Study of Two Families of Generalized Yager’s Implications for Describing the Structure of Generalized (h,e)-Implications

Raquel Fernandez-Peralta,Arnau Mir,Sebastia Massanet

doi:10.3390/math9131490

Raquel Fernandez-Peralta, Arnau Mir + Show 1 more

Open Access

https://doi.org/10.3390/math9131490

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Abstract

In this study, we analyze the family of generalized (h,e)-implications. We determine when this family fulfills some of the main additional properties of fuzzy implication functions and we obtain a representation theorem that describes the structure of a generalized (h,e)-implication in terms of two families of fuzzy implication functions. These two families can be interpreted as particular cases of the (f,g) and (g,f)-implications, which are two families of fuzzy implication functions that generalize the well-known f and g-generated implications proposed by Yager through a generalization of the internal factors x and 1x, respectively. The behavior and additional properties of these two families are also studied in detail.

Highlights

Fuzzy implication functions play a key role as operators that generalize the classical implications in crisp logic
In the same way classical implications are used in inference schemas such as modus ponens, modus tollens, etc., fuzzy implication functions play a similar role in the generalization of these schemas, which use fuzzy statements whose value is in [0, 1] instead of being in {0, 1}
It is of the utmost importance to study the additional properties that the operators of certain families satisfy and to provide an axiomatic characterization of the new operators in the literature in order to find its possible relation with respect to those already known

Summary

Introduction

Fuzzy implication functions play a key role as operators that generalize the classical implications in crisp logic. Apparently there is a huge amount of fuzzy implication functions available, these families can present intersection or even coincide with others already known [6] For this reason, it is of the utmost importance to study the additional properties that the operators of certain families satisfy and to provide an axiomatic characterization of the new operators in the literature in order to find its possible relation with respect to those already known. 1 x as a strictly decreasing function f : [0, 1] → [0, +∞] with f (0) = +∞ This approach is different from all the other generalization proposals, and the interest in these two families is twofold: to describe the structure of (h, e)-implications and to study these families in order to compare its properties with other generalizations considered in the literature.

Preliminaries

Generalized Yager’s Implications

Generalization of f -Generated Implications

Generalization of g-Generated Implications

Conclusions and Future Work