Pseudo-valuations and pseudo-metric on JU-algebras

1. Introduction

Pseudo-valuations in residuated lattices was introduced by Busneag [1] where many theorems based on pseudo-valuations in lattice terms and their extension theorem for residuated lattices to pseudo-valuation from valuations are shown using the model of Hilbert algebras [2]. But in fact Pseudo-valuations on a Hilbert algebras was initially introduced by Busneag [3] where it is proved that every pseudo-valuation induces a pseudometric on a Hilbert algebra. Further Busneag [2] proved many results on extensions of pseudo-valuation.

Logical algebras have become the keen interest for researchers in recent years and intensively studied under the influence of different mathematical concepts. Doh and Kang [4] introduced the concept of pseudo-valuation on BCK/BCI algebras and studied several results based on them. Ghorbani [5] defined congruence relations and gave quotient structure of BCI-algebras based on pseudo-valuation. Zhan and Jun [6] studied pseudo valuation on \(R_{0}\)-algebras. Based on the concept of pseudo-valuation in \(R_{0}\)-algebras, Yang and Xin [7] characterized pseudo pre-valuations on EQ-algebras. Mehrshad and Kouhestani studied Pseudo-Valuations on BCK-Algebras [8]. Pseudo-valuations on a BCC-algebra was introduced by Jun et al. [9], where they have shown that binary operation in BCC-algebras is uniformly continuous. Recently Moin et al. [16] introduced JU-algebras and their \(p\)-closure ideals.

UP-algebras were introduced by Iampan [10] as a new branch of logical algebras. Naveed et. al [11] introduced the concept of cubic KU-ideals of KU-algebras. Moin and Ali [12] have given the concept of roughness in KU-Algebras recently whereas rough set theory in UP-algebras have been introduced and studied by Moin et al. [13]. Next, graph associated to UP-algebras was introduced by Moin et al. [14]. Daniel studied pseudo-valuations on UP-algebras in [15].

In this paper, we focus on pseudo-valuation which is applied to JU-algebras and discuss related results. We define pseudo-valuations on JU-algebras using the model of Busneag and introduce a pseudo-metric on JU-algebras. We also prove that the binary operation defined on JU-algebras is uniformly continuous under the induce pseudo-metric.

2. Preliminaries and basic properties of JU-algebras

In this section, we shall introduce JU-algebras, JU-subalgebras, JU-ideals and other important terminologies with examples and some related results.

Definition 1. An algebra \((A,\diamond ,1)\) of type \((2,0)\) with a single binary operation \(\diamond \) is said to be JU-algebras satisfying the following identities: for any \(u,v,w\in X,\)
\((JU_1)\) \((u\diamond v)\diamond \lbrack (v\diamond w)\diamond (u\diamond w)]=1,\)
\((JU_2)\) \(1\diamond u=u,\)
\((JU_3)\) \(u\diamond v=v\diamond u=1\) implies \(u=v.\)

We call the constant \(1\) of \(X\) the fixed element of \(X.\) For the sake of convenience, we write \(X\) instead of \((X, \diamond , 1)\) to represent a JU-algebra. We define a relation \(“\leq “\) in \(X\) by \(v\leq u\) if and only if \(u\diamond v=1.\) If we add the condition \(u\diamond 1=1\) for all \(u\in X\) in the definition of JU-algebras, then we get that \(X\) is a KU-algebra. Therefore, JU-algebra is a generalization of KU-algebras.

Lemma 2. If \(X\) is a JU-algebra, then \((X, \leq )\) is a partial ordered set i.e.,
\((J_4)\) \(u\leq u,\)
\((J_5)\) \(u\leq v, v\leq u,\) implies \(u=v,\)
\((J_6)\) \(u\leq w, w\leq v,\) implies \(u\leq v.\)

Proof. Putting \(v=w=1\) in \((JU_{1})\) we get \(u\diamond u=1,\) i.e. \(u\leq u\) which proves \((J_4).\) \((J_5)\) directly follows from \((JU_3).\) For \((J_6)\) take \(u\leq w\) and \(w\leq v\) implies that \(w\diamond u=1\) and \(v\diamond w=1.\) By \((JU_1)\), we have \(v\diamond u=1\) implies that \(u\leq v.\)

