On a generalization of KU-algebras pseudo-KU algebras

As a generalization of KU-algebras, the notion of pseudo-KU algebras is introduced in 2020 by the author (D. A. Romano. Pseudo-UP algebras, An introduction. Bull. Int. Math. Virtual Inst., 10(2)(2020), 349-355). Some characterizations of pseudo-KU algebras are established in that article. In addition, it is shown that each pseudo-KU algebra is a pseudo-UP algebra. In this paper it is a concept developed of pseudo-KU algebras in more detail and it has identified some of the main features of this type of universal algebras such as the notions of pseudo-subalgebras, pseudo-ideals, pseudo-filters and pseudo homomorphisms. Also, it has been shown that every pseudo-KU algebra is a pseudo-BE algebra. In addition, a congruence was constructed on a pseudo-KU algebra generated by a pseudo-ideal and shown that the corresponding factor-structure is and pseudo-KU algebra as well.


Introduction
T he concept of pseudo-BCK algebras was introduce in [1] by Georgescu and Iorgulescu as an extension of BCK-algebras. The notion of pseudo-BCI algebras was introduced and analyzed in [2] by Dudek and Jun as a generalization of BCI-algebras. The concept of pseudo-BE algebras was introduced in 2013 and their properties were explored by Borzooei et al., in [3]. These algebraic structures has been in the focus of many authors (for example, see [4][5][6][7][8][9][10]). Pseudo BL-algebras are a non-commutative generalization of BL-algebras introduced in [11]. Pseudo BL-algebras are intensively studied by many authors (for example, [12][13][14]).  in [15,16] introduced a new algebraic structure which is called KU-algebras. They studied ideals and congruences in KU-algebras. They also introduced the concept of homomorphism of KU-algebras and investigated some related properties. Moreover, they derived some straightforward consequences of the relations between quotient KU-algebras and isomorphism. Many authors took part in the study of this algebraic structure (for example: [17,18]).
A detailed listing of the researchers and their contributions to these activities it can be found in [19]. Here, we will highlight the contribution of [20]. In [21], Kim and Kim introduced the concept of BE-algebras as a generalization of dual BCK-algebras. This class of algebra was also studied by Rezaei and Saeid 2012 in article [22]. In the article [20], the authors (Rezaei, Saeod and Borzooei) proved that a KU-algebra is equivalent to a commutative self-distributive BE-algebra. (A BE-algebra A is a self-distributive if x · (y · z) = (z · y) · (x · z) for all x, y, z ∈ A.) Additionally, they proved that every KU-algebra is a BE-algebra ( [20], Theorem 3.4), every Hilbert algebra is a KU-algebra ( [20], Theorem 3.5) and a self-distributive KU-algebra is equivalent to a Hilbert algebra ( [20]). Iampan constructed PU-algebra as a generalization of KU-algebra in [19] in 2017 and showed that each KU-algebra is a PU-algebra.
In article [23], the author designed the concepts of pseudo-UP ( [23], Definition 3.1) and pseudo-KU-algebras ( [23], Definition 4.1) and showed that each pseudo-KU algebra is a pseudo-UP algebra ( [23], Theorem 4.1). However, the term 'pseudo KU-algebra' and mark 'PKU' has already been used in [24] for different purposes. It should be noted here that this term 2019 has been renamed to 'JU-algebra' ( [25]). Although introducing the term 'pseudo-KU algebra' as a name for a structure constructed in the manner described here and using the abbreviation 'pKU' for this algebra can lead to confusion, we did it for needs of article [23] and of this paper.
In this paper we develop the concept in more detail of pseudo-KU algebras and we identify some of the main features of this type of universal algebras. The paper was designed as follows: After the Section 2, which outlines the necessary previous terms, Section 3 introduces the concept of pseudo-KU algebra and analyzes some of its important properties. In Section 4, the concept of pseudo-KU algebras is linked to the concepts of pseudo-UP and pseudo-BE algebras. Section 5 deals with some substructures of this class of algebras such as pseudo-subalgebras, pseudo-ideals and pseudo-filters. Finally, in Section 6, the concepts of pseudo-homomorphisms and congruences on pseudo-KU algebras are analyzed.

