On a generalization of KU-algebras pseudo-KU algebras

1. Introduction

The 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]).

Prabpayak and Leerawat 2009 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 \(x \cdot (y \cdot z) = (z \cdot y) \cdot (x \cdot z)\) for all \(x, y, z \in A\).) 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.

2. Preliminaries

In this section we will describe some elements of KU-algebras from the literature [1]Pra09, Pra09a} necessary for our intentions in this text.

Definition 1. ([15]) An algebra \(A = (A, \cdot, 0)\) of type \((2, 0)\) is called a KU-algebra where \(A\) is a nonempty set, \('\cdot '\) 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:
(KU-1) \((\forall x, y, z \in A)((x \cdot y) \cdot ((y \cdot z) \cdot (x \cdot z)) = 0)\),
(KU-2) \((\forall x \in A)(0 \cdot x = x)\),
(KU-3) \((\forall x \in A)(x \cdot 0 = 0)\), and
(KU-4) \((\forall x, y \in A)((x \cdot y = 0 \wedge y \cdot x = 0) \Longrightarrow x = y)\).

On a KU-algebra \(A = (A, \cdot, 0)\), we define the KU-ordering \(\leqslant\) on \(A\) as follows ([15], pp. 56): \[(\forall x, y\in A)(x \leqslant y \Longleftrightarrow y \cdot x = 0).\]

Lemma 1. In a KU-algebra \(A\), the following properties hold:
(1) \((\forall x \in A)(x \leqslant x)\),
(2) \((\forall x, y \in A)((x \leqslant y \wedge y \leqslant x), \Longrightarrow x = y)\),
(3) \((\forall x, y, z \in A)((x \leqslant y \wedge y \leqslant z) \Longrightarrow x \leqslant z)\),
(4) \((\forall x, y, z \in A)(x \leqslant y \Longrightarrow z \cdot x \leqslant z \cdot y)\),
(5) \((\forall x, y, z \in A)(x \leqslant y \Longrightarrow y \cdot z \leqslant x \cdot z)\),
(6) \((\forall x, y \in A)(x \cdot y \leqslant y)\) and
(7) \((\forall x \in A)(0 \leqslant x)\).

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 \((S,\cdot,0)\) is a KU-algebra.
(b) The subset \(S\) is said to be an ideal of \(A\) if it satisfies the following conditions:
(J1) \(0 \in S\), and
(J2) \((\forall x, y, z \in A)((x \cdot (y \cdot z) \in S \wedge y \in S) \Longrightarrow x \cdot z \in S)\).

As shown in [18], this kind of algebra satisfies one specific equality.

Lemma 2. [18] In a KU-algebra \(A\), the following holds:
(KU-5) \((\forall x, y, z \in A)(z \cdot (y \cdot x) = y \cdot (z \cdot x))\).

In the light of the previous equality, condition (J2) is transformed into condition:
(J3) \((\forall x, y \in A)((x \cdot y \in S \wedge x \in S) \Longrightarrow y \in S)\).
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 \(x, y, z \in A\) be arbitrary elements such that \(x \cdot (y \cdot z) \in J\) and \(y \in J\). Then \(y \cdot (x \cdot z) \in J\) by (KU-5). Thus \(x \cdot z \in J\) by (J3).
From (J3) it immediately follows:

Lemma 3. Let \(S\) be an ideal in a KU-algebra \(A\). Then
(J4) \((\forall x, y \in A)((x \leqslant y \wedge y \in S) \Longrightarrow x \in S)\).

We can introduce the concept of filters in KU-algebra if formula (J3) serves as a motivation.

Definition 2. The subset \(F\) is said to be a filter of \(A\) if it satisfies the following conditions:
(F1) \(0 \in F\), and
(F3) \((\forall x, y \in A)((x \cdot y \in F \wedge y \in F) \Longrightarrow x \in F)\).

A filter in KU-algebra, designed in this way, has the following property:

Lemma 4. Let \(F\) be a filter in a KU-algebra \(A\). Then
(F4) \((\forall x, y \in A)((x \leqslant y \wedge x \in F) \Longrightarrow y \in F)\).

3. Concept of pseudo-KU algebra

Definition 4.([23]) An algebra \(\mathfrak{A} = ((A,\leqslant),\cdot,\ast, 0)\) of type \((2,2,0)\) is called a pseudo-KU algebra if it satisfies the following axioms:
(pKU-1): \((\forall x, y, z \in A)((y \cdot x)\leqslant ((x \cdot z)\ast (y \cdot z)) \wedge (y \ast x)\leqslant ((x \ast z)\cdot (y \ast z))),\)
(pKU-2): \((\forall x \in A)((0 \cdot x = x) \wedge (0 \ast x = x))\),
(pKU-3): \((\forall x \in A)(x \leqslant 0)\),
(pKU-4): \((\forall x, y \in A)((x \leqslant y \wedge y \leqslant x) \Longrightarrow x = y)\), and
(pKU-5): \((\forall x, y\ \in A)((x \leqslant y \Longleftrightarrow x \cdot y = 0) \wedge (x \leqslant y \Longleftrightarrow x \ast y = 0))\).