Further we have the following Lemma for a JU-algebra \(X.\)

Lemma 3. If \(A\) is a JU-algebra, then following inequalities holds for any \(u,v,w\in A\):
\((J_7)\) \(u\leq v\) implies \(v\diamond w\leq u\diamond w,\)
\((J_8)\) \(u\leq v\) implies \(w\diamond u\leq w\diamond v,\)
\((J_9)\) \((w\diamond u)\diamond (v\diamond u)\leq v\diamond w,\)
\((J_{10})\) \((v\diamond u)\diamond u\leq v.\)

Proof. \((J_7),\;(J_8)\) and \((J_9)\) follows from \((JU_1)\) by adequate replacement of elements. \((J_{10})\) follows from \((JU_1)\) and Definition 1.

Next, we have the following Lemmas.

Lemma 4. Any JU-algebra \(X\) satisfies following conditions for any \(u, v, w\in A,\)
\((J_{11})\) \(u\diamond u=1,\)
\((J_{12})\) \(w\diamond (v\diamond u)=v\diamond (w\diamond u),\)
\((J_{13})\) If \((u\diamond v)\diamond v=1,\) then \(A\) is a KU-algebra,
\((J_{14})\) \((v\diamond u)\diamond 1=(v\diamond 1)\diamond (u\diamond 1).\)

Proof. Putting \(v=w=1\) in \(JU_1,\) we get; \(u\diamond u=1\) which proves (\(J_{11}\)). For \((J_{12})\), we have \((w\diamond u)\diamond u\leq w \). By putting \(v=1\) in \((JU_1)\) and using (\(J_7\)), we get

Replace \(w\) by \(w\diamond u\) in \((JU_1)\), we get \([v\diamond (w\diamond u)]\diamond [((w\diamond u)\diamond u)\diamond (v\diamond u)]=1\), which implies From (1), (2) and Lemma 2(\(J_6\)) we get, Further by replacing \(v\) with \(w\) and \(w\) with \(v\) in (3), we get Now (3), (4) and (\(J_5\)) yields, \(w\diamond (v\diamond u)=v\diamond (w\diamond u).\) In order to prove \((J_{13})\), we just needs to show that \(u\diamond 1=1, \;\; \forall \;\; u\in A.\) Replacing \(v\rightarrow 1, u\rightarrow 1, w\rightarrow u\) in \((JU_1),\) we obtained, \((1\diamond u)\diamond [(u\diamond 1)\diamond (1\diamond 1)]=1\Rightarrow u\diamond [(u\diamond 1)\diamond 1]=1\Rightarrow u\diamond 1=1\) (by using \(v=1\) in the given condition of \((J_{13})\)).
Using \((J_{12})\) for any \(u,v\in A\) in the followings we see that, \((v\diamond 1)\diamond (u\diamond 1)=(v\diamond 1)\diamond u\diamond [(v\diamond u)\diamond (v\diamond u)]=(v\diamond 1)\diamond [(v\diamond u)\diamond (u\diamond (v\diamond u))]= (v\diamond u)\diamond [(v\diamond 1)\diamond (v\diamond (u\diamond u))]= (v\diamond u)\diamond [(v\diamond 1)\diamond (v\diamond 1)]=(v\diamond u)\diamond 1,\) which shows that \((J_{14})\) holds.

Definition 5. A non-empty subset \(I\) of a JU-algebra \(A\) is called a JU-ideal of \(A\) if it satisfies the following conditions:
(1)\(\ 1\in I,\)
(2) \(\ u\diamond (v\diamond w)\in I,\) \(v\in I\) implies \(u\diamond w\in I,\) for all \(% u,v,w\in I.\)

3. Pseudo-valuations and pseudo-metric

Definition 6. A real-valued function \(\vartheta \) on a JU-algebra \(A\) is called a pseudo-valuation on \(A\) if it satisfies the following two conditions:
(1) \(\vartheta (1) = 0,\)
(2) \({\vartheta (u\diamond w)\leq \vartheta (u\diamond (v\diamond w))+\vartheta(v)}\) for all \( u, v, w\in A.\) A pseudo-valuation \(\vartheta \) on a JU-algebra \(A\) satisfying the following condition:
\(\vartheta (u)= 0\Rightarrow u=1\) for all \( u \in A\) is called a valuation on \(A\).

