Quotients with respect to similarity relations
Fuzzy Sets and Systems - Mathematics and Fuzziness, Part 1
Fuzzy Sets and Systems
Fuzzy homomorphism and algebraic structures
Fuzzy Sets and Systems
On a generalization of fuzzy algebras and congruences
Fuzzy Sets and Systems
Fuzzy partially ordered fuzzy subgroups
Fuzzy Sets and Systems
Fuzzy quotient algebras and fuzzy factor congruences
Fuzzy Sets and Systems
Fuzzy implicative ideals in BCK-algebras
Fuzzy Sets and Systems
Fuzzy Relational Systems: Foundations and Principles
Fuzzy Relational Systems: Foundations and Principles
Fuzzy radicals and prime fuzzy ideals of ordered semigroups
Information Sciences: an International Journal
Fuzzy homomorphisms of algebras
Fuzzy Sets and Systems
Fuzzy Sets and Systems
Weak Fuzzy Equivalence and Equality Relations
PReMI '09 Proceedings of the 3rd International Conference on Pattern Recognition and Machine Intelligence
EQ-algebras from the point of view of generalized algebras with fuzzy equalities
Fuzzy Sets and Systems
Fuzzy identities with application to fuzzy semigroups
Information Sciences: an International Journal
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An algebra with fuzzy equality is a set with operations on it that is equipped with similarity ~, 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).