An analysis of the logic of Riesz Spaces with strong unit

TL;DR

This paper explores the algebraic and logical structure of Riesz MV-algebras, extending Lukasiewicz logic with scalar multiplication, and investigates their connections with functional analysis and finitely presented algebras.

Contribution

It introduces Riesz MV-algebras as algebraic models, linking logic, algebra, and geometry, and deepens the understanding of their relation to functional analysis.

Findings

Riesz MV-algebras are characterized as unit intervals of Riesz spaces with a strong unit.

The paper establishes connections between Riesz MV-algebras and functional analysis.

It studies finitely presented MV-algebras, DMV-algebras, and Riesz MV-algebras from multiple perspectives.

Abstract

We study \L ukasiewicz logic enriched with a scalar multiplication with scalars taken in $[0,1]$. Its algebraic models, called {\em Riesz MV-algebras}, are, up to isomorphism, unit intervals of Riesz spaces with a strong unit endowed with an appropriate structure. When only rational scalars are considered, one gets the class of {\em DMV-algebras} and a corresponding logical system. Our research follows two objectives. The first one is to deepen the connections between functional analysis and the logic of Riesz MV-algebras. The second one is to study the finitely presented MV-algebras, DMV-algebras and Riesz MV-algebras, connecting them from logical, algebraic and geometric perspective.