Complete convergence and complete integral convergence of partial sums for moving average process under sub-linear expectations

1.   Introduction

${MS}$-algebras were considered by T. S. Blyth and J. C. Varlet^[15] as a common properties of de Morgan algebras and Stone algebras. T. S. Blyth and J. C. Varlet^[16] described the lattice $\Lambda{({MS})}$ of all subclasses of the class ${MS}$ of all $MS$-algebras by identities.

A. Badawy, D. Guffov$\acute{a}$ and M. Haviar^[10] introduced and characterized the class of principal $MS$-algebras and the class of decomposable $MS$-algebras by means of principal $MS$-triples and decomposable $MS$-triples, respectively. They obtained a one-to-one correspondence between principal $MS$-algebras (decomposable $MS$-algebras) and principal $MS$-triples (decomposable $MS$-triples). Moreover, they proved that the class of principal $MS$-algebras is a subclass of the class of decomposable $MS$-algebras.

A. Badawy^[6] established the relationship between de Morgan filters and congruences of a decomposable $MS$-algebra. A. Badawy^[3] introduced the notion of $d_{L}$-filters of a principal $MS$-algebra and characterized certain congruences in terms of $d_{L}$-filters. Recently, A. Badawy, M. Haviar and M. Plo$\check{s}\check{c}$ica^[11] introduced and described the notions of congruence pairs and perfect extensions of principal $MS$-algebras. They characterized the lattice of congruences of a principal $MS$-algebra in terms of congruence pairs. In 2021, S. El-Assar and A. Badawy^[19] characterized permutability of congruences and strong extensions of decomposable $MS$-algebras by means of congruence pairs.

We review in Section 2 many basic concepts and results that we are using throughout this article. We give an example (Example 2.1) to determine the principsl $MS$-triple $(L^{\circ\circ}, D(L), \varphi_{L})$ assosiated with a certain principal $MS$-algebra $L$. In Section $3$, we give equivalent conditions for a pair of congruence $(\theta_{1}, \theta_{2})\in Con(L^{\circ\circ})\times Con_{lat}(D(L))$ to become an $MS$-congruence pair of a principal $MS$-algebra $L$, where $\theta_{1}$ is a congruence on a de Morgan algebra $L^{\circ\circ}$ and $\theta_{2}$ is a lattice congruence on the lattice $D(L)$. Also, we characterize the lattice of all $MS$- congruence pairs which are induced by the Boolean elements of a principal $MS$-algebra. In Section $4$, we discuss many properties of the congruence relation $\Psi$ which defined on a principal $MS$-algebra $L$ by

We prove that $\Psi$ permutes with each congruence relation $\theta$ on $L$. Also, we characterize $2$-permutability of congruences (briefly permutability) by means of $MS$-congruence pairs. We illustrate Examples 3.1 and 4.1 to clarify Theorems 3.1 and 4.4, respectively. Also, Example 4.2 introduces certain principal $MS$-algebra $L$, which has not $2$-permutable congruences as well as $D(L)$ has not also $2$-permutable congruences. In Section $5$, A characterization of $n$-permutability of congruences of a principal $MS$-algebra is given, which is a generalization of the characterization of $2$-permutability of congruences of such algebras.

2.   Preliminaries

In this section, we recall some specific definitions and remarkable results which are discussed in previous articles ^{[10,14,15,16,17,19]}.

Definition 2.1. ^[16] A de Morgan algebra is an algebra $(L; \vee, \wedge, ^{-}, 0, 1)$ of type (2, 2, 1, 0, 0), where $(L; \vee, \wedge, 0, 1)$ is a bounded distributive lattice and the unary operation $^{-}$ satisfying:

Definition 2.2. ^[15] A Stone algebra is an algebra $(L; \vee, \wedge, ^{*}, 0, 1)$ of type $(2, 2, 1, 0, 0)$, where $(L; \vee, \wedge, 0, 1)$ is a bounded distributive lattice and the unary operation $^{*}$ satisfying:

Definition 2.3. ^[15] An $MS$-algebra is an algebra $(L; \vee, \wedge, ^{\circ}, 0, 1)$ of type (2, 2, 1, 0, 0), where $(L; \vee, \wedge, 0, 1)$ is a bounded distributive lattice and a unary operation $^{\circ}$ satisfying:

The class $MS$ of all $MS$-algebras is equational. A de Morgan algebra is an $MS$-algebra satisfying the identity, $a = a^{\circ\circ}$. The class $S$ of a Stone algebra is a subclass of $MS$ satisfying $a\wedge a^{\circ} = 0$.

The basic properties of $MS$-algebras which were shown in^[15] are given in the following theorem.

Theorem 2.1. Let $a, b$ be any two elements of an $MS$-algebra $\; L$. Then

(1) $0^{\circ\circ} = 0$ and $1^{\circ\circ} = 1$,

(2) $a\leq b\Rightarrow b^{\circ}\leq a^{\circ}$,

(3) $a^{\circ\circ\circ} = a^{\circ}$,

(4) $a^{\circ\circ\circ\circ} = a^{\circ\circ}$,

(5) $(a\vee b)^{\circ} = a^{\circ}\wedge b^{\circ}$,

(6) $(a\vee b)^{\circ\circ} = a^{\circ\circ}\vee b^{\circ\circ}$,

(7) $(a\wedge b)^{\circ\circ} = a^{\circ\circ}\wedge b^{\circ\circ}$.

An element $a$ of an $MS$-algebra $L$ is called a closed element of $L$ if $a = a^{\circ\circ}$ and an element $d\in L$ is called a dense element of $L$ if ${d}^{\circ} = 0$.

Theorem 2.2. Let $L$ be an $MS$-algebra. Then

(1) the set $L^{\circ\circ} = \{a\in L:a = a^{\circ\circ}\}$ of all closed elements of $L$ is a de Morgan subalgebra of $L$, see ^[17],

(2) the set $D(L) = \{a\in L:a^{\circ} = 0\}$ of all dense elements of $L$ is a principal filter of $L$, see ^[10].

Definition 2.4. ^[10] An $MS$-algebra $(L; \vee, \wedge, ^\circ, 0, 1)$ is called a principal $MS$-algebra if it satisfies the following conditions:

(1) the filter $D(L)$ is principal, that is, there exists an element $d_{L}\in L$ such that $D(L) = [d_{L})$,

(2) $a = a^{\circ\circ} \wedge (a\vee d_{L})$, for all $a\in L$.