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.

Lemma 5. If \(\mathfrak{A}\) is a pseudo-KU algebra, then (pKU-6) \((\forall x \in A)((x \cdot x = 0) \wedge (x \ast x = 0))\).

Proof. If we put \(x = 0\), \(y = 0\), and \(z = x\) in the formula (pKU-1), we get \[(0\cdot 0)\ast ((0\cdot x)\star (0\cdot x)) = 0 \wedge (0\ast 0)\cdot ((0\ast x)\cdot (0 \ast x)) = 0.\] From where we get \[x \cdot x = 0 \wedge x \ast x = 0\] with respect to (pKU-2).

Proposition 1. If \(\mathfrak{A}\) is a pseudo-KU algebra, then
(11) \((\forall x, y, z \in A)(x \leqslant y \Longrightarrow ((y \cdot z \leqslant x \cdot z) \wedge (y \ast z \leqslant x \ast z))\) and
(12) \((\forall x, y, z \in A)(x \leqslant y \Longrightarrow ((z \cdot x \leqslant z \cdot y) \wedge (z \ast x \leqslant z \ast y))\).

Proof. Let \(x, y, z \in A\) such that \(x \leqslant y\). Then \(x \cdot y = 0 = x \ast y\). If we put \( x = y \) and \( y = x \) in (pKU-1), we get \[0 = (x \cdot y)\ast ((y \cdot z)\ast (x \cdot z)) = 0 \ast ((y \cdot z)\ast (x \cdot z)) = (y \cdot z)\ast (x \cdot z).\] So, we have \(y \cdot z \leqslant x \cdot z\). Similarly, we have \[0 = (x \ast y)\cdot ((y \ast z)\cdot (x \ast z)) = 0 \cdot ((y \ast z)\cdot (x \ast z)) = (y \ast z)\cdot (x \ast z)\] and \(y \ast z \leqslant z \ast x\). On the other hand, if we put \( z = y \) and \( y = z \) in (pKU-1), we have \[0 = (z \cdot x)\ast ((x \cdot y)\ast (z \cdot y))) = (z \cdot x)\ast ((0 \ast (z \cdot y)) = (z \cdot x)\ast (z \cdot y).\] This means \(z \cdot x \leqslant z \cdot y\). It can be similarly proved that it is \(z \ast x \leqslant z \ast y\).

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 \(\mathfrak{A}\), then
(pKU) \((\forall x, y, z \in A)(x \ast (y \cdot z)= y \ast (x\cdot z) \wedge x \cdot (y \ast z) = y \cdot (x\ast z))\) is valid formula.

Proof. If we put \(y = 0\) in (pKU-1), we have \[0\cdot x \leqslant (x \cdot z)\ast (0\cdot z).\] Then, we have \(x \leqslant (x \cdot z)\ast z\). From here it follows \[((x \cdot z)\ast z)\cdot (y\ast z) \leqslant x \cdot (y\ast z)\] by (11). On the other hand, if we put \(x = z\cdot z\) in (pKU-1), we get \[y \ast (x \cdot z) \leqslant ((x \cdot z) \ast z)\cdot (y \ast z) \leqslant x \cdot (y\ast z).\] Since the variables \(x, y, z \in A\) 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.

4. 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].

Definition 5.([23]) A pseudo-UP algebra is a structure \(\mathfrak{A} = ((A,\leqslant),\cdot,\ast, 0)\), where \('\leqslant '\) is a binary relation on a set \(A\), \('\cdot '\) and \('\ast '\) are internal binary operations on \(A\) and \('0 '\) is an element of \(A\), verifying the following axioms:
(pUP-1) \((\forall x, y z \in A)(y \cdot z \leqslant (x \cdot y)\ast (x \cdot z) \wedge y \ast z \leqslant (x \ast y)\cdot(x \ast z))\);
(pUP-4) \((\forall x, y \in A)((x \leqslant y \wedge y \leqslant x) \Longrightarrow x = y)\);
(pUP-5) \((\forall x, y \in A)((y \cdot 0)\ast x = x \wedge (y\ast0)\cdot x = x)\) and
(pUP-6) \((\forall x,y \in A)((x\leqslant y \Longleftrightarrow x \cdot y = 0) \wedge (x \leqslant y \Longleftrightarrow x \ast y = 0))\).

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:

Definition 6. ([3]) An algebra \(A = (A,\cdot,\ast,1)\) of type \((2, 2, 0)\) is called a pseudo BE-algebra if satisfies in the following axioms:
(pBE-1) \((\forall x \in A)(x \cdot x = 1 \wedge x \ast x = 1)\);
(pBE-2) \((\forall x \in A)(x \cdot 1 = 1 \wedge x \ast 1 = 1)\);
(pBE-3) \((\forall x \in A)(1 \cdot x = x \wedge 1 \ast x = x)\);
(pBE-4) \((\forall x, y, z \in A)(x \cdot (y \ast z) = y \ast (x \cdot z))\); and
(pBE-5) \((\forall x, y \in A)(x \cdot y = 1 \Longleftrightarrow x \ast y = 1)\).

