Characterization of uninorms with continuous underlying t-norm and t-conorm by their set of discontinuity points

TL;DR

This paper characterizes uninorms with continuous underlying t-norms and t-conorms, analyzing their discontinuity points and providing conditions for continuity, supported by examples.

Contribution

It introduces a detailed analysis of the discontinuity set of such uninorms and offers a sufficient condition for their underlying operations to be continuous.

Findings

Discontinuity points form a subset of a specific symmetric, surjective, non-increasing multi-function graph.

A sufficient condition for the underlying t-norm and t-conorm to be continuous is established.

Several illustrative examples are provided to demonstrate the theoretical results.

Abstract

Uninorms with continuous underlying t-norm and t-conorm are discussed and properties of the set of discontinuity points of such a uninorm are shown. This set is proved to be a subset of the graph of a special symmetric, surjective, non-increasing multi-function. A sufficient condition for a uninorm to have continuous underlying operations is also given. Several examples are included.