Example 1. Let \(A=\{1, 2, 3, 4\}\) be a set with operation \(\diamond \). A Cayley table for \(A\) is defined as%

\(\diamond \)	\(1\)	\(2\)	\(3\)	\(4\)
\(1\)	\(1\)	\(2\)	\(3\)	\(4\)
\(2\)	\(1\)	\(1\)	\(1\)	\(4\)
\(3\)	\(1\)	\(2\)	\(1\)	\(4\)
\(4\)	\(1\)	\(2\)	\(1\)	\(1\)

Here \(A\) is a JU-algebra. We find that a real valued function defined on \(A\) by \(\vartheta (1)=0,\) \(\vartheta (2)=\vartheta (3)=1,\) and \(\vartheta (4)=3,\) is a pseudo-valuation on \(A\).

Proposition 7. Let \(\vartheta \) be a pseudo-valuation on a JU-algebras \(A\). Then we have
(1) \(u\leq v\Rightarrow \vartheta (v)\leq \vartheta (u).\)
(2) \(\vartheta ((u\diamond (v\diamond w))\diamond w)\leq \vartheta (u)+\vartheta (v)\) for all \( u, v, w\in A.\)

Proof. (1) Let \(u, v\in A \) be such that \(u \leq v\). Replacing \(u=1,\) \(v=u,\) \(w=v\) in Definition 6 and Definition 1, we get \(\vartheta (v)=\vartheta (1\diamond v)\leq \vartheta (1\diamond (u\diamond v))+\vartheta(u)=\vartheta (1\diamond 1)+\vartheta(u)=\vartheta(1)+\vartheta(u)=\vartheta(u).\)
(2) If we replace \(u\) by \(u\diamond (v\diamond w)\) in Definition 6(2), then we get $$ \vartheta ((u\diamond (v\diamond w))\diamond w)\leq \vartheta ((u\diamond (v\diamond w))\diamond (v\diamond w))+\vartheta (v),$$ again applying Definition 6 (2) by choosing \(u=u\diamond (v\diamond w)\) and \(w=v\diamond w\), we get $$ \vartheta ((u\diamond (v\diamond w))\diamond w)\leq \vartheta [(u\diamond (v\diamond w))\diamond (u\diamond(v\diamond w))]+\vartheta(u)+\vartheta (v)=\vartheta(1)+\vartheta(u)+\vartheta(v)$$ $$\Rightarrow \vartheta ((u\diamond (v\diamond w))\diamond w)\leq\vartheta(u)+\vartheta(v).$$

Corollary 8. A pseudo-valuation \(\vartheta\) on a JU-algebra \(A\) satisfies the inequality \(\vartheta (u)\geq 0\) for all \(u\in A.\)

Proposition 9. If \(\vartheta \) is a pseudo-valuation on a JU-algebra \(A\), then we have \(\vartheta ((u\diamond v)\diamond v)\leq \vartheta (u)\) for all \(u, v\in A.\)

Proof. It is easy to see that the required inequality holds by considering \(v=1\) and \(w=v\) in Proposition 7(2) and using Definition 1.

Following results are devoted to find conditions for a real valued function on a JU-algebra \(A\) to be a pseudo-valuation.

Theorem 10. Let \(\vartheta \) be a real valued function on a JU-algebra \(A\) satisfying the following conditions:
(a) If \(\vartheta (a)\leq \vartheta (u)\) for all \( u\in A\), then \(\vartheta (a)=0,\)
(b) \(\vartheta (u\diamond v)\leq \vartheta (v)\) for all \( u, v\in A,\)
(c) \(\vartheta ((u\diamond (v\diamond w))\diamond w)\leq \vartheta (u)+\vartheta (v),\)
(d) \(\vartheta (v\diamond (u\diamond w))\leq \vartheta (u\diamond (v\diamond w)).\)
Then \(\vartheta \) is a pseudo-valuation on \(A.\)

