Equality algebras

Motivated by EQ-algebras of V. Novák we introduce a new structure, called equality algebras. It has two connectives, a meet operation and an equivalence. As opposed to previous attempts in the literature, transitivity of the equivalence is formulated in a novel way, without reference to any conjunction. This lays down a new foundation for investigating fuzzy equivalence relations (fuzzy similarities). After listing its basic properties we introduce a closure operator on the class of equality algebras, and call the closed algebras equivalential. We show that equivalential equality algebras are term equivalent to BCK-algebras with meet. As a by-product we obtain a quite general generalization of a result of Kabziński and Wroński: we provide an equational characterization for the equivalential fragment of BCK-algebras with meet. Finally we investigate the congruence lattice of equality algebras and show that the variety of equality algebras is 1-regular and arithmetical.