The algebraic semantics for the one-variable monadic fragment of the predicate logic $\mathcal{G}\forall_{\sim}$

TL;DR

This paper characterizes the algebraic semantics of a specific one-variable monadic fragment of a fuzzy logic, introducing a new class of G"odel algebras with monadic and negation operators, and proves key properties including finite embeddability.

Contribution

It introduces the variety al{MG}_{\u00a7} as the algebraic semantics for the logic fragment, establishing its properties and representation, which was previously uncharacterized.

Findings

The variety al{MG}_{\u00a7} has the finite embeddability property.

A subvariety al{CMG}_{\u00a7} precisely captures the algebraic semantics.

Finite subdirectly irreducible algebras are characterized via functional representation.

Abstract

In this article we characterize the equivalent algebraic semantics for the one-variable monadic fragment of the first-order logic ${\cal G} \forall_{\sim}$ defined by F. Esteva, L. Godo, P. H\'ajek and M. Navara in Residuated fuzzy logics with an involutive negation, Archive for Mathematical Logic 39 (2000). To this end, we first introduce the variety $\mathbb{MG}_{\sim}$ as a certain class of G\"odel algebras endowed with two monadic operators and a De Morgan negation. We study its basic properties, determine its subdirectly irreducible members and prove that this variety has the finite embeddabilty property. In particular, we prove that a special subvariety $\mathbb{CMG}_{\sim}$ of $\mathbb{MG}_\sim$ is exactly the desired equivalent algebraic semantics; this is done via a functional representation of finite subdirectly irreducible algebras.