A few comments and some new results on JU-algebras

1. Introduction

In 1966, Imai and Iseki [1] introduced a notion of BCK-algebras. The concept of BE-algebra as a generalization of BCK-algebra was introduced in 2006 by Kim and Kum in [2]. The concept of KU-algebras was introduced and analyzed in 2009 in [3, 4]. KU-algebras are closely related to BE-algebras. Specifically, in the article [5], the authors have shown that 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 have shown that every KU-algebra is a BE-algebra [5]. The concept of UP-algebras as a generalization of KU-algebras was introduced by Iampan in [6]. The concept of JU-algebras, as a generalization of KU-algebras, was introduced and analyzed in [7, 8].
However, this concept was introduced in [9] by Leerawat and Prabpayak under the name ‘pseudo KU-algebra. In doing so, they used the PKU designation for this class of algebra. Since then, this type of generalization of KU-algebra has been in the focus of interest of the academic community (for example see [10, 11]).
We are more inclined to refer this concept as ‘weak KU-algebra’ in the same way as weak BCC-algebra [12]. However, due to the tight connection of this paper to the article [8], we will use the name ‘JU-algebra’ in what follows.
In this article, we revisit the axioms of JU-algebras and definitions of their substructures. We also link this class of algebras with the classes BE-algebras and UP-algebras. In addition, we introduce and analyze some new classes of ideals in this class of algebra such as closed ideal, ag-ideal, t-ideal, (\(\star\))-ideal and associative ideal.

2. Preliminaries

In this section, we take the definitions of JU-algebras, JU-subalgebras, JU-ideals and other important terminologies and some related results from literature [7, 8].

2.1. Definition and some comments

Definition 1.[8] An algebra \((A, \cdot, 1)\) of type \((2, 0)\) with a binary operation ”\(\cdot\)” and a fixed element \(1\) is said to be JU-algebras satisfying the following axioms:

• (JU-1) \((\forall x, y, z \in A)((y \cdot z) \cdot ((z \cdot x) \cdot (y \cdot x)) = 1)\),
• (JU-2) \((\forall x \in A)(1 \cdot x = x)\) and
• (JU-3) \((\forall x, y \in A)((x \cdot y \, \wedge \, y \cdot x = 1) \, \Longrightarrow \, x = y)\).

We denote this axiom system by [JU].

Lemma 1. [8] In the axioms system [JU], the following formulae are valid:

• [(J\(_{11}\))] \((\forall x\in A)(x \cdot x = 1),\)
• [(J\(_{12}\))] \( (\forall x, y, z \in A)(z \cdot (y \cdot x) = y \cdot (z \cdot x)) \).

Comment 1. In [3], a KU-algebra is defined as a system \((A,\cdot,0)\) by 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)] \((\forall x, y \in A)((x \cdot y = 0 \, \wedge \, y \cdot x = 0)\, \Longrightarrow \, x = y)\).

We denote this axiom system by [KU]. With [wKU] we denote axiomatic system [KU] without axiom (KU-3). So, [JU] \(\equiv\) [wKU] \(\equiv\) [PKU].
Recall that in the axiom system [KU], the formula (J\(_{12}\)) is a valid formula also [13].
If in the definition of KU-algebras we write \( 1 \) instead of \( 0 \), then we see that any KU-algebra \( A \) is a JU-algebra. Therefore, the concept of JU-algebras is a generalization of the concept of KU-algebras [8].
If we followed the formation of the concept of ‘weak BCC-algebras’ from the ‘concept of BCC-algebras’, then the name ‘weak KU-algebra‘ could also be used for a JU-algebra by analogy with the previous one.
If \(A\) is a JU-algebra, let us define \(\varphi: A \longrightarrow A\) as follows; $$(\forall x \in A) (\varphi(x) = x \cdot 1)$$ taking the idea from [14]. According to (JU-2), the equality \(\varphi(1) = 1 \) is valid for mapping \(\varphi\).

