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arXiv · 2110.10970

Fuzzy Algebraic Theories

Davide Castelnovo·Marino Miculan
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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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Davide Castelnovo, Marino Miculan. 2021-10-21. Fuzzy Algebraic Theories. https://arxiv.org/abs/2110.10970

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