The logical content of triangular bases of fuzzy sets in {\L}ukasiewicz infinite-valued logic

TL;DR

This paper demonstrates that { extbackslash}Lukasiewicz logic can express pseudo-triangular bases of fuzzy sets, providing a new logical perspective and an elementary characterization of these bases, extending previous work on related logics.

Contribution

It introduces the notion of pseudo-triangular bases within { extbackslash}Lukasiewicz logic and offers an elementary, logic-independent characterization of triangular bases of fuzzy sets.

Findings

{ extbackslash}Lukasiewicz logic can express pseudo-triangular bases.

Elementary characterization of triangular bases of fuzzy sets.

Extension of previous work on G"odel-Dummett logic and Ruspini partitions.

Abstract

Continuing to pursue a research direction that we already explored in connection with G\"odel-Dummett logic and Ruspini partitions, we show here that {\L}ukasiewicz logic is able to express the notion of pseudo-triangular basis of fuzzy sets, a mild weakening of the standard notion of triangular basis. En route to our main result we obtain an elementary, logic-independent characterisation of triangular bases of fuzzy sets.