A principal $MS$-algebra $L$ is a principal Stone algebra if $x^{\circ}\vee x^{\circ\circ} = 1$, for all $x\in L$.

Definition 2.5. ^[8] A principal $MS$-triple is $(M, D, \varphi)$, where

(1) $\; M$ is a de Morgan algebra,

(2) $\; D$ is a bounded distributive lattice,

(3) $\; \varphi$ is a $(0, 1)$-lattice homomorphism from $M$ into $D$.

Lemma 2.1. ^[10] Let $L$ be a principal $MS$-algebra. Define a map $\varphi_{L}:L^{\circ\circ}\rightarrow [d_{L})$ by the following rule

Then $\varphi_{L}$ is a $(0, 1)$-lattice homomorphism.

Example 2.1. Consider the $MS$-algebra $L$ in Figure 1.

It is clear that $L$ is a principal $MS$-algebra with the smallest dense element $z$. Then the principal triple $(L^{\circ\circ}, D(L) = [z), \varphi_{L})$ which associated with $L$ is given in Figure 2.

Definition 2.6. ^[22] An equivalence relation $\theta$ on a lattice $L$ is called a lattice congruence if $(a, b)\in \theta$ and $(c, d)\in \theta$ imply $(a\vee c, b\vee d)\in \theta$ and $(a\wedge c, b\wedge d)\in \theta$.

Definition 2.7. An equivalence relation $\theta$ on an $MS$-algebra $L$ is called a congruence on $L$ if

(1) ${\theta}$ is a lattice congruence,

(2) $(a, b)\in \theta$ implies $(a^{\circ}, b^{\circ})\in \theta$.

We use $Con_{lat}(L)$ for the lattice of all lattice congruences of a lattice $(L; \vee, \wedge)$ and $Con(L)$ for the lattice of all congruences of an $MS$-algebra $(L; \vee, \wedge, ^{\circ}, 0, 1)$.

Definition 2.8. Let $L$ be a lattice and $\theta\in Con_{lat}(L)$. Then we define the principal congruence generated by $(a, b)$ which denoted by $\theta(a, b)$ as follows:

If $L$ is an $MS$-algebra and $\theta\in Con(L)$, then

Let $L$ be an $MS$-algebra. Then $(\theta_{L^{\circ\circ}}, \theta_{D(L)})\in Con(L^{\circ\circ})\times Con_{lat}(D(L))$, where $\theta_{L^{\circ\circ}}$ and $\theta_{D(L)}$ are the restrictions of $\theta \in Con(L)$ to $L^{\circ\circ}$ and $D(L)$, respectively. It is clear that $\theta_{L^{\circ\circ}}$ is a congruence relation on a de Morgan algebra $L^{\circ\circ}$ and $\theta_{D(L)}$ is a lattice congruence on a lattice $D(L)$.

The symbols $\bigtriangledown_{L}$ and $\bigtriangleup_{L}$ will be used for the universal congruence $L\times L$ and the equality congruence on $L$, respectively.

The concept of $MS$-congruence pairs is given as follows:

Definition 2.9. ^[11] Let $L$ be a principal $MS$-algebra with a smallest dense element $d_{L}$. A pair of congruences $(\theta_{1}, \theta_{2})\in Con(L^{\circ\circ})\times Con_{lat}(D(L))$ will be called an $MS$-congruence pairs if

A characterization of congruences on a principal $MS$-algebra via $MS$-congruence pairs is given in the following theorem.

Theorem 2.3. ^[11] Let $L$ be a principal $MS$-algebra with the smallest dense element $d_{L}$. For every congruence $\theta$ on $L$, the restrictions of $\theta$ to $L^{\circ\circ}$ and $D(L)$ determine the $MS$-congruence pair $(\theta_{L^{\circ\circ}}, \theta_{D(L)})$.

Conversely, every $MS$-congruence pair $(\theta_{1}, \theta_{2})$ uniquely determines a congruence $\theta$ on $L$ satisfying $\theta_{L^{\circ\circ}} = \theta_{1}$ and $\theta_{D(L)} = \theta_{2}$. Such congruence can be defined by the rule

Throughout this paper, $d_{L}$ denotes to the smallest dense element of a principal $MS$-algebra $L$.

For extra information of $MS$-algebras, principal $MS$-algebras and decomposable $MS$-algebras, we refer the reader to^{[1,2,3,4,5,6,7,8,9,12,13,14,15,16,17,18,20,21,22,23,24]}.

3.   Characterization of $MS$-congruence pairs

In this section, we characterize $MS$-congruence pairs of a principal $MS$-algebra and a principal Stone algebra. Also, we describe the lattice of all $MS$-congruence pairs which induced by the Boolean elements of a principal $MS$-algebra, in fact such lattice forms a Boolean algebra on it is own.

Lemma 3.1. Let $L$ be a principal $MS$-algebra and $(\theta_{1}, \theta_{2})$ be an $MS$-congruence pair. Then

Proof. Let $(a, b)\in \theta_{1}$ and $(x, y)\in \theta_{2}.$ Then $(a\vee d_{L}, b\vee d_{L})\in \theta_{2}$ (by Definition 2.9) and $(x, y)\in \theta_{2}$ imply that $(a\vee x\vee d_{L}, b\vee y\vee d_{L})\in \theta_{2}.$ Therefore $(a\vee x, b\vee y)\in \theta_{2}$ as $x, y\geq d_{L}$. □

Now, we give an important characterization of $MS$-congruence pairs of a principal $MS$-algebra.

Theorem 3.1. Let $L$ be a principal $MS$-algebra with the smallest dense element $d_{L}.$ Then the following statements are equivalent:

(1) $\; (\theta_{1}, \theta_{2})$ is an $MS$-congruence pair of $L$,

(2) $Con_{\varphi_{L}}(L)\subseteq \theta_{2},$ where

Proof. $(1\Rightarrow2)$ Let $(\theta_{1}, \theta_{2})$ be an $MS$-congruence pair of $L$ and $(x, y)\in Con_{\varphi_{L}}(\theta_{1}).$ Then $(x, y) = (\varphi_{L}(a), \varphi_{L}(b)) = (a\vee d_{L}, b\vee d_{L}),$ where $(a, b)\in \theta_{1}.$ Since $(a, b)\in \theta_{1},$ then $(a\vee d_{L}, b\vee d_{L})\in \theta_{2},$ by $(1).$ Thus $(x, y)\in \theta_{2}$. So $Con_{\varphi_{L}}(\theta_{1})\subseteq \theta_{2}$.