Preliminaries
In this section we will describe some elements of KU-algebras from the literature [15,16] necessary for our intentions in this text. Definition 1. ( [15]) An algebra A = (A, ·, 0) of type (2, 0) is called a KU-algebra where A is a nonempty set, · is a binary operation on A, and 0 is a fixed element of A (i.e. a nullary operation) if it satisfies the following axioms: On a KU-algebra A = (A, ·, 0), we define the KU-ordering on A as follows ( [15], pp. 56): (∀x, y ∈ A)(x y ⇐⇒ y · x = 0).

Definition 2.
( [15]) Let S be a non-empty subset of a KU-algebra A.
(a) The subset S is said to be a KU-subalgebra of A if (S, ·, 0) is a KU-algebra. (b) The subset S is said to be an ideal of A if it satisfies the following conditions: (J1) 0 ∈ S, and (J2) (∀x, y, z ∈ A)((x · (y · z) ∈ S ∧ y ∈ S) =⇒ x · z ∈ S).
As shown in [18], this kind of algebra satisfies one specific equality.

Lemma 2 ([18]
). In a KU-algebra A, the following holds: In the light of the previous equality, condition (J2) is transformed into condition: Indeed, if we put x = 0, y = x and z = y in (J2), we immediately obtain (J3) by (KU-2). Conversely, let (J3) be a valid formula and let x, y, z ∈ A be arbitrary elements such that x · (y · z) ∈ J and y ∈ J. Then y · (x · z) ∈ J by (KU-5). Thus x · z ∈ J by (J3).
We can introduce the concept of filters in KU-algebra if formula (J3) serves as a motivation.

Definition 3.
The subset F is said to be a filter of A if it satisfies the following conditions: A filter in KU-algebra, designed in this way, has the following property:

Remark 1.
We emphasize that in pseudo-KU algebra the relation of the order is determined inversely with respect to the definition of the order in the KU-algebra.
Proof. If we put x = 0, y = 0, and z = x in the formula (pKU-1), we get From where we get with respect to (pKU-2).
Proof. Let x, y, z ∈ A such that x y. Then x · y = 0 = x * y. If we put x = y and y = x in (pKU-1), we get and y * z z * x.
On the other hand, if we put z = y and y = z in (pKU-1), we have This means z · x z · y. It can be similarly proved that it is z * x z * y.
In 2011, Mostafa, Naby and Yousef proved Lemma 2.2 in [18]. In the following Proposition, we show that analogous equality is also valid in pseudo-KU algebras.

Proposition 2. In pseudo-KU algebra
is valid formula.
Proof. If we put y = 0 in (pKU-1), we have Then, we have x (x · z) * z. From here it follows by (11). On the other hand, if we put x = z · z in (pKU-1), we get Since the variables x, y, z ∈ A are free variables, if we put x = y and y = x, we get an inverse inequality. From here it follows (pKU) by (pKU-4).
The other equality can be proved in an analogous way.

Correlation of pseudo-KU algebras with other types of pseudo algebras
The notion of pseudo-UP algebra as a generalization of the concept of UP-algebras was introduced and analyzed in [23].
is a binary relation on a set A, · and * are internal binary operations on A and 0 is an element of A, verifying the following axioms: The following theorem is an important result of pseudo-KU algebras for study in the connections between pseudo-UP algebras and pseudo-KU algebras.

Theorem 1. Any pseudo-KU algebra is a pseudo-UP algebra.
Proof. It only needs to show (pUP-1). By Proposition 2, we have that any pseudo-KU algebra satisfies (pUP-1).
Pseudo-BE algebra is defined by the follows: If we replace 1 with 0 in (BE-1), (BE-2), (BE-3) and (BE-5) and prove that the formula (pBE-4) is a valid formula in a pseudo-KU algebra A, we have proved that every pseudo-KU algebra A is a pseudo-BE algebra.
Theorem 2. Any pseudo-KU algebra is a pseudo-BE algebra.
Proof. It is sufficient to prove that the formula (pBE-4) is a valid formula in any pseudo-KU algebra. If we put y = 0 in the left-hand side of the formula (pKU-1), we get 0 · by the left part of formula (11). On the other hand, if we put x = x · z in the right-hand side of the formula (pKU-1), we get Which together with the previous inequality gives From this inequality by substituting the variables x and y, we obtain the necessary reverse inequality From these two inequalities follows the validity of the formula (pBE-4) in any pseudo-KU algebra by the axiom (pKU-4).
Since the formula previously proven is important below, we point it out in particular.

