Pseudovarieties of algebras with fuzzy equalities

We establish analogy of the Eilenberg–Schützenberger pseudovariety theorem in the setting of algebras with fuzzy equalities and Pavelka-style semantics using complete residuated lattices as structures of truth degrees. We show that under the assumption of countable reconstructibility, classes of finite algebras with fuzzy equalities closed under formations of subalgebras, homomorphic images, and finite direct products, are classes of finite algebras with fuzzy equalities which satisfy all but finitely many identities taken from fuzzy sets of identities which serve as graded equational theories. We also show that the assumption of countable reconstructibility is satisfied by model classes if one uses complete residuated lattices based on left-continuous triangular norms as structures of truth degrees.