$(2\Rightarrow 1)$ Let $Con(\theta_{1})\subseteq \theta_{2}.$ We prove that $(\theta_{1}, \theta_{2})$ is an $MS$-congruence pair of $L$. Let $(a, b)\in \theta_{1}.$ Then by (2), $(a\vee d_{L}, b\vee d_{L}) = (\varphi_{L}(a), \varphi_{L}(b))\in \theta_{2}$. □

A characterization of congruence pairs of a principal Stone algebra is given in the following.

Theorem 3.2. Let $L$ be a principal Stone algebra. Let $(\theta_{1}, \theta_{2})\in Con(L^{\circ\circ})\times Con_{lat}(D(L)).$ Then the following statements are equivalent:

(1) $(\theta_{1}, \theta_{2})$ is an $MS$-congruence pair of $L$,

(2) $Con_{\varphi_{L}}(\theta_{1})\subseteq \theta_{2}$,

(3) $(a, 1)\in \theta_{1}$ and $u\geq a, u\in D(L)$ imply $(u, 1)\in \theta_{2}$.

Proof. In Theorem 3.1, we proved that $(1)$ and $(2)$ are equivalent.

$(2\Rightarrow 3)$ Let $Con_{\varphi_{L}}(\theta_{1})\subseteq \theta_{2}$. Let $(a, 1)\in \theta_{1}$ and $u\geq a, u\in D(L)$. Then $(\varphi_{L}(a), \varphi_{L}(1))\in \theta_{2}$ by (2). Thus $(a\vee d_{L}, 1)\in \theta_{2}$.

Now $(a\vee d_{L}, 1)\in \theta_{2}$ and $(u, u)\in \theta_{2}, u\geq a, u\in D(L)$ imply $(a\vee u\vee d_{L}, 1\vee u)\in \theta_{2}$. Thus $(u, 1)\in \theta_{2}$ as $u\geq a, d_{L}$.

$(3\Rightarrow 1)$ Since $L$ is a Stone algebra then $L^{\circ\circ}$ is a Boolean subalgebra of $L$ and hence $a\vee a^{\circ} = 1$ for each $a\in L^{\circ\circ}$. Suppose $(3)$ holds and $(a, b)\in \theta_{1}.$ Then $(b^{\circ}, b^{\circ})\in \theta_{1}$ implies $(a\vee b^{\circ}, b\vee b^{\circ}) = (a\vee b^{\circ}, 1)\in \theta_{1}$ and $(a\vee a^{\circ}, b\vee a^{\circ}) = (b\vee a^{\circ}, 1)\in \theta_{1}$. Therefore $(\beta, 1)\in \theta_{1}$, where $\beta = (a\vee b^{\circ})\wedge (a^{\circ}\vee b)$. It is clear that $\beta\in L^{\circ\circ}$ and $\beta\wedge a = \beta\wedge b = a\wedge b$. Since $\beta\leq \beta\vee d_{L}\in D(L)$ and $(\beta, 1)\in \theta_{1}$ then $(\beta\vee d_{L}, 1)\in\theta_{2}$ by (3). Thus $(a\vee d_{L}, a\vee d_{L})\wedge (1, \beta\vee d_{L})\in \theta_{2}$ implies $(a\vee d_{L}, (a\wedge \beta)\vee d_{L}) = (a\vee d_{L}, (a\wedge b)\vee d_{L})\in \theta_{2}$. Similarly, we can get $(b\vee d_{L}, (b\wedge \beta)\vee d_{L}) = (b\vee d_{L}, (a\wedge b)\vee d_{L})\in \theta_{2}$. Then $(b\vee d_{L}, b\vee d_{L})\in \theta_{2}$. □

Example 3.1. Consider the principal $MS$-algebra $L$ in Example 2.1 (Figure 1). The lattices $Con(L)$ and $A(L)$ of all congruences of $L$ and all $MS$-congruence pairs of $L$ are given in Figure 3, respectively.

Where,

And

It is clear that $Con(L)$ isomorphic to $A(L)$ under the isomorphism $\theta \longmapsto (\theta_{L^{\circ\circ}}, \theta_{D(L)})$.

A subset $I$ of a lattice $L$ with $0$ is called an ideal of $L$ if

A principal ideal of $L$ generated by $a \in L$ is defined by

A subset $F$ of a lattice $L$ with $1$ is called a filter of $L$ if

A principal filter of $L$ generated by $a \in L$ is defined by

Definition 3.1. Let $\theta$ be a lattice congruence on a bounded lattice $L$. Then we have the following important subsets

(i) The Kernel of $\theta$ ($Ker \theta)$ is the set $\{x\in L:(x, 0)\in \theta\}$, which is an ideal of L,

(ii) The Cokernel of $\theta$ ($Coker \theta)$ is the set $\{x\in L:(x, 1)\in \theta\}$, which is a filter of L.

Definition 3.2. ^[7] An element $c$ of an $MS$-algebra $L$ is called a Boolean element of $L$ if $c\vee {c}^{\circ} = 1$.

It is ready seen that the set $B(L) = \{c:c\vee c^{\circ} = 1\}$ of all Boolean elements of $L$ forms a Boolean subalgebra of $L^{\circ\circ}$.

Lemma 3.2. Let $c$ be a Boolean element of a principal $MS$-algebra $L.$ Then

(1) $\theta(0, c)$ is a principal congruence relation on $L^{\circ\circ}$ with $Ker\; {\theta(0, c)} = (c]$, where

(2) $\theta(\varphi_{L}{(c)}, 1)$ is a principal congruence relation on $D(L)$ with $Coker\; {\theta(\varphi_{L}(c), 1)} = [\varphi_{L}(c))$, where

Proof.

(1) It is easy to check that $\theta(0, c)$ is a principal congruence relation on $L^{\circ\circ}$ with $Ker\; {\theta(0, c)} = (c]$.

