k-filters and k-$\{^*\}$-congruences of core regular double Stone algebras

Abstract

In this paper, we investigate various elegant filters and congruences of the class of core regular double Stone algebras (briefly CRD-Stone algebras). We define and characterize the concepts of k-filters and principal k-filters of a core regular double Stone algebra with the core element k, as well as their algebraic structures. We also look at k-$\{^*\}$-congruences and principal k-$\{^*\}$-congruences of a CRD-Stone algebra L that are induced by k-filters and principal k-filters of L, respectively. We find an isomorphism between the lattice $F_{k}(L)$ of all k-filters of L (the lattice $F_{k}^{p}(L)$ principal k-filters of L) and the lattice $Con^{*}_{k}(L)$ of all k-$\{^*\}$-congruences on L (the lattice $Con^{*}_{k}(L)$ of all principal k-$\{^*\}$-congruences) of a CRD-Stone algebra L.

1 Introduction

The notion of psudo-complement introduced and characterized in semi-lattices and distributive lattice by Frink (1962) and Birkhoff (1967), respectively. The class ${\textbf {S}}$ of Stone algebras was studied and characterized by several authors, like, Katriňák (1973), Badawy (2018), Chen and Grätzer (1969a, 1969b), Grätzer (1971), Frink (1962), Balbes and Horn (1970), and Katriňák (1974). The concepts of regular double p-algebras and regular double Stone algebras are established by Katriňák (1974) and Comer (1995), respectively.

The intersection of the set D(L) of dense elements and the set $\overline{D(L)}$ of dual dense elements of a double Stone algebra L is called the core of L and denoted by K(L). In a regular double Stone algebra L, the core K(L) is either an empty set or a singleton set, if a regular double Stone L has a nonempty core, then such core K(L) has exactly one element, which is denoted by k. Many properties of ideals and filters of a core regular double Stone algebra with core element k are established in Ravi Kumar et al. (2015) and Srikanth and Ravi Kumar (2018). Double Stone algebras are constructed by means of Stone quadruples in Badawy et al. (2019). Recently, Badawy (2020), Theorem 4.1 introduced and constructed a core regular double MS-algebra $M^{[2]}$ via a suitable de Morgan algebra M, where

$$\begin{aligned} M^{[2]}=\{(a,b)\in M\times M:~a\le b\}. \end{aligned}$$

Also, he constructed a core regular double Stone algebra

$$\begin{aligned} B^{[2]}=\{(a,b)\in B\times B:~a\le b\} \end{aligned}$$

from a Boolean algebra B (see Corollary 4.5. of Badawy 2020).

In Sect. 2, We recall the basic concepts and important results which are needed throughout this article. Also, we provide some examples of CRD-Stone algebras with core element k and RD-Stone algebras with empty core. We refer the reader to Badawy (2019), Badawy et al. (2023), Badawy and Helmy (2023), Badawy and Shum (2014), and Sambasiva Rao and Badawy (2014, 2017) for filters, ideals and Badawy (2017), Badawy and Atallah (2019), and Badawy and Shum (2017) for congruences of lattices and p-algebras.

In Sect. 3, we define a k-filter of a CRD-Stone algebra L and study many related properties. A set of equivalent statements for a filter F of a CRD-Stone algebra L to be a k-filter is obtained. It is observed that the class $F_{k}(L)$ of all k-filters of L forms a bounded distributive lattice.

In Sect. 4, we introduce and characterize the notion of principal k-filters of L. We verify that the class $F^{p}_{k}(L)$ of all principal k-filters of L forms a Boolean algebra and so a Boolean ring.

In Sect. 5, we investigate the k-$\{^*\}$-congruences via k-filters of L. Also, we observe that the set $Con^{*}_{k}(L)$ of all k-$\{^*\}$-congruences forms a bounded distributive lattice which is isomorphic to the lattice $F_{k}(L)$ of k-filters.

In Sect. 6, we obtain and characterize the principal k-$\{^*\}$-congruences of a CRD-Stone algebra L. Then, we study the properties and the algebraic structure of the class $Con^{p}_{k}(L)$ of all principal k-$\{^*\}$-congruences of L. Moreover, there is a one to one correspondence between the class $F^{p}_{k}(L)$ of all principle k-filters of L and the class $Con^{p}_{k}(L)$ of all principal k-$\{^*\}$-congruences of L.

2 Preliminaries

In this section, we recall certain definitions and important results which are needed throughout the paper, which are taken from the references Badawy (2018), Badawy and Atallah (2015), Blyth (2005), Comer (1995), Grätzer (1971), Ravi Kumar et al. (2015), and Srikanth and Ravi Kumar (2018, 2017).

