Variety theorem for algebras with fuzzy order

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.