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 \(0 \cdot x \leqslant ((x \cdot z) \ast (0 \cdot z)\). It means \(x \leqslant (x \cdot z)\ast z\). From here follows \[((x \cdot z) \ast z)\cdot (y \ast z) \leqslant x \cdot (y \ast z),\] by the left part of formula (11). On the other hand, if we put \(x = x \cdot z\) in the right-hand side of the formula (pKU-1), we get \[y \ast (x \cdot z) \leqslant ((x \cdot z) \ast z)\cdot (y \ast z).\] Which together with the previous inequality gives \[y \ast (x \cdot z) \leqslant x \cdot (y \ast z).\] From this inequality by substituting the variables \(x\) and \(y\), we obtain the necessary reverse inequality \[x \cdot(y \ast z) \leqslant y \ast (x \cdot z).\] 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 \(\mathfrak{ A }\),
(pKU-7) \((\forall x, y, z \in A)(x \cdot (y \ast z) = y \ast (x \cdot z))\) is a valid formula.

5. Some substructures in pseudo-KU algebras

5.1. Concept of pseudo-subalgebras

Definition 7. A nonempty subset \(S\) of a pseudo-KU algebra \(A\) is a pseudo-subalgebra in \(\mathfrak{A}\) if \[(\forall x, y \in A)((x \in S \wedge y \in S) \Longrightarrow (x \cdot y \in S \wedge x \ast y \in S)).\] holds.

Putting \(y = x\) in the previous definition, it immediately follows:

Lemma 6. If \(S\) is a pseudo-subalgebra of a pseudo-KU algebra \(\mathfrak{A}\), then \(0 \in S\).

Proof. Let \(S\) be a pseudo-subalgebra of a pseudo-KU algebra \(\mathfrak{A}\). It means that \(S\) is a nonempty subset of \(A\). Then there exists an element \(y \in S\). Thus \(0 = y \cdot y = y \ast y \in S\) by Definition 7.

It is clear that subsets \(\{0 \}\) and \(A\) are pseudo-subalgebras of a pseudo-KU algebras \(\mathfrak{A}\). So, the family \(\mathfrak{S}(A)\) of all pseudo-subalgebras of a pseudo-KU algebra \(\mathfrak{A}\) is not empty. Without major difficulties, the following theorem can be proved.

Theorem 3. The family \(\mathfrak{S}(A)\) of all pseudo-subalgebras of a pseudo-KU algebra \(\mathfrak{A}\) forms a complete lattice.

5.2. Concept of pseudo-ideals

Definition 8. The subset \(J\) is said to be a pseudo-ideal of a pseudo-KU algebra \(\mathfrak{A}\) if it satisfies the following conditions:
(pJ1) \(0 \in J\),
(pJ3a) \((\forall x, y \in A)((x \cdot y \in J \wedge x \in J) \Longrightarrow y \in J)\) and
(pJ3b) \((\forall x, y \in A)((x \ast y \in J \wedge x \in J) \Longrightarrow y \in J)\).

Proposition 4. Let \(J\) be a nonempty subset of a pseudo-KU algebra \(\mathfrak{A}\). Then the condition \((pJ3a)\) is equivalent to the condition:
(pJ4a) \((\forall x, y , z \in A)((x \ast (y \cdot z) \in J \wedge y \in J) \Longrightarrow x \ast z \in J)\).

Proof. Putting \(x = y\) and \(y = x \ast z\) in the condition (pJ3a), it immediately follows \[(\forall x, y, z \in A)((y \cdot (x \ast z)\in J \wedge y \in J) \Longrightarrow x \ast z \in J).\] Thus \[(\forall x, y, z \in A)((y \ast (x \cdot z)\in J \wedge y \in J) \Longrightarrow x \ast z \in J)\] by (pKU-7).
Conversely, let (pJ4a) it be. Let us choose \(x = 0\), \(y = x\) and \(z = y\) in (pJ4a). We get \((0 \ast (x \cdot y) \in J \wedge x \in J) \Longrightarrow 0 \ast y \in J\). Thus (pJ3a) by (pKU-2).

Corollary 4.Let \(J\) be a pseudo-ideal in a pseudo-KU-algebra \(\mathfrak{A}\). Then
(13) \((\forall x, y \in A)(y \in J \Longrightarrow x \ast y \in J)\).

Proof. Putting \(z = y\) in (pJ4a), with respect to (pKU-6), (pKU-3) and (pJ1), we obtain (13).

Proposition 5. Let \(J\) be a nonempty subset of a pseudo-KU algebra \(\mathfrak{A}\). Then the condition \((pJ3b)\) is equivalent to the condition
(pJ4b) \((\forall x, y, z \in A)((x \cdot (y \ast z) \in J \wedge y \in J) \Longrightarrow x \cdot z)\).

