A Note on One Less Known Class of Generated Residual Implications

TL;DR

This paper explores the construction and properties of residual implications derived from various conjunction models, including generated t-norms, means, and the Choquet integral, extending the understanding of logical implications in fuzzy logic.

Contribution

It introduces a new class of generated implications based on conjunctions modeled by t-norms, means, and the Choquet integral, and analyzes their properties and relations.

Findings

Implication based on a t-norm belongs to the class I^g_N.

Properties of generated implications are characterized.

Relations between classes of implications are established.

Abstract

This paper builds on our contribution [Havlena and Hlinena, 2016] which studied modelling of the conjunction in human language. We have discussed three different ways of constructing conjunction. We have dealt with generated t-norms, generated means and Choquet integral. In this paper we construct the residual operators based on the above conjunctions. The only operator based on a t-norm is an implication. We show that this implication belongs to the class of generated implications I^g_N which was introduced in [Smutna, 1999] and studied in [Biba and Hlinena, 2012]. We study its properties. More, we investigate this class of generated implications. Some important properties, including relations between some classes of implications, are given.