Proof. From Lemma 4 and given condition (b), we have \(\vartheta (1)= \vartheta (u\diamond u)\leq \vartheta (u)\) for all \( u\in A\) and hence \(\vartheta (1)=0,\) using given condition (a). Now, from Definition 1, Lemma 4 and given condition (c), we get \(\vartheta (v)=\vartheta (1\diamond v)= \vartheta (((u\diamond v)\diamond (u\diamond v))\diamond v)\leq \vartheta (u\diamond v)+ \vartheta (u)\) for all \( u, v\in A\). It follows from given condition (d) that \(\vartheta (u\diamond w)\leq \vartheta (v\diamond (u\diamond w))+ \vartheta (v)\leq \vartheta (u\diamond (v\diamond w))+ \vartheta (v)\) for all \( u, v, w\in A\). Therefore \(\vartheta \) is a pseudo-valuation on \(A\).

Corollary 11. Let \(\vartheta \) be a real-valued function on a JU-algebra \(A\) satisfying the following conditions:
(a) \(\vartheta (1)=0,\)
(b) \(\vartheta (u\diamond v)\leq \vartheta (v)\), for all \( u, v\in A,\)
(c) \( \vartheta ((u\diamond (v\diamond w)\diamond w))\leq \vartheta (u) + \vartheta (v)\) for all \( u, v, w\in A\),
(d) \(\vartheta (v\diamond (u\diamond w))\leq \vartheta (u\diamond (v\diamond w)).\)
Then \(\vartheta \) is a pseudo-valuation on \(A\).

Theorem 12. If \(\vartheta \) is a pseudo-valuation on a JU-algebra \(A\), then \(\vartheta (v)\leq \vartheta (u\diamond v) + \vartheta (u)\), for all \( u, v\in A\).

Proof. Let \(m = (u\diamond v)\diamond v\) for any \(u, v\in A\), and \(n = u\diamond v\). Then \(v = 1\diamond v= (((u\diamond v)\diamond v)\diamond ((u\diamond v)\diamond v))\diamond v = (m \diamond (n \diamond v))\diamond v\). It follows from Proposition{} \ref{p1}(2) and Proposition{} \ref{p2} that \(\vartheta (v) = \vartheta ((m \diamond (n\diamond v))\diamond v)\leq \vartheta (m)+ \vartheta (n) = \vartheta ((u\diamond v)\diamond v)+ \vartheta (u\diamond v) \leq \vartheta (u) + \vartheta (u\diamond v)\). This completes the proof.

Theorem 13. Let \(\vartheta \) be a real-valued function on a JU-algebra \(A\) satisfying the following conditions.
(1) \(\vartheta (1)=0\),
(2) \(\vartheta (v)\leq \vartheta (u\diamond v)+ \vartheta (u)\),
(3) \(\vartheta (v\diamond (u\diamond w))\leq \vartheta (u\diamond (v\diamond w))\) for all \( u, v, w \in A.\)
Then \(\vartheta \) is a pseudo-valuation on \(A\).

Proof. For any \(u,v,a,b\in A,\) and using 4 with given condition (2) and (3) we get, \(\vartheta (u\diamond v)\leq \vartheta (v\diamond (u\diamond v))+\vartheta (v) \leq \vartheta (u\diamond (v\diamond v))+\vartheta (v)=\vartheta (v\diamond (1))+\vartheta (v)=\vartheta (1)+\vartheta (v)=\vartheta(v).\) Also, \begin{eqnarray*}\vartheta [(b\diamond (a\diamond u))\diamond u] &\leq& \vartheta [a\diamond ((b\diamond (a\diamond u))\diamond u)]+ \vartheta (a)\\ &\leq& \vartheta [(b\diamond (a\diamond u))\diamond (a\diamond u)]+ \vartheta (a)\\ &\leq& \vartheta [b\diamond[(b\diamond (a\diamond u))\diamond (a\diamond u)]]+\vartheta(a)+\vartheta (b)\\ &\leq&\vartheta [(b\diamond (a\diamond u))\diamond(b\diamond (a\diamond u))] + \vartheta (a)+\vartheta (b)\\ &=&\vartheta (1)+ \vartheta (a)+\vartheta (b)\\ &=&\vartheta (a) + \vartheta (b). \end{eqnarray*} By Corollary 11, we get that \(\vartheta \) is a pseudo-valuation on \(A\).