Comment 2. The concept of UP-algebras was introduced in 2017 in article [6] as a \((A,\cdot, 0) \) system that satisfies the following axioms:

• (UP-1) \((\forall x, y, z)((y\cdot z)\cdot((x \cdot y)\cdot (x \cdot z))= 0)\),
• (UP-2) \((\forall x \in A)(0 \cdot x = x)\),
• (UP-3) \((\forall x \in A)(x \cdot 0 = 0)\) and
• (UP-4) \((\forall x, y \in A)((x \cdot y = 0 \, \wedge \, y \cdot x = 0)\, \Longrightarrow \, x = y)\).

We denote this axiom system by [UP]. With [wUP] we denote axiomatic system [UP] without axiom (UP-3).
We can transform the formula (JU-1) into the formula (UP-1) using valid equation (J\(_{12}\)) [8] and replacing the element \( 1 \) by the element \( 0 \). However, since formula (J\(_{12}\)) does not have to be a valid formula in [UP], we conclude that there is no direct connection between [JU] and [UP]. On the other hand, the system [UP] + (J\(_{12}\)) is equivalent to the system [KU] according to theorems in [6], so we conclude that the system [JU] is contained in the system [wUP] + (J\(_{12}\)). Therefore, any UP-algebra that additionally satisfies equality (J\(_{12}\)) is also a JU-algebra at the same time.

Comment 3. The concept of BE-algebras is defined in [2] as a system \((A,\cdot,1)\) satisfying the following axioms:

• (BE-1) \((\forall x \in A)(x \cdot x = 1)\),
• (BE-2) \((\forall x \in A)(x \cdot 1 = 1)\),
• (BE-3) \((\forall x \in A)(1 \cdot x = x)\) and
• (BE-4) \((\forall x, y, z \in A)(x \cdot (y \cdot z) = y \cdot (x \cdot z))\).

We denote this axiom system by [BE]. The axiomatic system generated by axioms (B-1), (BE-3) and (BE-4) is denoted by [wBE].
It is shown in [5] that every KU-algebra is a BE-algebra. Since any KU-algebra is a JU-algebra, by Comment 1, we get that every BE-algebra is a JU-algebra.

2.2. An order relation

Definition 2.[8] Let \(A\) be a JU-algebra. We define a relation ”\(\leqslant\)” in \(A\) as follows: $$\forall x, y \in A)(y \leqslant x \, \Longleftrightarrow \, x \cdot y = 1.$$

According to claims (J\(_{4}\)), (J\(_{5}\)), (J\(_{6}\)), and claims (J\(_{7}\)), (J\(_{8}\)), the relation “\(\leqslant\)” is a partial order in \( A \) left compatible and right reverse compatible with the internal operation in \(A\) [8].

Proposition 1. Let \(A\) be a JU-algebra. Then

• (1) \((\forall x, y \in A)(x \cdot \varphi(y) = y \cdot \varphi(x))\);
• (2) \((\forall x, y \in A)(\varphi(x)\cdot \varphi(y) \leqslant y \cdot x)\);
• (3) \((\forall x, y \in A)(x \cdot (y \cdot x)\leqslant \varphi(y))\);
• (4) \((\forall x, y \in A)(\varphi(x \cdot y) = \varphi(x) \cdot \varphi(y))\);
• (5) \((\forall x, y \in A)(x \leqslant y \, \Longrightarrow \, \varphi(y) \leqslant \varphi(x))\).

Proof. Relation \((1)\) is obtained directly from (J\(_{12}\)) where, we put \(z = 1\).
If we put \( x = 1 \) and \(z = x\) in (JU-1), we get \((y \cdot x)\cdot ((x \cdot 1)\cdot (y \cdot 1)) = 1\). This means \(\varphi(x)\cdot \varphi(y) \leqslant y \cdot x\) according to the Definition 2.
If we put \(z = 1\) in (JU-1), we get \((y \cdot 1)\cdot((1 \cdot x)\cdot (y \cdot x)) = 1\). Hence \(\varphi(y)\cdot (x \cdot (y \cdot x)) = 1\). So, we have \(x \cdot (y \cdot x) \leqslant \varphi(y)\).
Relation \((4)\) is proved in [8] as formula (J\(_{14}\)).
Relation \((5)\) is a direct consequence of the right inverse compatibility of order relations with an internal operation in \( A \) if we choose \( z = 1 \).