Definition 2.1

(Badawy 2018) An algebra $(L; \wedge , \vee )$ of type (2, 2) is called to be a lattice if for every $a, b, c \in L$, it satisfies the following properties:

• (1) $a \wedge a=a, a \vee a=a$ (Idempotency),

• (2) $a \wedge b=b \wedge a, a \vee b=b \vee a$ (Commutativity),

• (3) $(a \wedge b) \wedge c=a \wedge (b \wedge c),(a \vee b) \vee c=a \vee (b \vee c)$ (Associativity),

• (4) $(a \wedge b) \vee a=a,(a \vee b) \wedge a=a$ (Absorption identities).

Definition 2.2

(Blyth 2005) A lattice L is called a bounded if it has the greatest element 1 and the smallest element 0.

Definition 2.3

(Badawy 2018) A lattice L is called distributive if it satisfies either of the following equivalent distributive laws:

• (1) $a \wedge (b \vee c)=(a \wedge b) \vee (a \wedge c)$,

• (2) $a \vee (b \wedge c)=(a \vee b) \wedge (a \vee c)$, for all $a, b, c \in L$.

Definition 2.4

(Srikanth and Ravi Kumar 2018) A nonempty subset I of a lattice L is called a filter of L if

• (1) $x \wedge y \in F$ for all $x, y \in F$,

• (2) $x \in F$ and $z \in L$ be such that $z \ge x$ imply $z \in F$.

Definition 2.5

(Grätzer 1971) If $\phi \ne A\subseteq L$, then [A) is the smallest filter of a lattice L which contains A, where $[A)=\{x\in L: x\ge a_{1}\wedge a_{2}\wedge \cdots \wedge a_{i},~ a_{i}\in A,~ i=1,2,\ldots ,n\}$.

The case that $A=\{a\}$, we write [a) instead of $[\{a\})$, where [a) is called the principal filter of L generated by a and $[a)=\{x\in L: x\ge a\}$.

Let F(L) be the set of all filters of a lattice L. It is known that $(F(L); \wedge , \vee )$ forms a lattice,

where $F \wedge G=F \cap G$ and $F \vee G=\{x \in L: x \ge f \wedge g: f \in F, g \in G\}$.

Also, the set $F^{p}(L)$ of all principal filters of L is a sublattice of lattice F(L), where $[a)\wedge [b)=[a\vee b)$ and $[a)\vee [b)=[a\wedge b)$. It is known that the lattice F(L) is distributive if and only if L is distributive.

Definition 2.6

(Badawy 2018) For any element a of a bounded lattice L, the pseudo-complement $a^*$ (the dual pseudo-complement $a^+$) of a (if exists) is defined as follows:

$$\begin{aligned} a \wedge x=0 \Leftrightarrow x \le a^*~ (a \vee x=1 \Leftrightarrow a^+ \le x ). \end{aligned}$$

Definition 2.7

(Grätzer 1971) A distributive lattice L in which every element has a pseudo-complement is called a distributive pseudo-complement lattice or a distributive p-algebra. Dually, a distributive lattice L in which every element has a daul pseudo-complement is called a distributive daul pseudo-complement lattice or a dual distributive p-algebra.

Definition 2.8

(Badawy and Atallah 2015) A distributive p-algebra (distributive dual p-algebra) L is called a Stone algebra (dual Stone algebra) if $x^* \vee x^{ **}=1$ $(x^+ \wedge x^{ ++}=0)$ for all $x \in L$.

Theorem 2.9

(Badawy 2018) Let L be a distributive p-algebra (distributive dual p-algebra). Then for any two elements a, b of L, we have

• (1) $0^{**}=0$ and $1^{**}=1$ $(0^{++}=0~ and ~1^{++}=1)$,

• (2) $a \wedge a^*=0$ $(a \vee a^+=1)$,

• (3) $a \le b$ implies $b^* \le a^*$ $(a \ge b~ implies ~b^+ \ge a^+)$,

• (4) $a \le a^{**}$ $(a ^{++}\le a)$,

• (5) $a^{***}=a^*$ $(a^{+++}=a^+)$,

• (6) $(a \vee b)^*=a^* \wedge b^*$ $((a \wedge b)^+=a^+ \vee b^+)$,

• (7) $(a \wedge b)^{*}=a^{*} \vee b^{*}$ $((a \vee b)^{+}=a^{+} \wedge b^{+})$,

• (8) $(a\vee b)^{**}=a^{**}\vee b^{**}$ $((a\wedge b)^{++}=a^{++}\wedge b^{++})$,

• (9) $(a \wedge b)^{**}=a^{**} \wedge b^{**}$ $((a \vee b)^{++}=a^{++} \vee b^{++})$.

Definition 2.10

(Varlet 1972) A Double Stone-algebra L is an algebra $\langle L, ^*,^+ \rangle$, where

• (i) $(L,^*)$ is a Stone algebra,

• (ii) $(L,^+)$ is a dual Stone algebra.