(2) It is easy to show that $\theta(\varphi_{L}(c), 1)$ is an equivalence relation on $D(L)$. Let $(x, y), (m, n)\in \theta(\varphi_{L}(c), 1).$ Thus $x\wedge (c\vee d_{L}) = y\wedge (c\vee d_{L})$ and $m\wedge (c\vee d_{L}) = n\wedge(c\vee d_{L}).$ Then, we get $(x\wedge m, y\wedge n)\in \theta(\varphi_{L}(c), 1)$ and $(x\vee m, y\vee n)\in \theta(\varphi_{L}(c), 1).$ Therefore $\theta(\varphi_{L}(c), 1)$ is a principal congruence on the lattice $D(L)$. Also, we have

□

Now, we observe that every Boolean element $c$ of a principal $MS$-algebra $L$ associated with the $MS$-congruence pair of the form $(\theta(0, c), \theta(\varphi_{L}(c^{\circ}), 1))$.

Theorem 3.3. Let $L$ be a principal $MS$-algebra and $c\in L^{\circ\circ}.$ Then $c$ is a Boolean element of $L$ if and only if $(\theta(0, c), \theta(\varphi_{L}(c^{\circ}), 1))$ is an $MS$-congruence pair of $L$.

Proof. Let $c$ be a Boolean element of $L.$ We proved that $\theta(0, c)$ and $\theta(\varphi_{L}(c^{\circ}), 1)$ are $MS$-congruence on $L^{\circ\circ}$ and lattice congruence on $D(L)$, respectively (Lemma 3.2). To show that $(\theta(0, c), \theta(\varphi_{L}(c^{\circ}), 1))$ is an $MS$-congruence pair, let $(x, y)\in \theta(0, c).$ Then

Conversely, let $(\theta(0, c), \theta(\varphi_{L}(c^{\circ}), 1))$ be an $MS$-congruence pair. Since $(0, c)\in \theta(0, c)$ then $c\wedge c^{\circ}$ = $0\wedge c^{\circ} = 0.$ Now, $c\vee c^{\circ} = (c^{\circ}\wedge c)^{\circ} = 1$. Therefore $c$ is a Boolean element. □

The basic properties of principal congruence relations $\theta(0, a)$ and $\theta(\varphi_{L}(a), 1), \forall a\in B(L)$ are given in the following:

Lemma 3.3. Let $a, b$ be Boolean elements of a principal $MS$-algebra $L$. Then

(1) $a\leq b$ if and only if $\theta(0, a)\subseteq \theta(0, b)$,

(2) $a = b$ if and only if $\theta(0, a) = \theta(0, b)$,

(3) $\theta(0, 0) = \triangle_{L^{\circ\circ}}$ and $\theta(0, 1) = \triangledown_{L^{\circ\circ}}$,

(4) $\theta(0, a)\vee \theta(0, b) = \theta (0, a\vee b)$,

(5) $\theta(0, a)\cap \theta(0, b) = \theta (0, a\wedge b)$.

Lemma 3.4. Let $L$ be a principal $MS$-algebra. Then for every $a, b\in B(L)$, we have

(1) $a\leq b$ implies $\theta(\varphi_{L}(a^{\circ}), 1)\subseteq \theta (\varphi_{L}(b^{\circ}), 1)$,

(2) $\theta(\varphi_{L}(a^{\circ}), 1)\vee \theta(\varphi_{L}(b^{\circ}), 1) = \theta(\varphi_{L}(a\vee b)^{\circ}, 1)$,

(3) $\theta(\varphi_{L}(a^{\circ}), 1)\wedge \theta(\varphi_{L}(b^{\circ}), 1) = \theta(\varphi_{L}(a\wedge b)^{\circ}, 1)$,

(4) $\theta(\varphi_{L}(0), 1) = \triangledown_{D(L)}$ and $\theta(\varphi_{L}(1), 1) = \triangle_{D(L)}$.

Proof.

(1) Let $a\leq b$. Then $a^{\circ}\geq b^{\circ}$. Let $(x, y)\in \theta(\varphi_{L}(a^{\circ}), 1)$ then $x\wedge (a^{\circ}\vee d_{L}) = y\wedge (a^{\circ}\vee d_{L})$. Now

Then $(x, y)\in \theta(\varphi_{L}(b^{\circ}), 1)$. Therefore $\theta(\varphi_{L}(a^{\circ}), 1)\subseteq \theta(\varphi_{L}(b^{\circ}), 1)$.

(2) Since $a, b\leq a\vee b$ then by $(1)$, we have

Then $\theta(\varphi_{L}(a\vee b)^{\circ}, 1)$ is an upper bound of $\theta(\varphi_{L}(a^{\circ}), 1)$ and $\theta(\varphi_{L}(b^{\circ}), 1)$. Let $\theta(\varphi_{L}(c^{\circ}), 1)$ be an upper bound of $\theta(\varphi_{L}(a^{\circ}), 1)$ and $\theta(\varphi_{L}(b^{\circ}), 1)$.

Then $\theta(\varphi_{L}(a^{\circ}), 1), \theta(\varphi_{L}(b^{\circ}), 1)\subseteq \theta(\varphi_{L}(c^{\circ}), 1)$ implies $\varphi_{L}(c^{\circ})\leq \varphi_{L}(a^{\circ})$, $\varphi_{L}(b^{\circ}).$ Thus $\varphi_{L}(c^{\circ})\leq \varphi_{L}(a^{\circ}\wedge b^{\circ}) = \varphi_{L}(a\vee b)^{\circ}$ and hence $\theta(\varphi_{L}(a\vee b)^{\circ}, 1)\subseteq\theta(\varphi_{L}(c^{\circ}), 1).$ Therefore $\theta(\varphi_{L}(a\vee b)^{\circ}, 1)$ is the least upper bound of $\theta(\varphi_{L}(a^{\circ}), 1)$ and $\theta(\varphi_{L}(b^{\circ}), 1)$.