Proposition 3. In any pseudo-KU algebra
is a valid formula.

Concept of pseudo-subalgebras
holds.
Putting y = x in the previous definition, it immediately follows: Lemma 6. If S is a pseudo-subalgebra of a pseudo-KU algebra A, then 0 ∈ S.
Proof. Let S be a pseudo-subalgebra of a pseudo-KU algebra A. It means that S is a nonempty subset of A.
Then there exists an element y ∈ S. Thus 0 = y · y = y * y ∈ S by Definition 7.
It is clear that subsets {0} and A are pseudo-subalgebras of a pseudo-KU algebras A. So, the family S(A) of all pseudo-subalgebras of a pseudo-KU algebra A is not empty. Without major difficulties, the following theorem can be proved.

Concept of pseudo-ideals
Definition 8. The subset J is said to be a pseudo-ideal of a pseudo-KU algebra A if it satisfies the following conditions: Proposition 4. Let J be a nonempty subset of a pseudo-KU algebra A. Then the condition (pJ3a) is equivalent to the condition: (pJ4a) (∀x, y, z ∈ A)((x * (y · z) ∈ J ∧ y ∈ J) =⇒ x * z ∈ J).
Proof. Putting x = y and y = x * z in the condition (pJ3a), it immediately follows .

Proposition 5.
Let J be a nonempty subset of a pseudo-KU algebra A. Then the condition (pJ3b) is equivalent to the condition Proof. If we put x = y and y = x · z in (pJ3b), we get by (pKU-7). Conversely, if we put x = 0, y = x, and z = y in (pJ4b), we get Thus (pJ3b) with respect to (pKU-2).
The following important statement describes the connection between conditions (pJ3a) and (pJ3b).
Now, x · y ∈ J and (x · y) * y it follows y ∈ J. This proves the validity of the formula (pJ3a).

Proposition 7.
Any pseudo-ideal in a pseudo-KU-algebra A is a pseudo-subalgebra in A.
Proof. The proof of this proposition follows from (13) and (14).

Theorem 6. The family J(A) of all pseudo-ideals in a pseudo-KU algebra A forms a complete lattice and J(A) ⊆ S(A)
holds.
Proof. Let {J i } i∈I be a family of pseudo-ideals in a pseudo-KU algebra A. Clearly 0 ∈ i∈I J i is valid. Let x, y ∈ A be elements such that x · y ∈ i∈I J i , x * y ∈ i∈I J i and x ∈ i∈I J i . Then x · y ∈ J i , x * y ∈ J i and x ∈ F i for any i ∈ I. Thus y ∈ J i because J i is a pseudo-ideal in A and x ∈ i∈I J i . So, i∈I J i is a pseudo-ideal in A. If X is the family of all pseudo-ideals of A that contain the union i∈I J i , then ∩X is also a pseudo-ideal in A that contains i∈I J i by previous evidence.
If we put i∈I J i = i∈I J i and i∈I J i = ∩X, then (J(A), , ) is a complete lattice.
To round out this subsection we need the following lemma.

Lemma 7.
Let J be a pseudo-ideal in a pseudo-KU algebra A. Then Proof. The proof of this proposition follows from (pJ3a) (or (pJ3b)) with respect to (pKU-6) and (pJ1).

