Some results on pseudo-Q algebras

Abstract The notions of a dual pseudo-Q algebra and a dual pseudo-QC algebra are introduced. The properties and characterizations of them are investigated. Conditions for a dual pseudo-Q algebra to be a dual pseudo-QC algebra are given. Commutative dual pseudo-QC algebras are considered. The interrelationships between dual pseudo-Q/QC algebras and other pseudo algebras are visualized in a diagram.


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A. Rezaei, A. Borumand Saeid and A. Walendziak of pseudo-BE algebras and proved that the class of commutative pseudo-CI algebras coincides with the class of commutative pseudo-BCK algebras [16]. Recently, Y.B. Jun et al. defined and investigated pseudo-Q algebras [9] as a generalization of Q-algebras [14].
In this paper, we define dual pseudo-Q and dual pseudo-QC algebras. We investigate the properties and characterizations of them. Moreover, we provide some conditions for a dual pseudo-Q algebra to be a dual pseudo-QC algebra. We also consider commutative dual pseudo-QC algebras and prove that the class of such algebras coincides with the class of commutative pseudo-BCI algebras. Finally, the interrelationships between dual pseudo-Q/QC algebras and other pseudo algebras are visualized in a diagram.

Preliminaries
In this section, we review the basic definitions and some elementary aspects that are necessary for this paper.
A pseudo-CI algebra X = (X; →, , 1) verifying condition for all x ∈ X, is said to be a pseudo-BE algebra (see [1]).
In a pseudo-CI algebra X we can introduce a binary relation " ≤ " by x ≤ y ⇐⇒ x → y = 1 ⇐⇒ x y = 1 for all x, y ∈ X.
An algebra X = (X; →, , 1) of type (2, 2, 0) is called commutative if for all x, y ∈ X, it satisfies the following identities: From [2] (see Theorem 3.4) it follows that any commutative pseudo-BE algebra is a pseudo-BCK algebra. By Theorem 3.9 of [16], any commutative pseudo-CI algebra is a pseudo-BE algebra. Therefore we obtain Proposition 2.7 Commutative pseudo-CI algebras coincide with commutative pseudo-BE algebras and with commutative pseudo-BCK algebras (hence also coincide with commutative pseudo-BCI algebras and with commutative pseudo-BCH algebras).

Dual pseudo-Q algebras
Definition 3.1 An algebra X = (X; →, , 1) of type (2, 2, 0) is called a dual pseudo-Q algebra if, for all x, y, z ∈ X, it verifies the following axioms: In a dual pseudo-Q algebra, we can introduce two binary relations ≤ → and ≤ by x ≤ → y ⇐⇒ x → y = 1 and x ≤ y ⇐⇒ x y = 1.
Define binary operations → and on X by the following tables ( [16]): Then X = (X; →, , 1) is a dual pseudo-Q algebra which is not a pseudo-BCI algebra, since b = c and b → c = c b = 1 (that is, (psBCI 5 ) does not hold in X).
(ii) Let X = {1, a, b, c}. Define binary operations → and on X by the following tables: Then X = (X; →, , 1) is a dual pseudo-Q algebra which is not a pseudo-CI algebra, because b → c = 1 but b c = c. Proposition 3.6 Let X be a dual pseudo-Q algebra. If one of the following identities: holds in X, then X is a trivial algebra.
Proposition 3.7 Let X be a dual pseudo-Q algebra. If one of the following identities: holds in X, then X is a trivial algebra.
Proof. The proof is similar to the proof of Proposition 3.6.

Proposition 3.8
In a dual pseudo-Q algebra X, for all x, y, z ∈ X, we have: A. Rezaei, A. Borumand Saeid and A. Walendziak Similarly, if 1 ≤ x, then x = 1.
(6) The proof is similar to the proof of (5).
The converse is obvious.

Proposition 3.18
Let X be a dual pseudo-QC algebra and x, y, z ∈ X such that x ≤ y and y ≤ z.

Corollary 3.19
If a dual pseudo-QC algebra X satisfies the condition (psBCI 5 ), then (X; ≤) is a poset.
Theorem 3.20 If X is a commutative dual pseudo-QC algebra, then it is a pseudo-BCI algebra.
Proof. It is sufficient to prove that (psBCI 5 ) holds in X. Let x, y ∈ X and x → y = y x = 1. Then Therefore, X satisfies (psBCI 5 ). Thus X is a pseudo-BCI algebra.
By Proposition 2.7 and Corollary 3.21, commutative pseudo-QC algebras coincide with commutative algebras pseudo-BCK, -BCI, -BCH, -CI, -BE. Now, in the following diagram we summarize the results of this paper and the previous results in this filed. An arrow indicates proper inclusion, that is, if X and Y are classes of algebras, then X → Y denotes X ⊂ Y. The mark X C → Y means that every commutative algebra of X belongs to Y.
Is it true that every commutative dual pseudo-Q algebra is a pseudo-BCK algebra?