Rule mining algorithms are one of the fundamental techniques in data mining for discovering significant patterns in terms of linguistic rules expressed in natural language. In this paper, we revisit the concept of fuzzy implicative rules to provide a solid theoretical framework for any fuzzy rule mining algorithm interested in capturing patterns in terms of logical conditionals rather than the co-occurrence of antecedent and consequent. In particular, we study which properties the fuzzy operators should satisfy to ensure a coherent behavior of different quality measures. As a consequence of this study, we introduce a new property of fuzzy implications related to a monotone behavior of the generalized modus ponens for which we provide several valid solutions. Also, we prove that our modeling generalizes other approaches if an adequate choice of the fuzzy implication is made, so it can be seen as an unifying framework. We test the applicability and relevance of our framework on different real datasets and with different fuzzy operators.

The research on fuzzy implication functions has exponentially grown in the last decades becoming one of the main topics studied in the Fuzzy Logic community. These efforts have led to significant advances on the theoretical (mainly) and applied aspects, but the overwhelming number of papers hinder the track of the new results and the problems that remain unsolved. The goal of this paper is to determine the valuable practices in the research on classes of fuzzy implication functions and the desirable characteristics that a paper on this topic should have to be useful for a reader. Clearly appointing these features can accelerate the pace of the future research and be constructive for new researchers.

The invariance property with respect to powers of continuous t-norms is an additional property of fuzzy implication functions which is particularly interesting in the area of approximate reasoning. In this paper, we provide the characterization of all the binary functions which are invariant with respect to the powers of a nilpotent t-norm and, from the restriction of this result to the definition of fuzzy implication function, we introduce the family of nilpotent T-power invariant implications. We study when the members of this family do satisfy other additional properties apart from the invariance. Moreover, the intersection between the family of nilpotent T-power invariant implications and several families of fuzzy implication functions is investigated.

The continuous completions of conditionally cancellative pre-t-norms defined in the regions linked to the characterization of (S,N)-implications when S is a continuous t-conorm and N is a fuzzy negation with one point of discontinuity are determined. The results have been divided in two parts. In this first part we provide three main results: we determine the existence and expression of the additive generator of a cancellative pre-t-norm known above a level curve and, using this result as a basis, we provide the continuous completions of pre-t-norms determined in regions of the type [0,1]2∖(0,a)2 or [0,1]2∖(a,1)2 with a∈(0,1). It is highlighted that, differently from the cancellative case where all the results provide unique continuous completions, in the analogous situations in the conditionally cancellative case the corresponding continuous completions might not be unique. This fact causes more complex results for the remaining regions, which are discussed in the second paper of this study.

The continuous completions of conditionally cancellative pre-t-norms defined in the regions linked to the characterization of (S,N)-implications when S is a continuous t-conorm and N is a fuzzy negation with one point of discontinuity are determined. The results have been divided in two parts. In this second part, we use the results of the first part for providing the continuous completions of a conditionally cancellative pre-t-norm defined in the four remaining regions. The most significant contribution in this second part is the study of the case that corresponds to a region of the type B=([0,1]2∖(a,1)2)∪({c}×[0,1])∪([0,1]×{c}) with 0<a<c<1, which has to be separated in six different cases.

The characterization of (S,N)-implications when N is a non-continuous negation has remained one of the most significant open problems in fuzzy logic for the last decades. This paper constitutes the first progress in this topic. Namely, a general characterization of this family of fuzzy implication functions is presented, in which the central property is the existence of a completion of a binary function defined on a certain subregion of [0,1]2 to a t-conorm. In this paper, the dual problem of finding a completion of a binary function defined on a subregion of [0,1]2 to a continuous t-norm is studied and solved for the minimum and a cancellative function. These results are the basis for the novel axiomatic characterizations of (S,N)-implications in the case when N has one point of discontinuity and S is equal to the maximum t-conorm in a certain subregion of [0,1]2 or a strict t-conorm.