Generating Fuzzy Implications by Ordinal Sums

Abstract

ABSTRACT. This paper deals with ordinal sums of fuzzy implications. Some of the known constructions are recalled and new ways of generating fuzzy implications from given ones are proposed. Sufficient properties of fuzzy implications as summands for obtaining a fuzzy implication as a result are presented. IntroductionFuzzy implications find applications in many fields such as fuzzy control, approximate reasoning, and decision support systems. For that reason, new families of these connectives are the subject of investigation.One of the directions of such a research is to consider an ordinal sum of fuzzy implications on the pattern of the ordinal sum of t-norms. Some interesting results connected to representation of the residual implication corresponding to a fuzzy conjunction (for example continuous or at least left-continuous tnorm) given by an ordinal sum were obtained in [2], [4], [7]. In [9] S u et al. introduced a concept of an ordinal sum of fuzzy implications similar to the construction of the ordinal sum of t-norms. In [3], there were proposed other constructions of ordinal sums of fuzzy implications. In this contribution, some of the ideas are recalled and new possibilities of defining ordinal sums of fuzzy implications are proposed.First, in Section 2, we recall definitions and basic results concerning t-norms and fuzzy implications and we revise methods of constructing ordinal sums of fuzzy implications. Then, in Section 3, we propose new constructions of ordinal sums of fuzzy implication which generate fuzzy implications. Next, we suggest further research directions for the ordinal sums of fuzzy implications.

Construction methods for implications on bounded lattices

“…This means that the ordinal sum of every fuzzy implications need not be a fuzzy implication without some special conditions. In this sense, in [5,6], Drygaś and Król introduced some construction methods generating again a fuzzy implication by means of the ordinal sum of fuzzy implications having no additional conditions. In this paper, we introduced the ordinal sum of implications on bounded lattices based on [5,25].…”

“…In this sense, the ordinal sum construction method is one of the most commonly used generating methods for such operators. For more results on ordinal sum of triangular norms, triangular conorms, uninorms, fuzzy implications on the unit interval [0, 1], see [2,5,6,11,14,16,21,22,23,26,27].…”

“…This means that the construction method need not generate a fuzzy implication on [0, 1] without the additional conditions. In [1,5,6], new ordinal sum construction methods for fuzzy implication which generate fuzzy implications have been proposed. DOI: 10.14736/kyb-2019-4-0641…”

“…In the present paper, based on [5,6], the ordinal sum construction methods of implications on bounded lattices are studied. The paper is organized as follows.…”

Ordinal Sums of Fuzzy Negations: Main Classes and Natural Negations

“…In the context of fuzzy logic, the ordinal sums were first studied for triangular norms and triangular conorms in [37] in order to provide a method to construct new t-norms and t-conorms from other t-norms and t-conorms (for more details see [27]). However, the ordinal sums of several others important fuzzy connectives also has been studied, such as, for example, the ordinal sums of copulas [33], overlap functions [20], uninorms [30,31], fuzzy implications [17,39] and fuzzy negations [9]. In particular, the ordinal sums of fuzzy negations proposed in [9] were made in the context of Morgan's triples and they were not deeply studied.…”