(3) Since $a\wedge b\leq a, \; b$ then by (1) $\theta(\varphi_{L}(a\wedge b)^{\circ}, 1)\subseteq \theta(\varphi_{L}(a^{\circ}), 1), \theta(\varphi_{L}(b^{\circ}), 1)$. Then $\theta(\varphi_{L}(a\wedge b)^{\circ}, 1)$ is a lower bound of $\theta(\varphi_{L}(a^{\circ}), 1)$ and $\theta(\varphi_{L}(b^{\circ}), 1)$. Let $\theta(\varphi_{L}(c^{\circ}), 1)$ be a lower bound of $\theta(\varphi_{L}(a^{\circ}), 1)$ and $\theta(\varphi_{L}(b^{\circ}), 1)$. Then $\theta(\varphi_{L}(c^{\circ}), 1)\subseteq\theta(\varphi_{L}(a^{\circ}), 1), \theta(\varphi_{L}(b^{\circ}), 1)$ implies $\varphi_{L}(a^{\circ}), \varphi_{L}(b^{\circ})\leq\varphi_{L}(c^{\circ}).$ Thus $\varphi_{L}(a\wedge b)^{\circ} = \varphi_{L}(a^{\circ})\vee \varphi_{L}(b^{\circ})\leq \varphi_{L}(c^{\circ})$ and hence $\theta(\varphi_{L}(c^{\circ}), 1)\subseteq \theta(\varphi(a\wedge b)^{\circ}, 1).$ Therefore $\theta(\varphi_{L}(a\wedge b)^{\circ}, 1)$ is the greatest lower bound of $\theta(\varphi_{L}(a^{\circ}), 1)$ and $\theta(\varphi_{L}(b^{\circ}), 1)$.

(4) Since $d_{L}, 1\in\theta (d_{L}, 1)$, then $\theta (d_{L}, 1) = D(L)\times D(L) = \triangledown_{D(L)}$. Also, let $(x, y)\in \theta (\varphi_{L}(1), 1)$ then $x\wedge(1\vee d_{L}) = y\wedge(1\vee d_{L})$. It follows, $x = y$ and hence $\theta (\varphi_{L}(1), 1) = \triangle_{D(L)}$. □

Theorem 3.4. Let $L$ be a principal $MS$-algebra. Consider the subsets $H$ and $G$ of $Con(L^{\circ\circ})$ and $Con(D(L))$, respectively as follows:

Then we have

(1) $(H; \vee, \wedge, \; ^{\prime}\;, \triangle_{L^{\circ\circ}}, \triangledown_{L^{\circ\circ}})$ is a Boolean algebra, where $(\theta(0, a))^{\prime} = \theta(a^{\circ}, 0)$,

(2) $(G; \vee, \wedge, \; ^{\prime}\;, \triangle_{D_(L)}, \triangledown_{D_{L}})$ is a Boolean algebra, where $(\theta(\varphi_{L}(a), 1))^{\prime} = \theta(\varphi_{L}(a^{\circ}), 1)$.

Proof.

(1) Since $\theta(0, 0) = \triangle_{L^{\circ\circ}}$ and $\theta(0, 1) = \triangledown_{L^{\circ\circ}}$. Then $\triangledown_{L^{\circ\circ}}, \; \triangle_{L^{\circ\circ}} \in H$. Let $\theta(0, a), \theta(0, b)\in H$. Then we get

and

Therefore $H$ is a bounded lattice. Since $B(L)$ is a Boolean algebra, then $a^{\circ}\in B(L)$ for all $a\in B(L)$. Thus $\theta(0, a^{\circ})\in H$. Then we have

and

Therefore $(H; \vee, \wedge, ^{\prime}, \triangle_{L^{\circ\circ}}, \triangledown_{L^{\circ\circ}})$ is a Boolean algebra.

(2) We have $\triangle_{D(L)} = \theta(\varphi_{L}(1), 1)\in G$ and $\triangledown_{D(L)} = \theta(\varphi_{L}(0), 1) = \triangledown_{D(L)}\in G$. Let $\theta(\varphi_{L}(a), 1), \theta(\varphi_{L}(b), 1)\in G$. Then we get

and

Therefore $G$ is a bounded lattice. Since $B(L)$ is a Boolean algebra, then $a^{\circ}\in B(L)$ for all $a\in B(L)$. Thus $\theta(\varphi_{L}(a^{\circ}), 1)\in G$. Then we have

and

Therefore $(G; \vee, \wedge, ^{\prime}, \triangle_{D_{L}}, \triangledown_{D_{L}})$ is a Boolean algebra. □

Let $L$ be a principal $MS$-algebra. Let $A(L)$ be the lattice of all $MS$-congruence pairs of $L$. We consider a subset ${A}^{\prime}(L)$ of $A(L)$ as follows:

From the above results, we observe that the set ${A}^{\prime}(L)$ of all $MS$-congruence pairs induced by the Boolean elements of a principal $MS$-algebra forms bounded sublattice of the lattice $A(L)$. Moreover ${A}^{\prime}(L)$ is a Boolean algebra on its own.

Theorem 3.5. Let $L$ be a principal $MS$-algebra. Then $({A}^{\prime}(L); \vee, \wedge, \; ^{'}\;, 0_{A^{\prime}(L)}, 1_{A^{\prime}(L)})$ is a Boolean algebra, where

Example 3.2. Consider the principal $MS$-algebra $L$ in Example 2.1. The lattices $B(L)$ and ${A}^{\prime}(L)$ of all Boolean elements of $L$ and all $MS$-congruence pairs of $L$ of the form $(\theta(0, a), \theta(\varphi_{L}(a^{\circ}), 1)), \; a\in B(L)$ are given in the following Figure 4, respectively. It is clear that $B(L)$ and $A(L)$ are isomorphic Boolean algebras under the isomorphism $a\longmapsto (\theta(0, a), \theta(a^{\circ}\vee z, 1))$. Also, ${A}^{\prime}(L)$ is a bounded sublattice of $A(L)$.

4.   $2$-permutability of principal $MS$-algebras

In this section, we characterize the notion of $2$-permutability of congruences of a principal $MS$-algebra by means of $MS$-congruence pairs.

Let $L$ be an algebra. We say the congruences $\theta, \Phi \in Con(L)$ are 2-permutable if $\theta \circ \Phi = \Phi \circ \theta$, that is, $(x, a)\in\theta$ and $(a, y)\in\Phi$, $\; x, y, a \in L$ imply $(x, b) \in \Phi$ and $(b, y)\in \theta$ for some $\; b\in L$. We call the algebra $L$ has $2$-permutable congruences (briefly permutable) if every pair of congruences of $L$ permute.