Proof. If we put \( x = y \) and \( y = x \cdot z\) in (pJ3b), we get \[(y \ast (x \cdot z) \in J \wedge y \in J) \Longrightarrow x \cdot z \in J.\] Hence \[(x \cdot (x \ast z) \in J \wedge y \in J) \Longrightarrow x \cdot z \in J.\] by (pKU-7). Conversely, if we put \(x = 0\), \(y = x\), and \(z = y\) in (pJ4b), we get \[(0 \cdot (x \ast y) \in J \wedge x \in J) \Longrightarrow 0 \cdot y \in J.\] Thus (pJ3b) with respect to (pKU-2).

Corollary 5.Let \(J\) be a pseudo-ideal in a pseudo-KU-algebra \(\mathfrak{A}\). Then
(14) \((\forall x, y \in A)(y \in J \Longrightarrow x \cdot y \in J)\).

Proof. Putting \(z = y\) in (pJ4b), with respect to (pKU-6), (pKU-3) and (pJ1), we obtain (14).

The following important statement describes the connection between conditions (pJ3a) and (pJ3b).

Proposition 6. Let \(J\) be a pseudo-ideal of a pseudo-KU algebra \(\mathfrak{A}\). Then \[(pJ3a) \Longleftrightarrow (pJ3b).\]

Proof. \((pJ3a) \Longleftrightarrow (pJ3b)\). Suppose (pJ3a) holds and let \(x \ast y \in J\) and \(x \in J\). How obvious it is that the following \[x \ast ((x \cdot y) \ast y) = 0 \Longleftrightarrow x \cdot ((x \cdot y)\star x)= 0 \Longleftrightarrow (x \cdot y)\ast (x \cdot y) = 0\] is valid, we have \[(x \in J \wedge x \cdot ((x \ast u)\cdot y) = 0 \in J) \Longrightarrow (x \ast y) \cdot y \in J.\] Now \[(x \ast y \in J \wedge (x \ast y)\cdot y \in J) \Longrightarrow y \in J.\] We have proved that (pJ3b) is a valid implication. \textbf{\((pJ3b) \Longrightarrow (pJ3a)\).} Let (pJ3b) be a valid formula and let \(x, y \in A\) be such that \(x \in J\) and \(x \cdot y \in J\). As above, from \[x \ast ((x \cdot y)\ast y) = 0 \Longleftrightarrow x \cdot ((x \cdot y) \ast y) = 0 \Longleftrightarrow (x \cdot y)\ast (x \cdot y) = 0 \] it follows \[(x \in J \wedge x \ast ((x \cdot y) \ast y) = 0 \in J) \Longrightarrow (x \cdot y) \ast y \in J.\] Now, \(x \cdot y \in J\) and \((x \cdot y) \ast y\) it follows \(y \in J\). This proves the validity of the formula (pJ3a).

Proposition 7. Any pseudo-ideal in a pseudo-KU-algebra \(\mathfrak{A}\) is a pseudo-subalgebra in \(\mathfrak{A}\).

Proof. The proof of this proposition follows from (13) and (14).

Theorem 6. The family \(\mathfrak{J}(A)\) of all pseudo-ideals in a pseudo-KU algebra \(\mathfrak{A}\) forms a complete lattice and \(\mathfrak{J}(A) \subseteq \mathfrak{S}(A)\) holds.

Proof. Let \(\{J_{i}\}_{i \in I}\) be a family of pseudo-ideals in a pseudo-KU algebra \(\mathfrak{A}\). Clearly \(0 \in \bigcap_{i \in I}J_{i}\) is valid. Let \(x, y \in A\) be elements such that \(x \cdot y \in \bigcap_{i \in I}J_{i}\), \(x \ast y \in \bigcap_{i \in I}J_{i}\) and \(x \in \bigcap_{i \in I}J_{i}\). Then \(x \cdot y \in J_{i}\), \(x \ast y \in J_{i}\) and \(x \in F_{i}\) for any \(i \in I\). Thus \(y \in J_{i}\) because \(J_{i}\) is a pseudo-ideal in \(\mathfrak{A}\) and \(x \in \bigcap_{i \in I}J_{i}\). So, \(\bigcap_{i \in I}J_{i}\) is a pseudo-ideal in \(\mathfrak{A}\). If \(\mathfrak{ X }\) is the family of all pseudo-ideals of \(\mathfrak{ A }\) that contain the union \(\bigcup_{i \in I}J_{i}\), then \(\cap \mathfrak{X} \) is also a pseudo-ideal in \(\mathfrak{ A }\) that contains \(\bigcup_{i \in I}J_{i}\) by previous evidence. If we put \(\sqcap_{i \in I}J_{i} = \bigcap_{i \in I}J_{i}\) and \(\sqcup_{i \in I}J_{i} = \cap \mathfrak{X}\), then \((\mathfrak{J}(A),\sqcap,\sqcup)\) 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 \(\mathfrak{A}\). Then
(15) \((\forall x, y \in A)((x \leqslant y \wedge x \in J) \Longrightarrow y \in J)\).

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 \(\mathfrak{A}\) such that \(0 \in J\). Then, \(J\) is a pseudo-ideal in \(\mathfrak{A}\) if and only if the following holds
(pJ5) \((\forall x, y, z \in A)((x \in A \wedge y \in A \wedge x \leqslant y \cdot z) \Longrightarrow z \in J)\).