Remark 1. The relation (5) of Proposition 1 is a direct consequence of the Proposition 1(4). Indeed, if \( x \leqslant y \), then \( y\cdot x = 1 \). Thus \(\varphi(y \cdot x) = 1\). Hence \(\varphi(y)\cdot \varphi(x) = 1\) by Proposition 1(4). This means \(\varphi(x) \leqslant \varphi(y)\).

3. Some types of JU-ideals

Definition 3.[8] A non-empty subset \(J\) of a JU-algebra \(A\) is called a JU-ideal of \(A\) if

• (J-1) \(1 \in J\) and
• (J-2) \((\forall x, y \in A)((x \in J \, \wedge \, x \cdot y \in J)\, \Longrightarrow \, y \in J)\).

Lemma 2. Let \(J\) be a JU-ideal of a JU-algebra \(A\). Then

• (J-3) \((\forall x, y \in A)((x \leqslant y \, \wedge \, y \in J)\, \Longrightarrow \, x \in J)\).

Proof. Let \(x, y \in A\) be such that \(x \leqslant y\) and \(y \in J\). Then \(y\cdot x = 1 \in J\) and \(y \in J\). Thus \(x \in J\) by (J-2).

Proposition 2. Let \(J\) be a subset of a JU-algebra such that (J-1) holds. Then the condition (J-2) is equivalent to the condition;

• (J-4) \((\forall x, y, z \in A)((x \cdot (y \cdot z)\in J \, \wedge \, y \in J) \, \Longrightarrow \, x \cdot z \in J)\).

Proof. (J-4) \(\Longrightarrow\) (J-2). If we put \( x = 1 \), \( y = x \) and \( z = y \) in (J-4), then we get (J-2) with respect to (JU-2). (J-2) \(\Longrightarrow\) (J-4). Let \(x, y, z \in A\) such that \(x \cdots (y \cdot z) \in J\) and \(y \in J\). Then \(y \cdot (x \cdot z) \in J\) and \(y \in J\) by (J\(_{12}\)). Thus \(x \cdot z \in J\) by (J-2).

As a consequence of the Proposition 2, we can describe some of the features of the JU-ideals as follows:

Proposition 3. Let \(J\) be a JU-ideal of a JU-algebra \(A\). Then

• (6) \(\forall x, y \in A)((\varphi(x) \in J\, \wedge \, y \in J)\, \Longrightarrow \, x\cdot y \in J)\) and
• (7) \((\forall x, y \in A)((x \cdot \varphi(y) \in J \, \wedge \, y \in J)\, \Longrightarrow \, \varphi(x))\).

Proof. If we put \( z = y \) in (J-4), we get $$(x \cdot (y \cdot y) \in J \, \wedge \, y \in J) \Longrightarrow x \cdot y \in J.$$ From where, we get (J-5) using (J\(_{11}\)) in [8]. If we put \( z = 1 \) in (J-4), we get (7).

Definition 4. Let \(J\) be a JU-ideal of a JU-algebra \(A\), then

• (C) \(J\) is a closed ideal of \(A\) if \(\{x \in A : \varphi(x) \in J\} = \varphi(J) \,\subseteq \, J\) holds.

Theorem 1. An ideal \(J\) of a JU-algebra \(A\) is closed if and only if it is a subalgebra of \(A\).

Proof. Assume that an ideal \(J\) is a subalgebra of \(A\) and \(x \in J\). Then, from \(1 \in J\) and \(x \in J\) follows \(\varphi(x) = x \cdot 1 \in J\) because \(J\) is a subalgebra of \(A\). This means \(\varphi(J) \, \subseteq \, J\). Conversely, let an ideal \(J\) of \(A\) is closed and let \(x, y \in J.\) Then \(\varphi(x) \in \varphi(J) \, \subseteq \, J\) and \(y \in J\). Thus \(x \cdot y \in J\) by Proposition 3(6). This means that \(J\) is a subalgebra of \(A\).

As usual, we will write \(Ker\varphi = \{x \in A : \varphi(x) = 1 \}\). In [8], this set is labeled by \(B_{A}\).

Corollary 1. \(Ker\varphi\) is a closed JU-ideal of \(A\).

