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

We introduce the quasi-i-Boolean algebras and the quasi-m Boolean algebras and we prove their categorically equivalence.


Introduction
The departure point for the research was the introducing of the quasi-MV algebras in 2006 [15], as generalizations of MV algebras, following an investigation into the foundations of quantum computing (see [4]); since then, many papers investigated them [17,1,6,14]. 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. (1") Any algebra (structure) A = (A, τ ) which is term equivalent to a regular algebra (structure) A = (A, σ), is also called regular.
An M algebra is an algebra A = (A, →, 1) verifying (M) [13]. Note that the class of M algebras is the largest class of regular algebras. Other examples of regular algebras are the BCI, BCK algebras, the Hilbert algebras, the commutative implicative-groups [13], the Wajsberg algebras etc.
Let us consider the following subsets of A [10]: = 1 → x}. By (M), we have that: V M = V = U = A, and this is the basic, definable property of regular algebras (structures). In the theory of regular algebras (structures), we determine which properties depend on (M) (are proved by (M)) and which are independent of (M); this is the necessary step to develop further the theory of quasi-algebras (quasi-structures).

Implicative-Boolean algebras
There are many equivalent definitions of Boolean algebras (see [16]). We recall here the definition introduced in 2009 [8] and presented also in [3], motivated by the axioms system of the classical propositional logic: An implicative-Boolean algebra, or an i-Boolean algebra for short, is an algebra of type (2, 1, 0) verifying: for all x, y, z ∈ A, x → y = 1 = y → x =⇒ x = y (Antisymmetry). Let A be an i-Boolean algebra. The following properties hold, among many others [12]: Note that an i-Boolean algebra is a regular algebra, since (M) holds.

Introduction to a theory of regularm algebras (structures)
Let A = (A, , 1) be a commutative algebra of type (2, 0) or let A = (A, ≤, , 1) be a commutative structure, through this paper, where commutative means x y = y x, for all x, y ∈ A. Consider the property: for all x ∈ A, (U) 1 x = x.
(1"m) Any algebra (structure) A = (A, τ ) which is term equivalent to a regular-m algebra (structure) A = (A, σ), is also called regular-m.
A commutative unital magma is a commutative algebra A = (A, , 1) verifying (U). Note that their class is the largest class of regular-m algebras ("m" comes from magma). Other examples of regular-m algebras are the commutative monoids, the commutative groups, the MV algebras etc.
Let us consider the following subset of A [13]: Note that, if property (U) holds, then A ⊆ V U , hence V U = A, and this is the basic, definable property of regular-m algebras (structures). In the theory of regular-m algebras (structures), we determine which properties depend on (U) (are proved by (U)) and which are independent of (U); this is the necessary step to develop further the theory of quasi-m algebras (structures).

Boolean algebras
Boolean algebras were introduced in 1854 by George Boole. The most used definition is as a complemented, bounded, distributive lattice, namely: A Boolean algebra is an algebra x ∨ x − = 1. Note that a Boolean algebra (that can be defined equivalently as an algebra (A, ∧, − , 1) of type (2, 1, 0)) is a regular-m algebra, since (mUW), and the dual (mUV), hold.

Implicative-Boolean algebras vs. Boolean algebras
The i-Boolean algebras are categorically equivalent to the Boolean algebras just as the Wajsberg algebras are categorically equivalent to the MV algebras, namely we have: = (A, ∧, ∨, − , 0, 1) as follows: for every (2) The mappings Φ and Ψ are mutually inverse.
and this is the basic, definable property of quasi-algebras (quasi-structures).
For every quasi-algebra (quasi-structure) A, the subset V M = V = U of A will be called the regular set of A and will be denoted by R(A): A is a regular algebra (structure). The elements of R(A) are called the regular elements of A, the elements of A \ R(A) are called the quasi-elements of A. In a q-proper quasi-algebra, we shall call quasiproperty, a property that is dependent of (q-M), and we shall call regular property, a property that is independent of (q-M).

Quasi-implicative-Boolean algebras
We have introduced the notion of quasi-implicative-Boolean algebra in the preprint [12], as follows: A quasi-implicative-Boolean algebra, or a quasi-i-Boolean algebra for short, is an algebra Note that a quasi-i-Boolean algebra is a quasialgebra, since (q-M) and (M1) hold. Indeed, by [10],   [12], and the following quasi-properties, among others [12]: ⇐⇒ x → y = 1. Then, we have: for all x, y, z ∈ A,
A commutative quasi-m unital magma is a commutative algebra A = (A, , 1) verifying (qm-U) and (U1) [13]. Note that their class is the largest class of quasi-m algebras. Other examples of quasi-m algebras are the commutative quasi-m monoids, quasi-m groups [13], the quasi-MV algebras [15] (in fact quasi-m MV algebras, by [13]). For every quasi-m algebra (structure) A, the subset V U of A will be called the regular-m set of A and will be denoted by Rm(A):  In a qm-proper quasi-m algebra, we shall call quasim property, a property that is dependent of (qm-U), and we shall call regular-m property, a property that is independent of (qm-U).
(1'): First note that we have: Thus, Ψ(A) is a quasi-i-Boolean algebra. Then, for all x, y ∈ A, we have: Then, for all x, y ∈ A, we have:

Examples
Consider the standard i-Boolean algebra represented by the Hasse diagram given in Figure 1 and with the tables of → and − (involutive negation) recalled below: Boolean quasi-i-Boolean algebra algebras  We can build an infinity of quasi-i-Boolean algebras whose regular algebra be the standard i-Boolean algebra L 2 , by [10]. We present two examples.

Example 1
By adding two quasi-elements, a, b, to the above regular elements 0, 1, such that a 0 and b 1 (see [10]), we obtain the quasi-i-Boolean algebra represented by the quasi-Hasse diagram (see [10]) given also in Figure 1 and with the following tables of → 1 and − (involutive quasi-negation): .
Note that the line/column of a (b) coincides with the line/column of 0 (1, respectively).
By Theorem 3.11, the quasi-i-Boolean algebra L 1 2 is term-equivalent to the quasi-m Boolean algebra