Proof. Let \(J\) be a pseudo-ideal in \(\mathfrak{A}\) and let \(x, y , z \in A\) such that \(x \in J\), \(y \in J\) and \(x \leqslant y \cdot z\). Then \(x \cdot (y \cdot z) = 0 \in J\). Thus \(y \cdot z \in J\) by (pJ3a) and again, from here and \(y \in J\) it follows \(z \in J\). So, we have shown that (pJ5) is a valid formula.
Opposite, suppose that (pJ5) is a valid in \(\mathfrak{A}\). Let us show that \(J\) is a pseudo-ideal and \(\mathfrak{ A }\). Let \(x, y \in A\) be such that \(x \in J\) and \(x \cdot y \in J\). Then \(x \ast y \in J\) by Proposition 6. On the other hand, from \(x \cdot ((x \ast y) \cdot y) = 0\), i.e. from \(x \leqslant (x \ast y) \cdot y \) it follows \(y \in J\) by hypothesis. So, the set \(J\) is a pseudo-ideal in \(\mathfrak{A}\).

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 \(\sigma\) is a quasi-order relation on \(A\), then the relation \(\sigma \cap \sigma^{-1}\) is an equivalence on \(A\).

Theorem 8. Let \(J\) be a pseudo-ideal in a pseudo-KU algebra \(\mathfrak{A}\). Then the relation \('\preccurlyeq '\), defined by \[(\forall x, y \in A)(x \preccurlyeq y \Longleftrightarrow x \cdot y \in J),\] is a quasi-order in the set \(A\) left compatible and right reverse compatible with the internal operations in \(\mathfrak{A}\).

Proof. Since \(x \cdot x = 0 \in J \) is valid in \(\mathfrak{A}\) for any \(x \in A\), it is clear that \('\preccurlyeq '\) is a reflexive relation in the set \(A\). Let \(x, y, z \in A\) be arbitrary elements such that \(x \preccurlyeq y\) and \(y \preccurlyeq z\). This means \(x \cdot y \in J\) and \(y \cdot z \in J\). From inequality (pKU-1) in the form \(x \cdot y \leqslant (y \cdot z)\ast (x \cdot z)\) and \( x \cdot y \in J \) it follows \((y \cdot z)\ast (x \cdot z) \in J\) according to (15). From here and from \( y \cdot z \in J \) it follows \( x \cdot z \in J \) according to (pJ3a). Hence, the relation \('\preccurlyeq '\) is transitive. So, this relation is a quasi-order in \(A\). Let \(x, y, z \in A\) be such \(x \preccurlyeq y\). Then \(x \cdot y \in J\) and \(x \ast y \in J\).
(i) If we put \(x = y\) and \(y = x\) in the left part of the formula (pKU-1), we get \( x \cdot y \leqslant (y \cdot z) \ast (x \cdot z)\). Now, from here and \(x \cdot y \in J\) it follows \((y \cdot z) \ast (x \cdot z) \in J\) by (15). Thus \((y \cdot z) \cdot (x \cdot z) \in J\) by Proposition 6. Finally, we have \(y \cdot z \preccurlyeq x \cdot z\). So, the relation \('\preccurlyeq '\) is reverse right compatible with the internal operation \(' \cdot '\) in \(\mathfrak{A}\).
(ii) If we put \(x = y\) and \(y = x\) in the right part of the formula (pKU-1), we get \( x \ast y \leqslant (y \ast z) \cdot (x \ast z)\). Then \((y \ast z) \cdot (x \ast z) \in J\) by (15). Thus \(y \ast z \preccurlyeq x \ast z\). Therefore, the relation \('\preccurlyeq '\) is reverse right compatible with the internal operation \('\ast '\) in \(\mathfrak{A}\).
(iii) Let us put \(y = z\) and \(z = y\) in the left part of the formula (pKU-1). We get \((z \cdot x) \ast ((x \cdot y) \ast (z \cdot y)) = 0 \in J\). From here and from \(x \cdot y \in J\) it follows \((z \cdot x) \ast (z \cdot y) \in J\) by (pJ4a). Thus \(z \cdot x \preccurlyeq z \cdot y\). So, the relation \('\preccurlyeq '\) is left compatible with the operation \('\cdot '\).
(iv) Let us put \(y = z\) and \(z = y\) in the right part of the formula (pKU-1). We get \((z \ast x) \cdot ((x \ast y) \cdot (z \ast y)) = 0 \in J\). From here and from \(x \ast y \in J\) it follows \((z \ast x) \cdot (z \ast y) \in J\) by (pJ4b). Thus \(z \ast x \preccurlyeq z \ast y\). So, the relation \('\preccurlyeq '\) is left compatible with the operation \('\ast '\).

5.3. Concept of pseudo-filters

Definition 7. A non-empty subset \(F\) of a pseudo-KU algebra \(\mathfrak{A}\) is called a pseudo-filter of \(A\) if it satisfies in the following axioms:
(pF1) \(0 \in F\);
(pF3) \((\forall x, y \in A)((x \cdot y \in F \wedge x \ast y \in F \wedge y \in F) \Longrightarrow x \in F)\).