Theorem 7.
Let J be a subset of a pseudo-KU algebra A such that 0 ∈ J. Then, J is a pseudo-ideal in A if and only if the following holds Proof. Let J be a pseudo-ideal in A and let x, y, z ∈ A such that x ∈ J, y ∈ J and x y · z. Then x · (y · z) = 0 ∈ J. Thus y · z ∈ J by (pJ3a) and again, from here and y ∈ J it follows z ∈ J. So, we have shown that (pJ5) is a valid formula.
Opposite, suppose that (pJ5) is a valid in A. Let us show that J is a pseudo-ideal and A. Let x, y ∈ A be such that x ∈ J and x · y ∈ J. Then x * y ∈ J by Proposition 6. On the other hand, from x · ((x * y) · y) = 0, i.e. from x (x * y) · y it follows y ∈ J by hypothesis. So, the set J is a pseudo-ideal in A.
For a relation on the set A we say that it is a quasi-order relation on A if it is reflexive and transitive. It is easy to prove that if σ is a quasi-order relation on A, then the relation σ ∩ σ −1 is an equivalence on A.
Theorem 8. Let J be a pseudo-ideal in a pseudo-KU algebra A. Then the relation , defined by is a quasi-order in the set A left compatible and right reverse compatible with the internal operations in A.
Proof. Since x · x = 0 ∈ J is valid in A for any x ∈ A, it is clear that is a reflexive relation in the set A. Let x, y, z ∈ A be arbitrary elements such that x y and y z. This means x · y ∈ J and y · z ∈ J. From inequality (pKU-1) in the form x · y (y · z) * (x · z) and x · y ∈ J it follows (y · z) * (x · z) ∈ J according to (15). From here and from y · z ∈ J it follows x · z ∈ J according to (pJ3a). Hence, the relation is transitive. So, this relation is a quasi-order in A.
Let x, y, z ∈ A be such x y. Then x · y ∈ J and x * y ∈ J.
(i) If we put x = y and y = x in the left part of the formula (pKU-1), we get x · y (y · z) * (x · z). Now, from here and x · y ∈ J it follows (y · z) * (x · z) ∈ J by (15). Thus (y · z) · (x · z) ∈ J by Proposition 6. Finally, we have y · z x · z. So, the relation is reverse right compatible with the internal operation · in A. (ii) If we put x = y and y = x in the right part of the formula (pKU-1), we get x * y (y * z) · (x * z). Then (y * z) · (x * z) ∈ J by (15). Thus y * z x * z. Therefore, the relation is reverse right compatible with the internal operation * in A.
(iii) Let us put y = z and z = y in the left part of the formula (pKU-1). We get (z · x) * ((x · y) * (z · y)) = 0 ∈ J. From here and from x · y ∈ J it follows (z · x) * (z · y) ∈ J by (pJ4a). Thus z · x z · y. So, the relation is left compatible with the operation · .
(iv) Let us put y = z and z = y in the right part of the formula (pKU-1). We get (z * x) · ((x * y) · (z * y)) = 0 ∈ J. From here and from x * y ∈ J it follows (z * x) · (z * y) ∈ J by (pJ4b). Thus z * x z * y. So, the relation is left compatible with the operation * .

Concept of pseudo-filters
Definition 9. A non-empty subset F of a pseudo-KU algebra A is called a pseudo-filter of A if it satisfies in the following axioms: {0} and A are pseudo-filters of A. So, the family F(A) of all pseudo-filters in a pseudo-KU algebra A is not empty.
It is obviously the following is valid Lemma 8. Let F be a pseudo-filter in a pseudo-KU algebra A. Then Theorem 9. The family F(A) of all pseudo-ideals in a pseudo-KU algebra A forms a complete lattice.
Proof. Let {F i } i∈I be a family of pseudo-filters in a pseudo-KU algebra A. Clearly 0 ∈ i∈I F i is valid. Let x, y ∈ A be elements such that x · y ∈ i∈I F i , x * y ∈ i∈I F i and y ∈ i∈I F i . Then x · y ∈ F i , x * y ∈ F i and y ∈ F i for any i ∈ I. Thus x ∈ F i because F i is a pseudo-filter in A and x ∈ i∈I F i . So, i∈I F i is a pseudo-filter in A. If X is the family of all pseudo-filters of A that contain the union i∈I F i , then ∩X is also a pseudo-filter in A that contains i∈I F i by previous evidence.
If we put i∈I F i = i∈I F i and i∈I F i = ∩X, then (F(A), , ) is a complete lattice.

Remark 2.
Note that if f : A −→ B is a pseudo homomorphism, then f (0 A ) = 0 B . Indeed, if we chose y = x, from the previous formula we immediately get f (0 A ) = B 0 B with respect (pKU-6).
From here it immediately follows: Lemma 9. Any pseudo-homomorphism between pseudo-KU algebras is isotone mapping.
Proof. Let f : A −→ B be a pseudo-homomorphism between pseudo-KU algebras and let x, y ∈ A be such . This means f (x) B f (y).