Proof. Let \(x, y \in A\) be such \(x \in Ker\varphi\) and \(x \cdot y \in Ker\varphi\). Then \(\varphi(x) = 1 \) and \(1 = \varphi(x \cdot y) = \varphi(x) \cdot \varphi(y) = 1 \cdot \varphi(y) = \varphi(y)\) by Proposition 1(4) and (JU-2). Thus \(y \in Ker\varphi\). So, \(Ker\varphi\) is a JU-ideal of \(A\).
Suppose \(x \in Ker\varphi\) and \(y \in Ker\varphi\). Then, from \(\varphi(x) = 1\) and \(\varphi(y) = 1\) it follows \(\varphi(x \cdot y) = \varphi(x)\cdot \varphi(y) = 1 \cdot 1 = 1\) by Proposition 1(4). Thus \(x \cdot y \in Ker\varphi\). So, \(Ker\varphi\) is a subalgebra in \(A\). Therefore \(Ker\varphi\) is a closed JU-ideal of \(A\) by Theorem 1.

In what follows, we need the following lemma.

Lemma 3. Let \(J\) be a JU-ideal of a JU-algebra \(A\), then

• (8) \((\forall x \in A)(x \in J \, \Longrightarrow \, \varphi^{2}(x) \in J).\)

Proof. Let us first show that the following holds;

• (9) \((\forall x \in A)(\varphi^{2}(x) \leqslant x)\).

This inequality immediately follows from Proposition 1(2) if we put \( y = 1 \). Now, from \(x \in J\) and \(\varphi^{2}(x) \leqslant x \) it follows \(\varphi^{2}(x) \in J \) according to (J-3).

The preceding lemma is a motive for introducing the following concept.

Definition 5. Let \(J\) be a JU-ideal of JU-algebra \(A\), then

• (AG) \(J\) is ag-ideal of \(A\) if \((\forall x \in A)(\varphi^{2}(x) \in J \, \Longrightarrow \, x \in J)\) holds.

Proposition 4. Let \(A\) be a JU-algebra, then

• (10) \((\forall x \in A)(\varphi^{3}(x) = \varphi(x)).\)

Proof. It has already been shown that \( \varphi^{2}(x) \leqslant x \) is valid. Thus \(\varphi(x) \leqslant \varphi^{3}(x)\) by Proposition 1(5) and \(\varphi^{3}(x)\cdot \varphi(x) = 1\). On the other hand, from \( 1 = \varphi^{2}(x) \cdot \varphi^{2}(x) = \varphi^{2}(x) \cdot \varphi(\varphi(x))\), it follows \( \varphi(x) \cdot \varphi^{3}(x) = 1\) according to Proposition 1(1). Thus \(\varphi^{3}(x) = \varphi(x)\) by (JU-3).

Corollary 2. \(Ker\varphi\) is a closed ag-ideal of \(A\).

Proof. It has already been shown that \( Ker\varphi \) is a closed JU-ideal of \( A \). Let \(x\in A\) be an arbitrary element such that \(\varphi^{2}(x) \in Ker\varphi\). Then \(\varphi(x) = \varphi^{3}(x) = 1\) by (9) given in proof of Lemma 3. Thus \(x \in Ker\varphi\).

Lemma 4. If \(J\) is a JU-ideal of a JU-algebra \(A\), then \(\varphi^{2}(J) = \{x \in A : \varphi^{2}(x) \in J\}\) is a JU-ideal of \(A\) also. The reverse also applies: If \(\varphi^{2}(J)\) is a JU-ideal of JU-algerba \( A \), then \( J \) is a JU-ideal of \( A \) also.

Proof. Since \(1 \in J\), then \(\varphi^{2}(1) = 1 \in J\). Thus \(1 \in \varphi^{2}(J)\). Suppose \(x \in \varphi^{2}(J)\) and \(x \cdot y \in \varphi^{2}(J)\). Then \(\varphi^{2}(x) \in J\) and \(\varphi^{2}(x) \cdot \varphi^{2}(y) = \varphi^{2}(x \cdot y) \in J\) with respect to Proposition 1(4). Thus \(\varphi^{2}(y) \in J\) and \(y \in \varphi^{2}(J)\). So, \(\varphi^{2}(J)\) is a JU-ideal of JU-algebra \(A\).
It is obvious that \(1 = \varphi^{2}(1) \in J\) holds because \(1 \in \varphi^{2}(J)\) is valid. Let \(x, y \in A\) be such \(x \in J\) and \(x \cdot y \in J\). Then \(\varphi^{2}(x) \in \varphi^{2}(J)\) and \(\varphi^{2}(x)\cdot \varphi^{y} = \varphi^{2}(x \cdot y) \in \varphi^{2}(J)\). Thus \(\varphi^{2}(y) \in \varphi^{2}(J)\) by (J-2) because \(\varphi^{2}(J)\) is a JU-ideal of \(A\). Hence \(y \in J\). Therefore, \(J\) is a JU-ideal of \(A\).

