Variety theorem for algebras with fuzzy order

TL;DR

This paper generalizes the Bloom variety theorem to fuzzy ordered algebras, showing that classes of models defined by fuzzy inequalities are closed under key algebraic operations and are characterizable by such inequalities.

Contribution

It introduces algebras with fuzzy orders based on residuated lattices and proves a variety theorem in this fuzzy setting, extending classical algebraic results.

Findings

Classes of fuzzy ordered algebra models are closed under subalgebras, homomorphic images, and direct products.

Such classes are definable by fuzzy sets of inequalities.

The paper establishes a correspondence between closure properties and definability in fuzzy algebraic structures.

Abstract

We present generalization of the Bloom variety theorem of ordered algebras in fuzzy setting. We introduce algebras with fuzzy orders which consist of sets of functions which are compatible with particular binary fuzzy relations called fuzzy orders. Fuzzy orders are defined on universe sets of algebras using complete residuated lattices as structures of degrees. In this setting, we show that classes of models of fuzzy sets of inequalities are closed under suitably defined formations of subalgebras, homomorphic images, and direct products. Conversely, we prove that classes having these closure properties are definable by fuzzy sets of inequalities.