Proposition 14. If \(\vartheta \) is a pseudo-valuation on a JU-algebra \(A\), then

Proof. Suppose that \(a, b, u\in A\) such that \(a\leq b\diamond u\). Then by Proposition 7 (2) and Theorem 12, we have
\(\vartheta (u)\leq \vartheta ((a\diamond (b\diamond u))\diamond u)+ \vartheta (a\diamond (b\diamond u)) = \vartheta ((a \diamond (b\diamond u))\diamond u) + \vartheta (1) = \vartheta ((a\diamond (b\diamond u))\diamond u)\\ \leq \vartheta (a) + \vartheta (b).\)

Proposition 15. Suppose that \(A\) is JU-algebra. Then every pseudo-valuation \(\vartheta \) on \(A\) satisfies the following inequality: \(\vartheta (u\diamond w)\leq \vartheta (u\diamond v) + \vartheta (v\diamond w)\), for all \( u, v, w\in A.\)

Proof. It follows from \(JU_1\) and Proposition 14.

Theorem 16. If \(\vartheta \) is a pseudo-valuation on a JU-algebra \(A\), then the set \(I:=\{u\in A|\; \vartheta (u) = 0\}\) is an ideal of \(A\).

Proof. We have \(\vartheta (1) = 0\) and hence \(1\in I\). Next, \(u, v, w\in A\) be such that \(v\in I\) and \(u\diamond (v\diamond w)\in I\). Then \(\vartheta (v) = 0\) and \(\vartheta (u\diamond (v\diamond w))=0\). By 6(2), we get \(\vartheta (u\diamond w)\leq \vartheta (u\diamond (v\diamond w)) + \vartheta (v) = 0\) so that \(\vartheta (u\diamond w) = 0\). Hence \(u\diamond w\in I\), therefore \(I\) is an ideal of \(A\).

Example 2. [16] Let \(A=\{1,2,3,4,5\}\) in which \(\diamond \) is defined by the following table

\(\diamond \)	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)
\(1\)	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)
\(2\)	\(1\)	\(1\)	\(3\)	\(4\)	\(5\)
\(3\)	\(1\)	\(2\)	\(1\)	\(4\)	\(4\)
\(4\)	\(1\)	\(1\)	\(3\)	\(1\)	\(3\)
\(5\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)

It is easy to see that \(A\) is a JU-algebra. Now, define a real-valued function \(\vartheta \) on \(A\) by \(\vartheta (1)=\vartheta (2)=\vartheta (3)=0\), \(\vartheta (4)=3\), and \(\vartheta (5)=1.\) Then \(I:=\{u\in A \mid \vartheta (u) = 0\}\) = \(\{1,2, 3\}\) is the ideal of \(A.\) But \(\vartheta \) is not a pseudo-valuation as \(\vartheta (3\diamond 5)\nleq \) \(\vartheta (3\diamond (5\diamond 5))\) + \(\vartheta (5).\)

For a real-valued function \(\vartheta \) on a JU-algebra \(A\), define a mapping \(d_{\vartheta }: X\times X\rightarrow \mathbb{R} \) by \(d_{\vartheta} (u, v) = \vartheta (u\diamond v) + \vartheta (v\diamond u)\) for all \( (u, v)\in A\times A\). We have following result.

Theorem 17. Let \(A\) is a JU-algebra. If a real-valued function \(\vartheta \) on \(A\) is a pseudo-valuation on \(A\), then \(d_{\vartheta }\) is a pseudo-metric on \(A\), and so \((X, d_{\vartheta })\) is a pseudo-metric space. (The \(d_{\vartheta }\) is called pseudo-metric induced by pseudo-valuation \(\vartheta \).)

