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Tropicalization through the lens of Łukasiewicz logic, with a topos theoretic perspective

Antonio Di Nola, Giacomo Lenzi, Brunella Gerla

TL;DR

This work develops a cohesive framework linking Łukasiewicz logic, tropical structures, and algebraic geometry through idempotent semirings and a topos-theoretic lens. It introduces a functorial tropicalization via the $V(C)$-generated variety and the invariant $\\theta$, connecting MV-algebras to tropical geometry and enabling a categorical bridge to idempotent semirings. It further analyzes the special case of perfect MV-algebras, establishing equivalences with TOP and tying Perf$^\\mathbb{Q}$ to geometric structures, while relating MV-algebras to topos $\\widehat{\\mathbb N^\\times}$ through the $\\Delta$ equivalence. The results provide a robust, modular pathway for studying logical-algebraic models within tropical and topos-theoretic contexts, with potential implications for computability and logic of MV-algebras.

Abstract

The main aim of this paper is to show the interconnections between Łukasiewicz 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.

Tropicalization through the lens of Łukasiewicz logic, with a topos theoretic perspective

TL;DR

This work develops a cohesive framework linking Łukasiewicz logic, tropical structures, and algebraic geometry through idempotent semirings and a topos-theoretic lens. It introduces a functorial tropicalization via the -generated variety and the invariant , connecting MV-algebras to tropical geometry and enabling a categorical bridge to idempotent semirings. It further analyzes the special case of perfect MV-algebras, establishing equivalences with TOP and tying Perf to geometric structures, while relating MV-algebras to topos through the equivalence. The results provide a robust, modular pathway for studying logical-algebraic models within tropical and topos-theoretic contexts, with potential implications for computability and logic of MV-algebras.

Abstract

The main aim of this paper is to show the interconnections between Łukasiewicz 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.
Paper Structure (7 sections, 21 theorems, 5 equations)

This paper contains 7 sections, 21 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. MV-algebras, semirings and semifields
  3. Rump tropicalization and its generalization via $V(C)$ algebras: the functor $\theta$
  4. Tropicalization of Perfect MV-algebras
  5. MV-algebras and the topos $\widehat{\mathbb N^\times}$
  6. Flatness and $Pt(\widehat{\mathbb N^\times})$
  7. Conclusions

Key Result

Theorem 2.1

R There is an equivalence $\rho$ between the category of idempotent semifields and the category of abelian $\ell$-groups.

Theorems & Definitions (36)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Definition 3.1
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • Lemma 3.1
  • proof
  • Definition 3.2
  • ...and 26 more