Theorem 2. \(J\) is a closed a JU-ideal of a JU-algerba \(A\) if and only if \(\varphi^{2}(J)\) is a closed JU-ideal of \(A\).

Proof. Let \(J\) be a closed a JU-ideal of a JU-algebra \(A\). Let \(x \in A\) be an element such that \(x\in \varphi(\varphi^{2}(J))\). Then \(\varphi(x) \in \varphi^{2}(J)\) and \(\varphi^{2} (\varphi(x)) = \varphi(\varphi^{2}x))\in J\). Thus \(\varphi^{2}(x) \in \varphi(J) \, \subseteq \, J\) because \(J\) is a closed ideal of \(A\). Hence \(x \in \varphi^{2} (J)\). From this it has shown that the ideal \(\varphi^{2}(J)\) satisfies Definition 4 condition (C). Let \(J\) be a JU-ideal of a JU-algebra \(A\) such that \(\varphi^{2}(J)\) is a closed JU-ideal of \(A\). Then, we have \[\varphi(J) = \varphi^{3}(J) = \varphi(\varphi^{2}(J)) \,\subseteq \, \varphi^{2}(J) \, \subseteq \, J\] by Proposition 4(10) and since the inclusion \(\varphi^{2}(J) \, \subseteq \, J\) is obviously valid. So, \(J\) is a closed JU-ideal of \(A\).

Before presenting the following theorem, we will introduce the concept of JU-filters in a JU-algebra.

Definition 6. A subset \(F\) of a JU-algerba \(A\) is a JU-filter of \(A\) if the following hold:

• (F-1) \(1 \in F\) and
• (F-2) \((\forall x, y \in A)((x \cdot y \in F \, \wedge \, y \in F)\, \Longrightarrow \, x \in F)\).

Lemma 5. If \(F\) is a JU-filter of a JU-algerba \(A\), then

• [(F-3)] \((\forall x, y \in A)(x \in F \, \wedge \, x \leqslant y)\, \Longrightarrow \, y \in F)\).

Proof. Let \(x, y \in A\) be such that \(x \in F\) and \(x \leqslant y\). Then \(x \in F\) and \(y \cdot x = 1 \in F\). Thus \(y \in F\) by (F-3).

Lemma 6. If \(F\) is a JU-filter of a JU-algebra \(A\), then

• (11) \((\forall x \in A)(\varphi(x) \in F \, \Longrightarrow \, x \in F)\).

Proof. If we put \( y = 1 \) in (F-2), we immediately get (11).

Remark 2. In [8], the concept of strong JU-ideal of a JU-algebra was introduced in the way we introduced the concept of JU-filters.

Theorem 3. For a JU-ideal \(J\) of a JU-algerba \(A\) the following are equivalent:

• (a) \(J\) is an ag-ideal of \(A\);
• (b) \(J\) is a JU-filter of \(A\);
• (c) \((\forall x,y,z \in A)(((x \cdot z)\cdot(y\cdot z)\in J \, \wedge \, y \in J)\, \Longrightarrow \, x \in J)\); and
• (d) \((\forall x, y \in A)(\varphi(x \cdot y) \in J \,\wedge \, y \in J)\, \Longrightarrow \, x \in J)\).

Proof.

