Algebras with fuzzy equalities

An algebra with fuzzy equality is a set with operations on it that is equipped with similarity,(almost equal) i.e. a fuzzy equivalence relation, such that each operation f is compatible with . Described verbally, compatibility says that each f yields similar results if applied to pairwise similar arguments. On the one hand, algebras with fuzzy equalities are structures for the equational fragment of fuzzy logic and have been studied from this point of view before. On the other hand, they are the formal counterpart to the intuitive idea of having functions that are not allowed to map similar objects to dissimilar ones. The present paper aims at developing fundamental points of algebras with fuzzy equalities: we introduce the notion of an algebra with fuzzy equality, present natural examples, compare the notion with other approaches, and introduce and develop basic structural notions (subalgebras, morphisms, products, direct unions). In a follow-up paper [Vychodil, Direct limits and reduced products of algebras with fuzzy equalities, submitted for publication], we deal with advanced topics in algebras with fuzzy equalities (direct limits, reduced products).