TL;DR
This paper introduces a formal fuzzy algebraic reasoning system with a sequent calculus, semantics, and model theory, enabling the analysis of fuzzy algebraic structures and their categorical properties.
Contribution
It develops a novel fuzzy algebraic calculus with sound semantics, free models, and completeness, extending algebraic theory to fuzzy contexts with categorical characterizations.
Findings
Established soundness and completeness of the fuzzy calculus.
Constructed free models as Eilenberg-Moore algebras.
Provided HSP-like characterizations for categories of fuzzy algebraic models.
Abstract
In this work we propose a formal system for fuzzy algebraic reasoning. The sequent calculus we define is based on two kinds of propositions, capturing equality and existence of terms as members of a fuzzy set. We provide a sound semantics for this calculus and show that there is a notion of free model for any theory in this system, allowing us (with some restrictions) to recover models as Eilenberg-Moore algebras for some monad. We will also prove a completeness result: a formula is derivable from a given theory if and only if it is satisfied by all models of the theory. Finally, leveraging results by Milius and Urbat, we give HSP-like characterizations of subcategories of algebras which are categories of models of particular kinds of theories.
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