• (a)\(\Longrightarrow\) (b). Let \(J\) be an ag-ideal of \(A\) and \(x \leqslant y \, \wedge \, y \in J\). Then \(y \cdot x = 1 \in J\) and \(y \in J\).
Thus \(\varphi^{2}(y)\cdot \varphi^{2}(x) = \varphi^{2}(y \cdot x) = 1 \in J\) and \(\varphi^{2}(y)\in \varphi^{2}(J) \, \subseteq \, J\). Hence \(\varphi^{2}(x)J\) because \(J\) is an ideal of \(A\). From here, it is follows \(x \in J\) since \(J\) is an ag-ideal of \(A\). Therefore, subset \(J\) is a JU-filter of \(A\).
• (b)\(\Longrightarrow\) (c). From (JU-1), written in the form \((y \cdot x)\cdot ((x \cdot z)\cdot (y \cdot z)) = 1 \in J\), and from \((x \cdot z)\cdot (y \cdot z)\in J\) we get \(y \cdot x \in J\) according to (F-2).
Now, from \( y \cdot x \in J \) and \( y \in J \) it follows \(x \in J \) according to (J-2).
• (c)\(\Longrightarrow\) (d). If we put \( z = 1 \) in (c), we get \(\varphi(x) \cdot \varphi (y) \in J \, y \in J\). hence (d) with respect to (4).
• (d)\(\Longrightarrow\) (a). If we put \( y = 1 \) in (d), we get (a) with respect to (J-1).

Let’s introduce the following type of JU-ideals.

Definition 7. A subset \(J\) of a JU-algebra \(A\) is a t-ideal of \(A\) if the following hold:

• (J-1) \(1 \in J\) and
• (t) \((\forall x, y, z \in A)(((x\cdot y)\cdot z \in J \, \wedge \, y \in J)\, \Longrightarrow \, x \cdot z \in J).\)

Lemma 7. A t-ideal of a JU-algebra \(A\) is a JU-ideal of \(A\).

Proof. If we put \( x = 1 \), \( y = x \) and \( z = y \) in Dentition 7(t), we get condition (J-2).

In the following proposition we give some of the characteristics of this type of JU-ideals.

Theorem 4. Let \(J\) be a t-ideal of a JU-algerba \(A\), then

• [(e)] \((\forall x , z \in A)(\varphi(x) \cdot z \in J \, \Longrightarrow \, x \cdot z \in J)\);
• [(f)] \((\forall x, z \in A)(\varphi(x) \cdot z \in J \, \Longrightarrow \, \varphi^{2}(x) \cdot z \in J)\); and
• [(g)] \((\forall x \in A)(x \cdot \varphi(x) \in J)\).

Proof. If we put \( y = 1 \) in Definition 7(t), we get (e) with respect to (J-1). Let \(x, z \in A\) be such that \(\varphi(x) \cdot z \in J\). Then \[\varphi(\varphi^{2}(x))\cdot z = \varphi^{3}(x)\cdot z = \varphi(x)\cdot z \in J \,\Longrightarrow \, \varphi^{2}(x)\cdot z \in J\] by (e). To prove (g), we note that \((x \cdot 1)\cdot (x \cdot 1) = 1 \in J \) and \( 1 \in J\) follows \( x \cdot (x \cdot 1) \in J\) by Definition 7(t). So we have \(x \cdot \varphi(x) \in J\).

Proposition 5. The condition (e) of Theorem 4 is equivalent to the condition (t) of Definition 7.

Proof. Theorem 4 already proves that (t) \(\Longrightarrow\) (e), the inverse implication (e) \(\Longrightarrow\) (t) remains to be proved. We can write (JU-1) in the form \((y \cdot z) \cdot (y \cdot x) \leqslant z \cdot x\) according to (J\(_{12}\)). If we put \( z = 1 \), \( x = y \) and \( y = x \) in the previous inequality, we get \(\varphi(x) \cdot (x \cdot y) \leqslant y\). From here, it follows \(\varphi(x) \cdot (x \cdot y) \in J\) by (J-3) and the hypothesis \(y \in J\). On the other hand, if we put \( z = x \cdot y \), \( x = z \) and \( y = \varphi(x) \) in (JU-1) written in the form \((y \cdot z)\cdot (y \cdot x) \leqslant z \cdot x\), then we get \((\varphi(x) \cdot (x \cdot y))\cdot (\varphi(x) \cdot z) \leqslant (x \cdot y) \cdot z\). Thus \((\varphi(x) \cdot (x \cdot y))\cdot (\varphi(x) \cdot z) \in J\) by (J-3) and the hypothesis \((x \cdot y) \cdot z \in J\).
Now, from \((\varphi(x) \cdot (x \cdot y))\cdot (\varphi(x) \cdot z) \in J\) and \(\varphi(x) \cdot (x \cdot y) \in J\) it follows \(\varphi(x) \cdot z \in J\). Thus \(x \cdot z \in J\) by (e).

