Valuations in Nilpotent Minimum Logic

TL;DR

This paper introduces a new valuation called the idempotent Euler characteristic for NM algebras, which are algebraic structures of Nilpotent Minimum logic, and demonstrates its ability to encode propositional formulas.

Contribution

It proposes the idempotent Euler characteristic as a novel valuation for NM algebras, extending the classical Euler characteristic to better capture logical information.

Findings

The idempotent Euler characteristic encodes propositional formulas in NM logic.

It extends the understanding of valuations on finite distributive lattices.

Provides a new tool for analyzing Nilpotent Minimum logic structures.

Abstract

The Euler characteristic can be defined as a special kind of valuation on finite distributive lattices. This work begins with some brief consideration on the role of the Euler characteristic on NM algebras, the algebraic counterpart of Nilpotent Minimum logic. Then, we introduce a new valuation, a modified version of the Euler characteristic we call idempotent Euler characteristic. We show that the new valuation encodes information about the formul{\ae} in NM propositional logic.