Lemma 4.1. A principal $MS$-algebra $L$ has $2$-permutable congruences if and only if every pair of principle congruences of $L$ permute.

Proof. Assume that every pair of principal congruences on $L$ permute. Let $\theta$ and $\Phi$ be arbitrary congruences on $L$ and $(a, b)\in (\theta\circ \Phi), \; a, b\in L$. Then $(a, t)\in \theta$ and $(t, b)\in \Phi$ for some $t\in L$. Then $(a, t)\in \theta(a, t)$ and $(t, b)\in \theta(t, b)$ imply that $(a, b)\in \theta(a, t)\circ \theta(t, b)$. Thus $(a, b)\in \theta(t, b)\circ \theta(a, t).$ Since $\theta(a, t)\subseteq \theta$ and $\theta(t, b)\subseteq \Phi$, then $(a, b)\in (\Phi\circ \theta).$ Therefore $\theta$ and $\Phi$ permute. The second implication is obvious. □

Define a relation $\Psi$ on a principal $MS$-algebra $L$ as follows:

Theorem 4.1. Let $L$ be a principal $MS$-algebra with the smallest dense element $d_{L}$. Then

(1) $\Psi$ is a congruence relation on $L$ with $Ker \; \Psi = \{0\}$ and $Coker\; \Psi = D(L),$

(2) $\Psi$ is closed congruence on $L$, that is, $(x, x^{\circ\circ})\in {\Psi}, \forall x\in L$,

(3) max $[x]_{\Psi} = x^{\circ\circ}, x\in L$, where $[x]_{\Psi} = \{y\in L:y^{\circ\circ} = x^{\circ\circ}\}$ is the congruence class of $x$ modulo $\Psi$.

Proof.

(1) One can check that $\Psi$ is a congruence relation on $L$. Now, we have

and

(2) Since $x^{\circ\circ\circ\circ} = x^{\circ\circ}$ then $x^{\circ\circ}\in [x]_{\Psi}$. Thus $(x^{\circ\circ}, x)\in \Psi$.

(3) Let $y\in [x]_{\Psi}.$ Then $y\leq y^{\circ\circ} = x^{\circ\circ}$ and $x^{\circ\circ}\in [x]_{\Psi}.$ Thus $x^{\circ\circ}$ is the greatest member of the congruence class $[x]_{\Psi}$. □

Theorem 4.2. Let $L$ be a principal $MS$-algebra. Then

(1) $L/{\Psi}$ is a de Morgan algebra,

(2) $L/{\Psi}$ and $L^{\circ\circ}$ are isomorphic de Morgan algebras.

Proof.

(1) It is ready seen that $(L/\Psi; \vee, \wedge, \{0\}, D(L))$ is a bounded distributive lattice, where $L/\Psi = \{[a]_{\Psi}:a\in L\}$ is the set of all congruence classes module $\Psi$ and

Define $^\square$ on $L/\Psi$ by $([a]_{\Psi})^{\square} = [a^{\circ}]_{\Psi}, \forall a\in L.$ We have

Then $L/\Psi$ is a de Morgan algebra.

(2) Define $g:L^{\circ\circ}\rightarrow L/{\Psi}$ by

Let $a, b \in L^{\circ\circ}$ and $a = b.$ Then $a^{\circ\circ} = b^{\circ\circ}$ implies $[a]_{\Psi} = [b]_{\Psi}$. It follows that $g$ is well defined map of $L^{\circ\circ}$ into $L/\Psi$. Let $a, b\in L^{\circ\circ}$, we get

and

Let $[a]_{\Psi} = [b]_{\Psi}.$ Then ${a}^{\circ\circ} = b^{\circ\circ}$ implies $a = b$. Therefore $g$ is an injective map. Let $[a]_{\Psi}\in L/{\Psi}$. Then $[x]_{\Psi} = [x^{\circ\circ}]_{\Psi} = g(x)$. Then $g$ is a surjective map. This deduce that $g$ is an isomorphism of de Morgan algebras. □

Now, we observe that ${\Psi}$ satisfies the following property,

Theorem 4.3. Let $L$ be a principal $MS$-algebra. Then $\Psi$ permutes with each congruence of $L$.

Proof. We prove that $\Psi\circ \theta = \theta\circ \Psi$ for all $\theta \in Con(L)$. Let $(a, b)\in \Psi\circ \theta.$ Then $(a, z)\in \Psi$ and $(z, b)\in \theta$ for some $z\in L$. It follows that $a^{\circ\circ} = z^{\circ\circ}$ and $(z, b)\in \theta$. Now

Since $[b^{\circ\circ}\wedge (a\vee d_{L})]^{\circ\circ} = b^{\circ\circ},$ then $(b^{\circ\circ}\wedge (a\vee d_{L}), b)\in \Psi.$ Therefore $(a, b^{\circ\circ}\wedge (a\vee d_{L}))\in \theta$ and $(b^{\circ\circ}\wedge (a\vee d_{L}), b)\in \Psi.$ imply $(a, b)\in \theta\circ \Psi$. Then $\Psi\circ \theta = \theta\circ\Psi, \; \forall \theta\in Con(L)$. □

Lemma 4.2. Let $L$ be a principal $MS$-algebra. Then

(1) $\triangle_{L}$ permutes with every congruence on $L$,

(2) $\triangledown_{L}$ permutes with every congruence on $L$.

Proof.

(1) Let $(a, b)\in \theta\circ \triangle_{L}$. Then $(a, t)\in \theta$ and $(t, b)\in \triangle_{L}$ for some $t\in L.$ Thus $(a, t)\in \theta$ and $t = b$. Then $(a, a)\in \triangle_{L}$ and $(a, b)\in \theta$ imply $(a, b)\in\triangle_{L}\circ \theta.$ Therefore $\theta \circ \triangle_{L} = \triangle_{L}\circ \theta$.

(2) It is obvious. □

In the following theorem, we characterize $2$-permutability of congruences of a principal $MS$-algebra $L$ using $MS$-congruence pairs.

Theorem 4.4. Let $L$ be a principal $MS$-algebra$.$ Then the following conditions are equivalent:

(1) $L$ has $2$-permutable congruences.

(2) $L^{\circ\circ}$ and $D(L)$ have $2$-permutable congruences.