Proposition 6. The condition (f) of Theorem 4 is equivalent to the condition (e) of Theorem 4.

Proof. Let \(x, z \in A\) be elements such that \(\varphi(x)\cdot z \in J\). Then \(\varphi^{2}(x) \cdot z \in J\) by (f). On the other hand, it follows \(x \cdot z \leqslant \varphi^{2}(x) \cdot z\) from (9) with respect to right reverse compatibility of the operation in \(A\) with the order in \(A\). Now, from \(x \cdot z \leqslant \varphi^{2}(x) \cdot z\) and \(\varphi^{2}(x) \cdot z \in J\) it follows \(x \cdot z \in J\) by (J-3). This has shown that formula (e) is a consequence of formula (f).
As Theorem 4 already shows that (e) \(\Longrightarrow\) (f), we conclude that the condition (e) is equivalent to the condition (f).

Proposition 7. The condition (g) of Theorem 4 is equivalent to the condition (e) of Theorem 4.

Proof. Assume (g). Let \(x, y, z \in A\) be elements such that \(\varphi(x) \cdot z \in J\). If we put \( y = x \), \( x = z \) and \( z = \varphi(x) \) in (JU-1), written in the form \((z \cdot x) \cdot (t \cdot x) \leqslant y \cdot z\), we get \((\varphi(x) \cdot z)\cdot (x \cdot z) \leqslant x \cdot \varphi(x)\). From here, it follows \((\varphi(x) \cdot z)\cdot (x \cdot z) \in J\) by (J-3) and the hypothesis (g): \(\varphi(x) \cdot x \in J\). Now, from \((\varphi(x) \cdot z)\cdot (x \cdot z) \in J\) it follows \(x \cdot z \in J\) by (J-2) and hypothesis \(\varphi(x) \cdot z \in J\). We have shown by this that (g) \(\Longrightarrow\) (e).
In Theorem 4 it is shown that (t) \(\Longrightarrow\) (g). Since (t) \(\Longleftrightarrow\) (g) is a valid formula, by Proposition 5, we conclude that the equivalence of (e) \(\Longleftrightarrow\) (g) is a valid formula, too.

The concept of p-ideals of a JU-algebra was introduced and analyzed in [8].

Definition 8.[8] A subset \(J\) of a JU-algebra \(A\) is called a p-ideal of \(A\) if

• [(J-1)] \(1 \in J\) and
• [(p)] \((\forall x, y, z \in A)((z \cdot x) \cdot (z \cdot y) \in J \, \wedge \, y \in J)\, \Longrightarrow \, x \in J).\)

Lemma 8. Any p-ideal \( J \) of a JU-algebra \( A \) is a JU-ideal of \( A \).

Proof. If we put \( z = 1 \) in Definition 8(p), we get (JU-2).

Theorem 5. Let \(J\) be a p-ideal of a JU-algerba \(A\). Then

• (h) \((\forall y, z \in A)((z \cdot x) \cdot \varphi(z) \in J \, \Longrightarrow \, x \in J)\);
• (k) \((\forall x, z \in A)(z \cdot \varphi(z \cdot x) \in J \, \Longrightarrow \, x \in J)\);
• (m) \((\forall x, z \in A)((\varphi(z \cdot x)\in J \, \wedge \, z \in J)\, \Longrightarrow \, x \in J)\);
• (n) \((\forall x \in A)(\varphi(x) \in J \, \Longrightarrow \, x \in J)\).

Proof. If we put \( y = 1 \) in Definition 8(p), we get (h).
The condition (k) is obtained from condition (h) by applying equality (1) of Proposition 1.
If we put \( y = z \) in in Definition 8(p), we get (m).
If we put \(z = 1\) in (m), we get (n).