\(\{0\}\) and \(A\) are pseudo-filters of \(\mathfrak{A}\). So, the family \(\mathfrak{F}(A)\) of all pseudo-filters in a pseudo-KU algebra \(\mathfrak{A}\) is not empty. It is obviously the following is valid

Lemma 6. Let \(F\) be a pseudo-filter in a pseudo-KU algebra \(\mathfrak{A}\). Then
(16) \((\forall x, y \in A)((x \leqslant y \wedge y \in F) \Longrightarrow x \in F)\).

Theorem 3. The family \(\mathfrak{F}(A)\) of all pseudo-ideals in a pseudo-KU algebra \(\mathfrak{A}\) forms a complete lattice.

Proof. Let \(\{F_{i}\}_{i \in I}\) be a family of pseudo-filters in a pseudo-KU algebra \(\mathfrak{A}\). Clearly \(0 \in \bigcap_{i \in I}F_{i}\) is valid. Let \(x, y \in A\) be elements such that \(x \cdot y \in \bigcap_{i \in I}F_{i}\), \(x \ast y \in \bigcap_{i \in I}F_{i}\) and \(y \in \bigcap_{i \in I}F_{i}\). Then \(x \cdot y \in F_{i}\), \(x \ast y \in F_{i}\) and \(y \in F_{i}\) for any \(i \in I\). Thus \(x \in F_{i}\) because \(F_{i}\) is a pseudo-filter in \(\mathfrak{A}\) and \(x \in \bigcap_{i \in I}F_{i}\). So, \(\bigcap_{i \in I}F_{i}\) is a pseudo-filter in \(\mathfrak{A}\).
If \(\mathfrak{ X }\) is the family of all pseudo-filters of \(\mathfrak{ A }\) that contain the union \(\bigcup_{i \in I}F_{i}\), then \(\cap \mathfrak{X} \) is also a pseudo-filter in \(\mathfrak{ A }\) that contains \(\bigcup_{i \in I}F_{i}\) by previous evidence.
If we put \(\sqcap_{i \in I}F_{i} = \bigcap_{i \in I}F_{i}\) and \(\sqcup_{i \in I}F_{i} = \cap \mathfrak{X}\), then \((\mathfrak{F}(A),\sqcap,\sqcup)\) is a complete lattice.

6. Concept of pseudo-homomorphisms

Definition 10.\(((A,\leqslant_{A}),\cdot_{A},\ast_{A}, 0_{A})\) and \(((B,\leqslant_{B}),\cdot_{B},\ast_{B}, 0_{B})\) be pseudo-KU algebras. A mapping \(f : A \longrightarrow B\) of pseudo-KU algebras is called a pseudo-homomorphism if \[(\forall x,y \in A)(f(x \cdot_{A} y) =_{B} f(x)\cdot_{B} f(y) \wedge f(x \ast_{A} y) =_{B} f(x)\ast_{B} f(y)).\]

Remark 2. Note that if \(f : A \longrightarrow B\) 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 10. Any pseudo-homomorphism between pseudo-KU algebras is isotone mapping.

Proof. Let \(f : A \longrightarrow B\) be a pseudo-homomorphism between pseudo-KU algebras and let \(x, y \in A\) be such \(x \leqslant_{A} y\). Then \(x \cdot_{A} y =_{A} 0_{A}\). Thus \(0_{B}=_{B} f(x \cdot_{A}y) =_{B} f(x)\cdot_{B}f(y)\). This means \(f(x) \leqslant_{B} f(y)\).

Lemma 11. Let \(f : A \longrightarrow B\) be a pseudo-homomorphism between pseudo-KU algebras. Then the set \(Ker(f) =_{A} \{x \in A : f(x) =_{B} 0_{B}\}\) is a pseudo-ideal in \(\mathfrak{A}\).

Proof. It is obvious \(0_{A}\in Ker(f)\).
Let \(x, y \in A\) be such \(x \cdot_{A} y \in Ket(f)\) and \(x \in Ker(f)\). Then \(f(x) =_{B} 0_{B}\) and \(0 =_{B}f(x \cdot_{A} y) =_{B} f(x)\cdot_{B}f(y) =_{B} 0_{B}\cdot_{B} f(y) =_{B} f(y)\). Thus \(y \in Ker(f)\).
The implication of \(x \ast_{A} y \in Ker(f) \wedge x \in Ker(f) \Longrightarrow y \in Ker(f)\) can be proved by analogy with the previous proof.

The following statement is easy to prove:

Lemma If \(f : A \longrightarrow B\) is a pseudo-homomorphism between pseudo-KU algebras, then \(f(A)\) is a pseudo-subalgebra in \(B\).

