On good EQ-algebras

A special algebra called EQ-algebra has been recently introduced by Vilém Novák. Its original motivation comes from fuzzy type theory, in which the main connective is fuzzy equality. EQ-algebras have three binary operations meet, multiplication, fuzzy equality and a unit element. They open the door to an alternative development of fuzzy (manyvalued) logic with the basic connective being fuzzy equality instead of implication. This direction is justified by the idea due to G. W. Leibniz that “a fully satisfactory logical calculus must be an equational one”. In this paper, we continue the study of EQ-algebras and their special cases. We introduce and study the prefilters and the filters of separated EQ-algebras. We give great importance to the study of good EQ-algebras. As we shall see in this paper that the “goodness” property (and thus also separateness) is necessary for reasonably behaving algebras. We enrich good EQ-algebras with an unary operation (the so-called Baaz delta) fulfilling some additional assumptions, which is heavily used in fuzzy logic literature. We show that the characterization theorem obtained till now for representable good EQalgebras hold also for the enriched algebra. Abstract