In what follows, we introduce and analyze a new type of JU-ideal in JU-algebras.

Definition 9. Let \(J\) be a JU-ideal of a \(A\). \(J\) is a (\(\star\))-ideal of \(A\) if

• [(\(\star\))] \((\forall x, y \in A)(\neg(x \in J) \,\wedge \, y \in J)\, \Longrightarrow \, x \cdot y \in J)\).

Proposition 8. Let \(J\) be a (\(\star\))-ideal of a JU-algebra \(A\). Then

• [(q)] \((\forall x \in A)(\neg(x \in J) \, \Longrightarrow \, \varphi(x) \in J)\); and
• [(r)] \((\forall x, y \in A)((\neg(x \cdot y \in J) \, \wedge \, y \in J)\, \Longrightarrow \, x \in J)\).

Proof. The Condition (q) is obtained by putting \( y = 1 \) in Definition 9(\(\star\)). The Condition (q) can be obtained from the contraposition of Definition 9(\(\star\)).

Theorem 6. An ideal \(J\) of a JU-algebra \(A\) is a closed (\(\star\))-ideal if and only if \(\varphi(A) = \{\varphi(x) : x\in A \} \, \subseteq \, J\).

Proof. Let \(J\) be an JU-ideal of \(A\). If \(\varphi(A) \, \subseteq \, J\), then obviously \(\varphi(J) \, \subseteq \, J\) holds, i.e., the ideal \(J\) is closed.
Let \(x, y \in A\) be arbitrary elements. Then \(y \cdot (x \cdot y) \leqslant \varphi(x)\in J\) by Proposition 1(3) and hypothesis \(\varphi(A) \, \subseteq \, J\). Thus \(y \cdot (x \cdot y) \in J\) by Lemma 2. Hence \(x \cdot y \in J\) by (J-2). So, \(J\) is a (\(\star\))-ideal of \(A\).
Let \( J \) be a closed (\(\star\))-ideal of \( A \). For \(x \in A\), we have \(x \in A \, \vee \, \neg(\in J)\). If \(x \in J\), then \(\varphi(x) \in J\) because \( J \) is a closed ideal of \( A \).
If \(\neg(x \in J)\), then from \(\neg(x \in J) \, \wedge \, 1 \in J\) it follows \(\varphi(x) = x \cdot 1 \in J\) because \( J \) is a (\(\star\))-ideal of \( A \).

Before we finish this section, let us introduce another type of JU-ideal.

Definition 10. Let \(J\) be a JU-ideal of a JU-algebra \(A\). \(J\) is an associative JU-ideal of \(A\) is

• [(A)] \((\forall x, z \in A)(\varphi(z) \cdot x \in J \, \Longrightarrow \, x \cdot z \in J)\).

Theorem 7. An associative ideal \(J\) of a JU-algebra \(A\) is closed and ag-ideal of \(A\).

Proof. If we put \( x = 1 \) in Definition 10(A), we get \(\varphi(z) \cdot 1 \in J \, \Longrightarrow \, 1 \cdot z \in J\), i.e. we get \(\varphi^{2}(z) \in J \, \Longrightarrow \, z \in J\). This means that \(J\) is an ag-ideal of \(A\). If we put \( z = 1 \) in Definition 10(A), we get \(\varphi(1) \cdot x \in J \, \Longrightarrow \, x \cdot 1 \in J\), i.e. we get \(x \in J \, \Longrightarrow \, \varphi(x) \in J\). So, \(J\) is a closed JU-ideal of \(A\).

Conclusion

In [7, 8], the concept of JU-algebras is introduced and analyzed. However, this concept was introduced earlier in [9] under the name ‘pseudo KU-algebra’. This author is more inclined to use the term ‘weak KU-algebra’ for this generalization of KU-algebras. In this paper, we have introduced and analyzed the concepts of a few new types of JU-ideals of a JU-algebra such as closed ideal, ag-ideal, t-ideal, (\(\star\))-ideal and associative ideal. This article opens the possibility of introducing and analyzing several different types of JU filters (Definition 6) in these algebras.

Acknowledgments

The author would like to thank the College of Natural Sciences, Jimma University for funding this research work.

Conflict of Interests

The authors declare no conflict of interest.