Preideals in EQ-algebras

Abstract

EQ-algebras were introduced by Novak (2006) as an algebraic structure of truth values for fuzzy-type theory (FFT). Novák and De Baets (2009) introduced various kinds of EQ-algebras such as good, residuated, and IEQ-algebras. In this paper, we define the notion of (pre)ideal in bounded EQ-algebras (BEQ-algebras) and investigate some properties. Then, we introduce a congruence relation on good BEQ-algebras by using ideals, and then, we solve an open problem in Paad (2019). Moreover, we show that in IEQ-algebras, there is a one-to-one correspondence between congruence relations and the set of ideals. In the following, we characterize the generated preideal in BEQ-algebras, and by using this, we prove that the family of all preideals of a BEQ-algebra is a complete lattice. Then, we show that the family of all preideals of a prelinear IEQ-algebras is a distributive lattice and becomes a Heyting algebra. Finally, we show that we can construct an MV-algebra from the family of all preideals of a prelinear IEQ-algebra.