Variety theorem for algebras with fuzzy orders

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.

Fuzzy inequational logic

“…(b) Note that both (2) and (3) involve ⊗, i.e., the conditions of transitivity and compatibility of M with the functions in M are formulated in terms of the multiplication ⊗ in L. Condition (1) ensures that the symmetric interior of M is a compatible fuzzy equality relation, see [5,Theorem 3].…”

“…In our considerations on algebras with fuzzy orders, we utilize homomorphisms and factor algebras with fuzzy orders [5].…”

“…for all a, b ∈ M is called a homomorphism [5,Section 5] and is denoted by h : M → N. Therefore, homomorphisms are maps which are compatible with the functional parts of M and N and the L-orders of M and N.…”

“…One can show that factor algebras with L-orders are well defined algebras with L-orders, see [5,Lemma 4] for details. The notions of homomorphic images and factor algebras preserve desirable properties of their classic counterparts [21].…”

“…The map is called a natural homomorphism [5,Section 5] induced by ξ . Conversely, for a surjective homomorphism h : M → N, we may introduce a binary L-relation ξ h on M by putting…”