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

The main aim of this paper is to show that the topics of {\L}ukasiewicz logic, semirings and tropical structures fruitfully meet. This gives rise to a topos theoretic perspective to {\L}ukasiewicz logic. A functorial tropicalization of MV-algebras in the variety V(C) is proposed. We further consider a logic based on perfect MV-algebras and having truth values that are perturbations of boolean values, and we show how this logic can exhibit models functorially connected with points of a non-commutative geometry.