Representable good EQ-algebras

Abstract

Recently, a special algebra called EQ-algebra (we call it here commutative EQ-algebra since its multiplication is assumed to be commutative) has been introduced by Novák (Proceedings of the Czech-Japan seminar, ninth meeting, Kitakyushu and Nagasaki, 18-22 August, 2006), which aims at becoming the algebra of truth values for fuzzy type theory. Its implication and multiplication are no more closely tied by the adjunction and so, this algebra generalizes commutative residuated lattice. One of the outcomes is the possibility to relax the commutativity of the multiplication. This has been elaborated by El-Zekey et al. (Fuzzy Sets Syst 2009, submitted). We continue in this paper the study of EQ-algebras (i.e., those with multiplication not necessarily commutative). We introduce prelinear EQ-algebras, in which the join-semilattice structure is not assumed. We show that every prelinear and good EQ-algebra is a lattice EQ-algebra. Moreover, the f^; _; !; 1g-reduct of a prelinear and separated lattice EQ-algebra inherits several lattice-related properties from product of linearly ordered and separated EQ-algebras. We show that prelinearity alone does not characterize the representable class of all good (commutative) EQ-algebras. One of the main results of this paper is to characterize the representable good EQ-algebras. This is mainly based on the fact that f!; 1g-reducts of good EQ-algebras are BCK-algebras and run on lines of van Alten's (J Algebra 247:672-691, 2002) characterization of representable integral residuated lattices. We also supply a number of potentially useful results, leading to this characterization.

On EQ-Fuzzy Logics with Delta Connective

“…The completeness theorem has been proved in its basic form. With respect to the results from [3] it is clear that it the representability of prelinear EQ-algebras can be extended also to prelinear EQ ∆ -algebras. Consequently, extension of the completeness theorem to hold with respect to all linearly ordered EQ ∆ -algebras will be possible.…”

“…The latter are special algebras in which the fundamental operation is that of fuzzy equality. The concept of EQ-algebra was introduced in [1] and in more detail elaborated in [2] and [3,4]. It was motivated by the paper of L. Henkin [5] who introduced type theory (higherorder logic) in which equality is the sole connective.…”

Hesitant fuzzy prefilters and filters of EQ-algebras

“…(1) The inclusion H x⊗y ⊆ H y x follows from Proposition 3.8 (2). Note that y ≤ 1 → y for all y ∈ L. It follows from Proposition 3.7(1) and (8) that…”

“…For more details of EQ-algebras, we refer the reader to [2], [3], [6], [7], [8], and [10]. Definition 2.1 An EQ-algebra is an algebra L := (L, ∧, ⊗, ∼, 1) of type (2, 2, 2, 0) in which the following axioms are valid:…”

Boolean center of lattice ordered EQ-algebras with bottom element

“…Special classes of EQ-algebras have been studied in [2,3]. Moreover the {∧, →, 1}-reduct of good EQ-algebras are BCK-meet-semilattices.…”