Proof. To prove the conditions (1) and (2) are equivalent, we show that the two congruences $\theta\;, \Phi\; \in Con(L)$ are $2$-permutable if and only if their restrictions $\theta_{L^{\circ\circ}}, \; \Phi_{L^{\circ\circ}}$ and $\theta_{D(L)}, \; \Phi_{D(L)}$ have $2$-permutable congruences on $L^{\circ\circ}$ and $D(L)$, respectively. Firstly, suppose that $\theta\;, \Phi$ have $2$-permutable congruences on $L$. Let $(x, z)\in \theta_{L^{\circ\circ}}\circ {\Phi_{L^{\circ\circ}}}$. Then there exist $y\in {L^{\circ\circ}}$ such that $(x, y)\in \theta_{L^{\circ\circ}}$ and $(y, z)\in\Phi_{L^{\circ\circ}}$. Then we have $(x, y)\in\theta$ and $(y, z)\in \Phi$. Since $\theta, \Phi$ are $2$-permutable, then there exists $a\in L$ such that $(x, a)\in \Phi$ and $(a, y)\in\theta.$ We get $(x, a^{\circ\circ}) = (x^{\circ\circ}, a^{\circ\circ})\in \Phi_{L^{\circ\circ}}$ and $(a^{\circ\circ}, z^{\circ\circ})\in \theta_{L^{\circ\circ}}$. Therefore $\theta_{L^{\circ\circ}}, \; \Phi_{L^{\circ\circ}}$ are $2$-permutable. Also, to prove that $\theta_{D(L)}, \; \Phi_{D(L)}$ are $2$-permutable, let $(x, z) \in \theta_{D(L)}\circ {\Phi_{D(L)}}$. Then there exist $y\in D(L)$ such that $(x, y)\in \; \theta_{D(L)}$ and $(y, z)\in\Phi_{D(L)}$. Then we have $(x, y)\in\theta$ and $(y, z)\in \Phi$. Since $\theta, \Phi$ are $2$-permutable, there exists $a\in L$ such that

Thus $\theta_{D(L)}, \; \Phi_{D(L)}$ have $2$-permutable congruences on $D(L).$

Conversely, let $\theta\;, \Phi\; \in Con(L)$. Assume that $\theta_{L^{\circ\circ}}, \; \Phi_{L^{\circ\circ}}$ and $\theta_{D(L)}, \; \Phi_{D(L)},$ have $2$-permutable congruences on $L^{\circ\circ}$ and $D(L)$, respectively. Let $(x, z)\in\theta \circ \Phi$. Suppose that $x, \; y, \; z\in L$ with $(x, y)\in\theta$ and $(y, z)\in\Phi$. Then, by using Theorem 2.3, we get the following statements.

Since $\theta_{L^{\circ\circ}}, \; \Phi_{L^{\circ\circ}}$ have $2$-permutable congruences on $L^{\circ\circ}$, there exists an element $a\in L^{\circ\circ}$ such that

Since $\theta_{D(L)}, \Phi_{D(L)}$ have $2$-permutable congruences on $D(L)$, there exists $e\in D(L)$ such that $(x\vee d_{L}, e)\in \Phi_{D_{L}}$ and $(e, z\vee d_{L})\in \theta_{D_{L}})$. It follows that $(x^{\circ\circ}, a)\in \Phi, (a, z^{\circ\circ})\in\theta$ and $(x\vee d_{L}, e)\in \Phi, (e, z\vee d_{L})\in \theta$. Since $L$ is a principal $MS$-algebra we have

Since $\theta, \Phi$ are compatible with the $\wedge$ operation, then $(x^{\circ\circ}, a)\in \Phi$ and $(x\vee d_{L}, e)\in \Phi$ imply $(x, a\wedge e) = (x^{\circ\circ} \wedge (x\vee d_{L}), a\wedge e)\in \Phi$. Also $(a, z^{\circ\circ})\in\theta$ and $(e, z\vee d_{L})\in \theta$ imply $(a\wedge e, z) = (a\wedge e, z^{\circ\circ} \wedge (z\vee d_{L}))\in \theta$. Consequently, we deduce that $(x, a\wedge e)\in \Phi$ and $(a\wedge e, z)\in \theta\;, a\wedge e\in L$. Then $(x, z)\in \Phi\circ \theta$, and hence $\theta, \; \Phi$ are $2$-permutable. □

Now, we construct two examples to clarify the above theorem.

Example 4.1. Consider the principal $MS$-algebra $L$ in Example 2.1 (Figure 1). From Table 1, we show that $L$ has $2$-permutable congruences. Also, Tables 2 and 3 show that $L^{\circ\circ}\text{and}\; D(L)$ have $2$-permutable congruences, respectively. Where $\alpha_{L^{\circ\circ}} = \beta_{L^{\circ\circ}}$ and $\delta_{L^{\circ\circ}} = \triangle_{{L}^{\circ\circ}}$.

In the following example, we give a principal $MS$-algebra $L$ which has not $2$-permutable congruences as well as $D(L)$ has not also $2$-permutable congruences.

Example 4.2. Consider the principal $MS$-algebra $L$ in Figure 5.

It is clear that $D(L) = \{z, z_{1}, 1\}$ and $L^{\circ\circ} = \{0, c, a, b, d, 1\}$. Consider $\theta, \Phi\in Con(L)$ as follows:

We observe that $\theta\circ \Phi \neq \Phi\circ \theta$ as $(z, 1)\in \Phi\circ\theta$ but $(z,1)\notin \theta\circ \Phi$.

Then $L$ has not $2$-permutable congruences and $D(L)$ has not $2$-permutable congruences as $\theta_{D(L)}\circ \Phi_{D(L)}\neq \Phi_{D(L)}\circ \theta_{D(L)}$.

5.   $n$-permutability of principal $MS$-algebras

We generalize the concept of $2$-permutability to the concept of n-permutability of congruences of principal $MS$-algebras.

Definition 5.1. A principal $MS$-algebra $L$ is said to have n-permutable congrunces if every two congruences $\theta, \Phi$ of $L$ are n-permutable, that is,

At first, we need the following two lemmas.