Proof. Clearly, \(d_{\vartheta }\) \((u, v)\) \(\geq 1\), \(m_{\vartheta }\) \((u, u)=1\) and \(m_{\vartheta }\) \((u, v)\) = \(m_{\vartheta }(v, u)\) for all \( u, v\in A\). For any \(u, v, w\in A\) from Proposition 15, we get \(d_{\vartheta }(u, v)+ d_{\vartheta }(v, w) =[\vartheta (u\diamond v)+ \vartheta (v\diamond u)]+[\vartheta (v\diamond w)+ \vartheta (w\diamond v)] = [\vartheta (u\diamond v) + \vartheta (v\diamond w)]+[\vartheta (w\diamond v)+ \vartheta (v\diamond u)]\geq \vartheta (u\diamond w) + \vartheta (w\diamond u) = d_{\vartheta }(u, w)\). Hence \((X, d_{\vartheta })\) is a pseudo-metric space.

Proposition 18. Let \(A\) is a JU-algebra. Then every pseudo-metric \(d_{\vartheta }\) induced by pseudo-valuation \(\vartheta \) satisfies the following inequalities:
(1) \(d_{\vartheta }(u, v) \geq d_{\vartheta } (x\diamond u, x\diamond v)\),
(2) \(d_{\vartheta }(u \diamond v, x\diamond y) \leq d_{\vartheta }(u\diamond v, x\diamond v) + d_{\vartheta }(x\diamond v, x\diamond y)\)
for all \( u, v, x, y\in A\).

Proof. (1) Let \(u, v, a\in A\). By \(JU_1\) \(u\diamond v\leq (x\diamond v)\diamond (x\diamond u)\) and \(v\diamond u\leq (x\diamond u)\diamond (x\diamond v)\). It follows from Proposition \ref{p1}(1) that \(\vartheta (u\diamond v)\geq \vartheta ((x\diamond v)\diamond (x\diamond u))\) and \(\vartheta (v\diamond u)\geq \vartheta ((x\diamond u)\diamond (x\diamond v))\). So \(d_{\vartheta }(u, v) = \vartheta (u\diamond v)+ \vartheta (v\diamond u)\geq \vartheta ((x\diamond u) \diamond (x\diamond u))\)+ \(\vartheta ((x\diamond u)\diamond (x\diamond v)) = d_{\vartheta }(x\diamond u, x\diamond v).\)
(2) Followed by definition of pseudo-metric.

Theorem 19. Let \(\vartheta \) be a real-valued function on a JU-algebra \(A\), if \(d_{\vartheta }\) is a pseudo-metric on \(A\), then \((X\times X, d_{\vartheta }^\diamond)\) is a pseudo-metric space, where $$d^\diamond _{\vartheta }((u, v), (a, b)) = \max\{d_{\vartheta } (u, a), d_{\vartheta }(v, b)\} \hbox{ for all } (u, v), (a, b) \in A\times A.$$

Proof. Suppose \(d_{\vartheta }\) is a pseudo-metric on \(A\). For any \((u, v), (a, b)\in A\times A\), we have \(d^\diamond _{\vartheta }((u, v), (u, v))\) = \(\max\{d_{\vartheta }(u, u), d_{\vartheta }(v, v)\} = 0\) and $$d^\diamond _{\vartheta }((u, v), (a, b)) = \max \{d_{\vartheta }(u, a), d_{\vartheta }(v, b)\} = \max \{d_{\vartheta }(a, u), d_{\vartheta }(b, v)\} = d^\diamond ((a, b), (u, v)).$$ Now let \((u, v), (a, b), (u, v)\in A\times A\). Then we have \begin{eqnarray*}d^\diamond _{\vartheta }((u, v), (u, v))+ d^\diamond _{\vartheta }((u, v), (a, b)) &=& \max \{d_{\vartheta }(u, u), d_{\vartheta }(v, v)\} + \max \{d_{\vartheta }(u, a), d_{\vartheta }(v, b)\}\\ &\geq& \max\{d_{\vartheta }(u, u)+ d_{\vartheta }(u, a), d_{\vartheta }(v, v) + d_{\vartheta }(v, b)\}\\&\geq& \max\{d_{\vartheta }(u, a), d_{\vartheta }(v, b)\} = d^\diamond _{\vartheta }((u, v), (a, b)).\end{eqnarray*} Hence \((X\times X, d^\diamond _{\vartheta })\) is a pseudo-metric space.

