formula>-generated implications to other well-known classes of fuzzy implications are clarified: they will unify <inline-formula><tex-math notation="LaTeX">g</tex-math></inline-formula>-generated implications with <inline-formula><tex-math notation="LaTeX">g(1)< \infty</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">h</tex-math></inline-formula>-generated implications with <inline-formula><tex-math notation="LaTeX">h(1)>0</tex-math></inline-formula> and hence, have intersections with <inline-formula><tex-math notation="LaTeX">(S,N)</tex-math></inline-formula>-implications; the only intersection with <inline-formula><tex-math notation="LaTeX">R</tex-math></inline-formula>-implications is the Goguen implication and they have no intersections with <inline-formula><tex-math notation="LaTeX">f</tex-math></inline-formula>-generated implications. The main results of this article are several characterizing theorems for (sometimes subclasses of) <inline-formula><tex-math notation="LaTeX">k</tex-math></inline-formula>-generated implications by means of aforementioned desirable properties from the respective perspectives, of which some partially solved or enriched related open problems in the literature. Finally, the superiorities of <inline-formula><tex-math notation="LaTeX">k</tex-math></inline-formula>-generated implications for modeling strict fuzzy preference relations and for constructing novel fuzzy reasoning methods are analyzed.
implications on finite scales Qiao, Junsheng
International journal of approximate reasoning,
August 2023, 2023-08-00, Letnik:
159
Journal Article
Recenzirano
Fuzzy implications, as a kind of fuzzy logical connectives, play an important role both in the theory and applications involving fuzzy sets and systems. Accordingly, the theoretical study of various ...fuzzy implications generated by common aggregation operators has become an attractive topic, especially on finite scales. In this paper, we analyse residual implications induced from discrete overlap functions latest proposed by the author, which belong to the category of discrete implications. Explicitly, in the beginning, we present the definition of residual implications generated by discrete overlap functions ▪ and call them ▪-implications. In the next place, we analyse a number of elementary properties of ▪-implications. Most of all, it proves that ▪-implications satisfy most of algebraic properties of discrete implications spontaneously, such as the left neutrality property, identity principle, ordering property, etc., which are different from the circumstance on other truth values sets. At last, via ▪-implications, we reveal correlations of discrete implications and discrete quasi-overlap functions extended from discrete overlap functions.
Overlap and grouping functions are special kinds of non-necessarily associative aggregation operators recently proposed for applications in classification problems involving the overlap problem ...and/or when the associativity property is not strongly required, as in image processing and decision making based on fuzzy preference relations, respectively. The concepts of indifference and incomparability defined in terms of overlap and grouping functions may allow their application in several different contexts. This paper introduces the concept of (G,N)-implication, for a grouping function G and a fuzzy negation N. (G,N)-implications are weaker then (S,N)-implications for positive and continuous t-conorms S, in the sense that (G,N)-implications do not necessarily satisfy certain properties, as the exchange and the left neutrality principles, which are not demanded for applications in decision making based on fuzzy preference relations. We analyze several related important properties, providing a characterization of (G,N)-implications.
Some open problems on fuzzy implications dealing with the so-called importation law are studied and totally or partially solved in this work. A weaker version of the law of importation, called the ...weak law of importation, is introduced. The relationships of these two properties and the exchange principle are studied. In particular, it is proved that the law of importation is stronger than the exchange principle. On the other hand, the three properties are equivalent for some kind of fuzzy implications, those that satisfy a boundary property. Along this study, new characterizations of (
S,
N)-implications,
R-implications and their counterparts for uninorms based on the weak law of importation are showed.
In this paper, characterizations of Yager’s f- and g-implications are presented. Since their introduction in 2004, the properties of these implications have been studied in detail but they have not ...been characterized yet. The characterizations given here are based on the law of importation, a functional equation that has been extensively studied. Moreover, particular characterizations of Reichenbach, Yager and Goguen implications are derived from these results.
The main purpose of this paper is to apply the generic extension methods of uplifting and downlifting associative operations in our previous work for fuzzy negations and implications. We also ...characterize when the NG-uplift (resp. GH-uplift) of a fuzzy negation N (resp. fuzzy implication) on a bounded subposet of a complete lattice is a fuzzy negation (resp. fuzzy implication) on the entire complete lattice. The NG-uplift (resp. GH-uplift) of a fuzzy negation N (resp. fuzzy implication) involves the use of a set-valued mapping G (resp. two set-valued mappings G and H) between that complete lattice and the subposet. Moreover, we discuss about which properties of fuzzy implications are preserved through the use of this extension method. Furthermore, we discuss the relationship between these two extensions of fuzzy negations and implications, and investigate the extensions of R-implications and (S,N)-implications. Finally, we formulate the dual results on the NG-downlift (resp. GH-downlift) of a fuzzy negation N (resp. fuzzy implication).
A new class of fuzzy implications called the
h-implications is introduced. They are implications generated from an additive generator of a representable uninorm in a similar way of Yager’s
f- and
...g-implications which are generated from additive generators of continuous Archimedean t-norms and t-conorms. Basic properties of these implications are studied in detail. Modifications and generalizations of the initial definition are presented and their properties studied and compared between them. One of the modifications, called (
h,
e)-implications, is another example of a fuzzy implication satisfying the exchange principle but not the law of importation for any t-norm, in fact for any function
F : 0,
1
2
→
0,
1.
The present article proposes two construction ways to study the general forms of ordinal sums of fuzzy implications with the intent of unifying the ordinal sums existing in the literature. The first ...ordinal sum construction way, which we call "Implication Complementing," is to study how to complement a specific fuzzy implication to the linear transformations of given fuzzy implications defined on respective disjoint subsquares whose principal diagonals are segments of the principal diagonal of the unit square, in order that the resulting ordinal sum on the unit square is a fuzzy implication. The second way, which we call "Implication Reconstructing," is to study how to reconstruct an initial fuzzy implication through replacing its some values on given rectangular regions of the unit square with the linear transformations of respective given fuzzy implications such that the redefined function is a new fuzzy implication. In both ways, necessary and sufficient conditions for the final reconstructed functions to be fuzzy implications are given and several new constructions of ordinal sums of fuzzy implications are obtained, which would generalize the existing ordinal sums from several aspects. In particular, by adopting the idea behind the second way, the generalized ordinal sums of fuzzy implications are proposed, in which the regions where the linear transformations of given fuzzy implications are defined are neither necessarily subsquares nor necessarily along the principal or minor diagonal of the unit square.
This paper deals with two sorts of interval fuzzy implications derived from interval overlap and grouping functions, viz., interval RO- and (G,O,N)-implications. Firstly, interval RO-implications, ...preserving the residuation property, are the interval generalization of RO-implications induced by overlap functions. We investigate their properties and their correlations with interval automorphisms. Secondly, interval (G,O,N)-operations are generalized from D-operations induced by tuples (O,G,N). We obtain a necessary and sufficient condition for the interval (G,O,N)-operation to be an interval fuzzy implication, namely interval (G,O,N)-implication. And then, we investigate vital properties and conclusions regarding interval (G,O,N)-operations and interval (G,O,N)-implications. Finally, we study the intersections between families of interval fuzzy implications, including interval RO-, (G,N)-, (O,G,N)- and (G,O,N)-implications.
