On good EQ-algebras

Abstract

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, and fuzzy equality – and a unit element. They open the door to an alternative development of fuzzy (many-valued) logic with the basic connective being a fuzzy equality instead of an implication. This direction is justified by the idea presented by 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 filters of separated EQ-algebras. We give great importance to the study of good EQ-algebras. As we shall see in this paper, the “goodness” property (and thus separateness) is necessary for reasonably behaving algebras. We enrich good EQ-algebras with a unary operation Δ (the so-called Baaz delta), fulfilling some additional assumptions that are heavily used in fuzzy logic literature. We show that the characterization theorem obtained until now for representable good EQ-algebras also hold for the enriched algebra.