Proof. It is obvious 0
The implication of x * A y ∈ Ker( f ) ∧ x ∈ Ker( f ) =⇒ y ∈ Ker( f ) can be proved by analogy with the previous proof.
The following statement is easy to prove: Lemma 11. If f : A −→ B is a pseudo-homomorphism between pseudo-KU algebras, then f (A) is a pseudo-subalgebra in B.
Proposition 8. Let f : A −→ B be a pseudo homomorphism between pseudo-KU algebras A and B.
(i) If K is a pseudo-ideal in B, then f − (K) is a pseudo-ideal in A.
(ii) If G is a pseudo-filter in B, then f −1 (G) is a pseudo-filter in A.
Proof. (i) Assume that K is a pseudo-fulter of B. Obviously 0 A ∈ f −1 (K). Let x, y ∈ A be such x · y ∈ f −1 (K) and x ∈ f −1 (K). Then f (x) · B f (y) = B f (x · A y) ∈ K and f (x) ∈ K. It follows that f (y) ∈ K by (pJ3a) since K is a pseudo-ideal in B. Therefore, y ∈ f −1 (K). Thus, the set f −1 (K) satisfies the implication (pJ3a). That the set f −1 (K) satisfies the implication (pJ3b) can be proved in an analogous way. Therefore, the set In the following definition, we will introduce the concept of congruence on pseudo-KU algebras. Since we have two unitary operations on this algebra, it is possible to determine three different types of congruences. A = ((A, ), ·, * , 0) be a pseudo-KU algebra.

Definition 11. Let
For the equivalence relation q on the set A we say that it is a congruence of type · on A if it compatible with the operations · in A in the following sense (17) (∀x, y, z ∈ A)((x, y) ∈ q =⇒ ((x · z, y · z) ∈ q ∧ (z · x, z · y) ∈ q))).
For the equivalence relation q on the set A we say that it is a congruence of type * on A if it compatible with the operations * in A in the following sense (18) (∀x, y, z ∈ A)((x, y) ∈ q =⇒ ((x * z, y * z) ∈ q ∧ (z * x, z * y) ∈ q))).
For the equivalence relation q on the set A we say that it is a congruence of common type on A if it is compatible with both operations in A.
Let f : A −→ B be a pseudo homomorphism between pseudo-KU algebras. By direct check without difficulty, it can be proved that the relation q f , defined by is a congruence (all three types) on A.

Theorem 10.
The relation q f is a congruence of type · (type * , common type) on the pseudo-KU algebra A.
Proof. We will only demonstrate the proof that q f is a congruence of type · on A because the evidence that q f is a congruence of type * can obtain by analogy with the previous one, and the proof of common type is obtained by combining this two evidences.
Clearly, q f is an equivalence relation on the set A. It remains to verify that (16) is a valid formula in A. Let x, y, u, v ∈ A be such that (x, y) ∈ q f and (u, v) ∈ q f . Then f (x) = B f (y) and f (u) = B f (v). Thus Hence, (x · A u, y · A v) ∈ q f . We proved that (17a) is a valid formula. So q f is a congruence of type · on A.
Theorem 11. Let J be a pseudo-ideal in a pseudo-KU algebra A. Then the relation q J , defined by q J = ∩ −1 , is a congruence of common type in A.
Proof. The relation q is an equivalence relation on the set A. It is sufficient to prove that q is compatible with operations in A. Since the relation is left compatible and right reverse compatible with the internal operations in A, by Theorem 8, it is clear that the relation q J is a congruence on A.
For a congruence q on a pseudo-KU algebra A we denote qx = {y ∈ A : (x, y) ∈ q} = [x]. Let Without much difficulty it can be verified that the functions • and , defined in this way, are well-defined internal binary operations in A/q. Also, one can check that the set A/q with the operations • and , determined as above, satisfies all the axioms of Definition 4 except the axiom (pKU-4). However, if we take the relation q J , defined by an pseudo-ideal J of a pseudo-KU algebra A, then we have Analogous to the previous one may be shown that y · A x ∈ J holds. Thus, (x, y) ∈ q f =⇒ (x, y) ∈ q J is valid.
We end this section with the following theorem. Since this theorem can be proven by direct verification, we will omit evidence for it.
Theorem 13. Let f : A −→ B be pseudo-homomorphism between pseudo-KU algebras ((A, A ), · A , * A , 0 A ) and ((B, B ), · B , * B , 0 B ). Then there exists the unique epimorphism π : A −→ A/q f , defined by π(x) = [x] for any x ∈ A, and the unique monomorphism g : A/q f −→ B, defined by g([x]) = B f (x) for any x ∈ A such that f = g • π.