Proposition 8. Let \(f : A \longrightarrow B\) be a pseudo homomorphism between pseudo-KU algebras \(\mathfrak{A}\) and \(\mathfrak{B}\).
(i) If \(K\) is a pseudo-ideal in \(\mathfrak{B}\), then \(f^{-}(K)\) is a pseudo-ideal in \(\mathfrak{A}\).
(ii) If \(G\) is a pseudo-filter in \(\mathfrak{B}\), then \(f^{-1}(G)\) is a pseudo-filter in \(\mathfrak{A}\) s.

Proof. (i) Assume that \(K\) is a pseudo-fulter of \(\mathfrak{B}\). Obviously \(0_{A}\in f^{-1}(K)\). Let \(x, y \in A\) be such \(x \cdot y \in f^{-1}(K)\) and \(x \in f^{-1}(K)\). Then \(f(x)\cdot_{B}f(y) =_{B}f(x \cdot_{A}y)\in K\) and \(f(x)\in K\). It follows that \(f(y)\in K\) by (pJ3a) since \(K\) is a pseudo-ideal in \(\mathfrak{B}\). Therefore, \(y \in f^{-1}(K)\). 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 \(f^{-1}(K)\) is a pseudo-ideal in \(\mathfrak{A}\).
(ii) It is obvious \(0_{A}\in f^{-1}(G)\) again. Let \(x, y \in A\) be elements such that \(x \cdot_{A}y \in f^{-1}(G)\), \(x \ast_{A}y \in f^{-1}(G)\) and \(y \in f^{-1}(G)\). Then \(f(x)\cdot_{B}f(y) =_{B} f(x \cdot_{A} y)\in G\), \(f(x)\ast_{B}f(y) =_{B} f(x \ast_{A} y)\in G\) and \(f(y) \in G\). Thus \(f(x)\in G\) because \(G\) is a pseudo-filter in \(\mathfrak{B}\). This means \(x \in f^{-1}(G)\). So, the set\(f^{-1}(G)\) is a pseudo-filter in \(\mathfrak{A}\).

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.

Definition 11. Let \(\mathfrak{A} = ((A,\leqslant),\cdot,\ast,0)\) be a pseudo-KU algebra.
For the equivalence relation \(q\) on the set \(A\) we say that it is a congruence of type \('\cdot '\) on \(\mathfrak{A}\) if it compatible with the operations \('\cdot '\) in \(\mathfrak{A}\) in the following sense
(17) \((\forall x, y, z \in A)((x,y) \in q \Longrightarrow ((x \cdot z,y \cdot z) \in q \wedge (z \cdot x,z \cdot y)\in q)))\).
For the equivalence relation \(q\) on the set \(A\) we say that it is a congruence of type \('\ast '\) on \(\mathfrak{A}\) if it compatible with the operations \('\ast '\) in \(\mathfrak{A}\) in the following sense
(18) \((\forall x, y, z \in A)((x,y) \in q \Longrightarrow ((x \ast z,y \ast z) \in q \wedge (z \ast x,z \ast y)\in q)))\).
For the equivalence relation \(q\) on the set \(A\) we say that it is a congruence of common type on \(\mathfrak{A}\) if it is compatible with both operations in \(\mathfrak{A}\).

Lemma 12. Let \(q\) be a relation on a pseudo-KU algebra \(\mathfrak{A}\). Then:
(i) The condition (17) is equivalent to the condition
(17a) \((\forall x, y, u,v \in A)(((x,y) \in q \wedge (u,v) \in q) \Longrightarrow (x \cdot u, y \cdot v)\in q)\).
(ii) The condition (18) is equivalent to the condition .
(18a) \((\forall x, y, u,v \in A)(((x,y) \in q \wedge (u,v) \in q) \Longrightarrow (x \ast u, y \ast v)\in q)\).

Proof. \((17a) \Longrightarrow (17)\). If we choose \( v = z \) in (17a), we get the implication \((x,y)\in q \Longrightarrow (x \cdot z,y \cdot z)\in q\). On the other hand, if we put \( x = y = z \), \(u = x\) and \(v = y\) in (17a), we get the implication \((x,y)\in q \Longrightarrow (z \cdot x,z \cdot t).\)
\((17) \Longrightarrow (17a)\). Suppose (17) and let \(x,y, u, v\in A\) such that \((x,y)\in q\) and \((u,v)\in q\). Thus \((x \cdot u,x \cdot v)\in q\) and \((x \cdot v, y \cdot v)\in q\) by (16). Hence \((x \cdot u, y \cdot v)\in q\) by transitivity of \(q\).
Equivalence \((18) \Longleftrightarrow (18a)\) can be proved analogous to the previous proof.

Let \(f : A \longrightarrow B\) be a pseudo homomorphism between pseudo-KU algebras. By direct check without difficulty, it can be proved that the relation \(q_{f}\), defined by \[(\forall x, y \in A)((z,y) \in q_{f} \Longleftrightarrow f(x) =_{B} f(y)),\] is a congruence (all three types) on \(\mathfrak{ A }\).

Theorem 10. The relation \(q_{f}\) is a congruence of type \('\cdot '\) (type \('\ast '\), common type) on the pseudo-KU algebra \(\mathfrak{A}\).

