Quasi-Algebras versus Regular Algebras - Part I

Abstract

Starting from quasi-Wajsberg algebras (which are generalizations of Wajsberg algebras), whose regular sets are Wajsberg algebras, we introduce a theory of quasi-algebras versus, in parallel, a theory of regular algebras. We introduce the quasi-RM, quasi-RML, quasi-BCI, (commutative, positive implicative, quasi-implicative, with product) quasi-BCK, quasi-Hilbert and quasi-Boolean algebras as generalizations of RM, RML, BCI, (commutative, positive implicative, implicative, with product) BCK, Hilbert and Boolean algebras respectively.In Part I, the first part of the theory of quasi-algebras -versus the first part of a theory of regular algebras -is presented. We introduce the quasi-RM and the quasi-RML algebras and we present two equivalent definitions of quasi-BCI and of quasi-BCK algebras.

Quasi-i-Boolean algebras vs. quasi-m Boolean algebras

“…In paper [10], starting from the quasi-Wajsberg algebras, whose regular algebras are the Wajsberg algebras, we have introduced a theory of quasi-algebras (of logic) vs. a theory of regular algebras (of logic), in the commutative case. We then have developed the theory in the preprints [11,12]. In [12], we have introduced the quasi-i-Boolean algebras, whose regular algebras are the i-Boolean algebras.…”

“…Definitions 3.1 [10] (q1) The algebra A = (A, →, 1) (structure A = (A, ≤, →, 1)) is called quasi-algebra (quasi-structure, respectively), if it satisfies the properties (q-M) and (M1).…”

“…Then, the introducing of the quasi-Wajsberg algebras in 2010 [2], as generalizations of Wajsberg algebras; they are categorically equivalent to quasi-MV algebras, just as Wajsberg algebras are categorically equivalent to MV algebras.In paper [10], starting from the quasi-Wajsberg algebras, whose regular algebras are the Wajsberg algebras, we have introduced a theory of quasi-algebras (of logic) vs. a theory of regular algebras (of logic), in the commutative case. We then have developed the theory in the preprints [11,12]. In [12], we have introduced the quasi-i-Boolean algebras, whose regular algebras are the i-Boolean algebras.In [13], we have noted that, in fact, there are two kinds of "quasi"-generalizations, that we have called: "quasi-" and "quasi-m", corresponding to the two kinds of algebras: M algebras (inside the commutative algebras of logic) and commutative unital magmas, respectively ("m" comes from magma).…”

“…Thus, (in [13] is the non-commutative case) we have generalized the M algebras to quasi-M algebras, as the most general quasi-algebras, by generalizing the principal, defining property (M) to (q-M), and we have generalized the commutative unital magmas to commutative quasi-m unital magmas, as the most general quasi-m algebras, by generalizing the principal, defining property (U) to (qm-U).The Wajsberg algebra and the i-Boolean algebra belong to the "world of commutative algebras of logic", more precisely to the class of M algebras; the MV algebra and the Boolean algebra belong to the "world of commutative algebras", more precisely to the class of commutative unital magmas.This small paper is organized as follows. In Section 2, we recall some basic things about regular algebras (structures), from [10], and about regular-m algebras (structures), from [13]; we recall the i-Boolean algebras and the Boolean algebras, and their categorically equivalence. In Section 3, we recall some basic things about quasi-algebras (quasi-structures), from [10], and about quasi-m algebras (structures), from [13]; we introduce the quasi-i-Boolean algebras and the quasim Boolean algebras and we prove their categorically equivalence (the main result); finally, we present some examples.Let A = (A, →, 1) be an algebra of type (2, 0) through this paper, where a binary relation ≤ can be defined by: x ≤ y def.…”

“…In Section 2, we recall some basic things about regular algebras (structures), from [10], and about regular-m algebras (structures), from [13]; we recall the i-Boolean algebras and the Boolean algebras, and their categorically equivalence. In Section 3, we recall some basic things about quasi-algebras (quasi-structures), from [10], and about quasi-m algebras (structures), from [13]; we introduce the quasi-i-Boolean algebras and the quasim Boolean algebras and we prove their categorically equivalence (the main result); finally, we present some examples.Let A = (A, →, 1) be an algebra of type (2, 0) through this paper, where a binary relation ≤ can be defined by: x ≤ y def.…”