Corollary 20. If \(\vartheta : X\to \mathbb{R}\) is a pseudo-valuation on a JU-algebra \(A\), then \((X\times X, d^\diamond _{\vartheta })\) is a pseudo-metric space.

Theorem 21. Let \(A\) is a JU-algebra. If \(\vartheta : X\to \mathbb{R}\) is a valuation on \(A\), then \((X, d_{\vartheta })\) is a metric space.

Proof. Suppose \(\vartheta \) is a valuation on \(A\), then \((X, d_{\vartheta })\) is a pseudo-metric space by Theorem 19. Further consider \(u, v\in A\) be such that \(d_{\vartheta }(u, v) = 0\), then \(0 = d_{\vartheta }(u, v) = \vartheta (u\diamond v)+ \vartheta (v\diamond u)\), and hence \(\vartheta (u\diamond v) = 0\) and \(\vartheta (v\diamond u) = 0\) since \(\vartheta (u)\geq 0\) for all \( u\in A\) and, since \(\vartheta \) is a valuation on \(A\), it follows that \(u\diamond v = 1\) and \(v\diamond u = 1\) so from (condition in the given theorem) that \(u = v\). Hence \((X, d_{\vartheta })\) is a metric space.

Theorem 22. Let \(A\) is a JU-algebra. If \(\vartheta : X\to \mathbb{R}\) is a valuation on \(A\), then \((X\times X, d^\diamond _{\vartheta })\) is a metric space.

Proof. From Corollary 20, we know that \((X\times X, d^\diamond _{\vartheta })\) is a pseudo-metric space. Suppose \((u, v), (a, b)\in A\times A\) be such that \(d^\diamond _{\vartheta }((u, v), (a, b)) = 0\), then \(0 = d^\diamond _{\vartheta }((u, v), (a, b))\) = \(\max \{d_{\vartheta }(u, a), d_{\vartheta }(v, b)\}\), and so \(d_{\vartheta }(u, a) = 0\) = \(d_{\vartheta }(v, b)\). Since \(d_{\vartheta }(u, v) \geq 0\; \) for all \( (u, v)\in A\times A\). Hence \(0 = d_{\vartheta }(u, a) = \vartheta (u\diamond a) + \vartheta (a\diamond u)\) and \(0 = d_{\vartheta }(v, b) = \vartheta (v\diamond b) + \vartheta (b\diamond v)\). It follows that \(\vartheta (u\diamond a) = 0 = \vartheta (a\diamond u)\) and \(\vartheta (v\diamond b) = 0 = \vartheta (b\diamond v)\) so that \(u\diamond a = 1 = a \diamond u\) and \(v\diamond b = 0 = b\diamond v\). Now we have \(a = u\) and \(b = v\), and so \((u, v) = (a, b)\), therefore \((X\times X, d^\diamond _\vartheta )\) is a metric space.

Theorem 23. Let \(A\) is a JU-algebra. If \(\vartheta \) is a valuation on \(A\), then the operation \(\diamond \) in \(A\) is uniformly continuous.

Proof. Consider for any \(\epsilon \geq 0\), if \(d^\diamond _{\vartheta }((u, v), (a, b)) < \frac {\epsilon} {2}\) then \(d_{\vartheta }(u, a) < \frac {\epsilon} {2}\) and \(d_{\vartheta }(v, b) < \frac {\epsilon} {2}.\) This implies that \(d_{\vartheta }(u\diamond v, a\diamond b) \leq d_{\vartheta }(u\diamond v, a\diamond v)+ d_{\vartheta }(a\diamond v, a\diamond b)\leq d_{\vartheta }(u, a)+ d_{\vartheta }(v, b) < \frac {\epsilon} {2}\)+ \(\frac {\epsilon} {2}=\epsilon \) (from 18). Therefore the operation \(\diamond :X\times X\to A\) is uniformly continuous.

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Competing Interests

The author(s) do not have any competing interests in the manuscript.