Proof. We will only demonstrate the proof that \(q_{f}\) is a congruence of type \( '\cdot ' \) on \(\mathfrak{ A }\) because the evidence that \(q_{f}\) is a congruence of type \('\ast '\) 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 \(\mathfrak{ A }\). Let \(x,y,u,v \in A\) be such that \((x,y)\in q_{f}\) and \((u,v)\in q_{f}\). Then \(f(x) =_{B} f(y)\) and \(f(u) =_{B} f(v)\). Thus \[f(x \cdot_{A} u) =_{B} f(x)\cdot_{B}f(u) =_{B} f(y)\cdot_{B}f(v) =_{B}f(y \cdot_{A}u).\] Hence, \((x \cdot_{A}u,y \cdot_{A}v)\in q_{f}\). We proved that (17a) is a valid formula. So \(q_{f}\) is a congruence of type \('\cdot '\) on \(\mathfrak{ A }\).

Theorem 11. Let \(J\) be a pseudo-ideal in a pseudo-KU algebra \(\mathfrak{A}\). Then the relation \(q_{J}\), defined by \(q_{J} = \preccurlyeq \cap \preccurlyeq^{-1}\), is a congruence of common type in \(\mathfrak{A}\).

Proof. The relation \(q\) is an equivalence relation on the set \(A\). It is sufficient to prove that \(q\) is compatible with operations in \(\mathfrak{A}\). Since the relation \(\preccurlyeq\) is left compatible and right reverse compatible with the internal operations in \(\mathfrak{ A }\), by Theorem 8, it is clear that the relation \(q_{J}\) is a congruence on \(\mathfrak{ A }\).

For a congruence \(q\) on a pseudo-KU algebra \(\mathfrak{A}\) we denote \(qx = \{y \in A : (x,y)\in q\} = [x]\). Let's define \('\bullet '\) and \('\star '\) in \(A/q\) on this way \[(\forall x, y \in A)([x]\bullet [y] = [x \cdot y])\;\; and \;\;(\forall x, y \in A)([x] \star [y] = [x \ast y])\] Without much difficulty it can be verified that the functions \('\bullet '\) and \('\star '\), 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 \('\bullet '\) and \('\star '\), 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 \(\mathfrak{A}\), then we have

Theorem 12. Let \(J\) be a pseudo-ideal in a pseudo-KU algebra \(\mathfrak{A}\). Then the structure \(((A/q,\leqq),\bullet,\star,[0])\), where \('\leqq '\) is defined by \[(\forall x, y \in A)([x] \leqq [y] , \Longleftrightarrow x \preccurlyeq y),\] is a pseudo-KU algebra, too.

Proof. According to the commentary preceding this theorem, to prove this theorem it suffices to show that the structure \(((A/q,\leqq),\bullet,\star,[0])\) satisfies the axiom (pKU-4).
Let \(x, y \in A\) be such \([x] \leqq [y]\) and \([y] \leqq [x]\). Then \(x \preccurlyeq y\) and \(y \preccurlyeq x\) by definition. Thus \((x,y)\in q_{J}\) and \([x] = [y]\).

Let \(f : A \longrightarrow B\) be pseudo-homomorphism between pseudo-KU algebras \(((A,\leqslant_{A}),\cdot_{A},\ast_{A},0_{A})\) and \(((B,\leqslant_{B}),\cdot_{B},\ast_{B},0_{B})\). Then the set \(f(A)\) is a pseudo-subalgebra of \(\mathfrak{B}\) by and the set \(J= Ker(f)\) is a pseudo-ideal in \(\mathfrak{A}\) by Lemma 10 and the relation \(q_{f}\) is a congruence on \(\mathfrak{A}\) by Theorem 10. If \((x,y)\in q_{f}\) holds soe some \(x, y \in A\), we have \(f(x) =_{B} f(y)\). Thus \(f(x \cdot_{A} y) =_{B} f(x) \cdot_{B} f(y) =_{B} f(x) \cdot_{B} f(x) =_{B} 0_{B}\), i.e. \(x \cdot_{A}y \in J\). Analogous to the previous one may be shown that \(y \cdot_{A} x \in J\) holds. Thus, \((x,y) \in q_{f} \Longrightarrow (x, y) \in q_{J}\) 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 \(f : A \longrightarrow B\) be pseudo-homomorphism between pseudo-KU algebras \(((A,\leqslant_{A}),\cdot_{A},\ast_{A},0_{A})\) and \(((B,\leqslant_{B}),\cdot_{B},\ast_{B},0_{B})\). Then there exists the unique epimorphism \(\pi : A \longrightarrow A/q_{f}\), defined by \(\pi(x) = [x]\) for any \(x \in A\), and the unique monomorphism \(g : A/q_{f} \longrightarrow B\), defined by \(g([x]) =_{B} f(x)\) for any \(x \in A\) such that \(f = g \circ \pi\).

Acknowledgments

The author thanks the reviewers for helpful suggestions. The author also thanks the editors of the journal for their assistance in the technical preparation of the manuscript.

Conflict of Interests

The author declares no conflict of interest.