Lemma 5.1. Let $L$ be a principal $MS$- algebra with the smallest dense element $d_{L}$. Let $\theta, \Phi$ be two congruences on $L.$ Then we have

(i) $(\theta \circ \Phi \circ \theta \circ...)_{L^{\circ\circ}} = \theta _{L^{\circ\circ}}\circ \Phi_{L^{\circ\circ}}\circ...$(n times),

(ii) $(\theta \circ \Phi \circ \theta \circ...)_{D(L)} = \theta _{D(L)}\circ \Phi_{D(L)}\circ...$(n times).

Proof.

(i) Recall that $\theta_{L^{\circ\circ}}, \Phi_{L^{\circ\circ}}$ are the restrictions of $\theta, \Phi$ on $L^{\circ\circ}$, respectively. Let $a, b \in L^{\circ\circ}$ and $(a, b)\in (\theta _{L^{\circ\circ}}\circ \Phi_{L^{\circ\circ}}\circ...).$ Then there exist $c_{1}, c_{2}, ..., c_{n-1}\in L^{\circ\circ}$ such that

Then

Then $(a, b)\in (\theta \circ \Phi \circ \theta \circ...)$ and hence $(a, b)\in (\theta \circ \Phi \circ \theta \circ...)_{L^{\circ\circ}}$. Therefore

Conversely, let $a, b\in L^{\circ\circ}$ such that $(a, b)\in (\theta \circ \Phi \circ \theta \circ...)_{L^{\circ\circ}}$. Then $(a, b)\in (\theta\circ \Phi\circ \theta\circ...$). Then there exist $c_{1}, c_{2}, ..., c_{n-1}\in L$ such that

Then we get

Since ${c_{1}}^{\circ\circ}, {c_{2}}^{\circ\circ}, ..., {c_{n-1}}^{\circ\circ}\in L^{\circ\circ}$, we have

Therefore

and hence

Then we get

(ii) Take $x, y\in D(L)$, let $(x, y)\in(\theta_{D(L)}\circ \Phi_{D(L)}\circ \Phi_{D(L)}\circ\theta_{D(L)}\circ...)$. Then $(x, y)\in(\theta\circ\Phi\circ\theta\circ...)$ and hence $(a, b)\in (\theta\circ\Phi\circ\theta\circ...)_{D(L)}$. Then $(\theta\circ\Phi\circ\theta\circ...)\subseteq (\theta\circ\Phi\circ\theta\circ...)_{D(L)}$.

Conversely, let $(x, y)\in (\theta \circ \Phi \circ \theta \circ...)_{D(L)}$, then $(x, y)\in (\theta\circ \Phi\circ...)$. There exist $d_{1}, d_{2}, ...d_{n-1}\in L$, such that

Since $x, y \geq d_{L}$, we get,

Hence $(x, y)\in(\theta _{D(L)}\circ \Phi_{D(L)}\circ \theta_{D(L)}\circ...)$. Then the required equality is proved. □

Lemma 5.2. Let $\theta$ and $\Phi$ be congruences on a principal $MS$-algebra $L$. Let $\psi$ denoted to the relation $\theta \circ \Phi \circ \theta \circ...$(n times). Then $(a, b)\in \psi$ and $(c, d)\in \psi$ imply $(a\wedge c, b\wedge d)\in \psi$.

Proof. Let $(a, b)\in \psi$ and $(c, d)\in \psi$. Then there exist $a_{1}, a_{2}, ..., a_{n-1}, c_{1}, c_{2}, ..., c_{n-1}\in L$ such that

And

Since $\theta, \Phi$ are congruence on $L$ then we get,

Thus $(a\wedge c, b\wedge d)\in \psi$. □

Now, a characterization of $n$-permutability of principal $MS$-algebras is given.

Theorem 5.1. Two congruences $\theta, \Phi$ on a principal $MS$-algebra $L$ are n-permutable if and only if their restrictions $\theta_{L^{\circ\circ}}, \Phi_{L^{\circ\circ}}$ and $\theta_{D(L)}, \Phi_{D(L)}$ with respect to $L^{\circ\circ}$ and $D(L)$ respectively are n-permutable.

Proof. Let $L$ has n-permutable congruences. Let $\theta, \Phi$ be any two congruences on $L$. Then by $(i)$ and $(ii)$ of Lemma 5.1, respectively, we have

and

Therefore $L^{\circ\circ}$ and $D(L)$ have n-permutable congruences.

Conversely, let $L^{\circ\circ}$ and $D(L)$ have n-permutable congruences. If $(a, b)\in (\theta\circ\Phi\circ\theta\circ...)$, using Theorem 2.3, we get

and

By lemma 5.1, we get

and

Since $\theta_{L^{\circ\circ}}, \Phi_{L^{\circ\circ}}$ and $\theta_{D(L)}, \Phi_{D(L)}$ are n-permutable on $L^{\circ\circ}, D(L),$ respectively, we have

and

It follows that, $(a^{\circ\circ}, b^{\circ\circ})\in(\Phi\circ\theta\circ\Phi\circ...)$ and $(a\vee d_{L}, b\vee d_{L})\in(\Phi\circ\theta\circ\Phi\circ...)$. Since $L$ is a principal $MS$-algebra, then $a = a^{\circ\circ}\wedge (a\vee d_{L})$ and $b = b^{\circ\circ}\wedge (b\vee d_{L})$. By Lemma 5.2, we get

Thus $(\theta\circ\Phi\circ\theta\circ...)\subseteq(\Phi\circ\theta\circ\Phi\circ...)$. Similarly we can show that $(\Phi\circ\theta\circ\Phi\circ...)\subseteq (\theta\circ\Phi\circ\theta\circ...)$. Then $L$ has n-permutable congruences. □

6.   Conclusions

With the aid of the technique of $MS$-congruence pairs, many properties were investigated for principal $MS$-algebras, deal with a characterization of congruence pairs of a principal $MS$-algebra (a Stone algebra), a description of the lattice $A(L)$ of all $MS$-congruence pairs of $L$ via Boolean elements of $L$, characterizations of $2$-permutability and $n$-permutability of congruences of a principal $MS$-algebra. This work leads us in the future to study many aspects of principal $MS$-algebras and related structures, for instance, it can be applied to triple construction of principal $MS$-algebra, affine and locally complete of principal $MS$-algebras.

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The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Conflict of interest

The authors declare no conflict of interest.