Tropicalization through the lens of {\L}ukasiewicz logic, with a topos theoretic perspective

TL;DR

This paper explores the deep connections between { extL}ukasiewicz logic, algebraic geometry, and tropical geometry using categorical and topos-theoretic frameworks, revealing new algebraic structures and equivalences.

Contribution

It introduces a novel algebraic geometric approach to tropicalization through the lens of { extL}ukasiewicz logic and topos theory, establishing categorical equivalences.

Findings

Categorical equivalence between tropical geometry and many-valued logics.

Description of algebraic structures related to MV-algebras.

Connections between algebraic geometry, logic, and topos theory.

Abstract

The main aim of this paper is to show the interconnections between {\L}ukasiewicz logic and algebraic geometry using algebraic, geometric and logical instruments. We continue our investigation into a new algebraic geometry based on idempotent semifields, in particular those related with MV-algebras, describing categorical equivalence between different structures related to tropical geometry and many valued logics. Further, we describe such connections in terms of topoi.