Definition 2.11

(Comer 1995) A regular double Stone algebra (briefly RD-Stone algebra) L is a double Stone algebra which satisfying

$$\begin{aligned} x^{**}=y^{**} \text { and }x^{++}=y^{++} \text { imply }x=y. \end{aligned}$$

Let L be a double Stone algebra. The element $a\in L$ is called a closed element of L if $a^{**}=a$ and the element $a\in L$ is called a dual closed element of L if $a^{++}=a$. An element $d\in L$ is called dense if $d^*=0$ and an element $a\in L$ is called dual dense if $a^+=1$.

Lemma 2.12

(Srikanth and Ravi Kumar 2018) Let L be double Stone algebra. Then,

• (1) the set $D(L)=$ $\left\{ a \in L \mid a^*=0\right\}$=$\left\{ a \vee a^* \mid a \in L\right\}$ of all dense elements of L is a filter of L,

• (2) the set $\overline{D(L)}=$ $\left\{ a \in L \mid a^{+}=1\right\}$=$\left\{ a \wedge a^+ \mid a \in L\right\}$ of all dual dense elements of L is an ideal of L,

• (3) the set $B(L)=\{a^*: a\in L\}=\{a^+: a\in L\}$ of all closed elements of L forms a Boolean subalgebra of L,

• (4) the set $K(L)= D(L) \cap \overline{D(L)}$ is called the core of L, we have two cases of K(L), namely, $K(L)= \phi$ or $K(L)\ne \phi$.

Definition 2.13

A RD-Stone algebra L is called a core regular double Stone algebra (CRD-Stone algebra) if $K(L)\ne \phi$.

Lemma 2.14

The core K(L) of a CRD-Stone algebra L has exactly one element.

Proof

Let $x,y \in K(L)$. Then, $x^{++}=y^{++}=0$ and $x^{**}=y^{**}=1$. By regularity of L, $x=y$. Then, K(L) has exactly one element and hence $\mid K(L)\mid =1$. $\square$

We well use k for the core element of a CRD-Stone algebra L, that is, $K(L)=\{k\}$.

Lemma 2.15

Let L be a CRD-Stone algebra with core k. Then,

• (1) $D(L)=[k)$, that is, D(L) is a principal filter of L generated by k,

• (2) $\overline{D(L)}=(k]$, that is, $\overline{D(L)}$ is a principal ideal of L generated by k.

Example 2.16

• (1) Let $L=\{ 0,x,y,1: 0<x<y< 1\}$ be the four element chain. It is clear that $\langle L, ^*,^+ \rangle$ is a double Stone algebra, where $x^*=y^*=1^*=0,~ 0^*=1$ and $0^+=x^+=y^+=1,~ 1^+=0$. Then, the core $K(L)= D(L) \cap \overline{D(L)}=\{x, y, 1\}\cap \{x, y, 0\}=\{x, y\}$ is a non empty set. We observe that L is not regular as $x^{++}=y^{++}$ and $x^{**}=y^{**}$, but $x\ne y$.

• (2) The double Stone algebra $L=\{0,k,1: 0<k< 1\}$ is the smallest non-trival core regular double Stone algebra with core k.

• (3) Every Boolean algebra $(B, \vee , \wedge , ^{\prime }, 0, 1 )$ can be regarded as a RD-Stone algebra with empty core, where $x^{*}=x^{+}=x^{\prime }$, for all $x\in B$ and $K(B)=\{1\}\cap \{0\}=\phi$.

Fig. 1

$L_1$ is a RD-Stone algebra with core k

Fig. 2

$L_2$ is a RD-Stone algebra with empty core

Example 2.17

• (1) Consider the bounded distributive lattice $L_1$ in Fig. 1. It is clear that $L_1$ is a core regular double Stone algebra with core k, where $k^*=1^*=y^*=x^*=0,~ c^*=a^*=b,~ d^*=b^*=a, ~1^*=0$ and $k^+=c^+=d^+=0^+=1,~ b^+=y^+=a,~ x^+=a^+=b, ~0^+=1$.

• (2) Consider the bounded distributive lattice $L_2$ in Fig. 2. We observe that $L_2$ is a regular double Stone algebra with empty core as $K(L)= D(L) \cap \overline{D(L)}=\{d,1\}\cap \{0,y\}=\phi$, where $0^*=d^*=1^*$, $c=x^*$, $x=c^*=y^*$, $1=0^*$ and $0=1^+$, $c=x^+=d^+$, $x=c^+$, $1=y^+=0^+$.

Lemma 2.18

If L is a CRD-Stone algebra with core element k, then every element x of L can be written by each of the following formulas:

• (1) $x=x^{* *} \wedge \left( x^{++} \vee k\right)$,

• (2) $x=x^{++} \vee \left( x^{* *} \wedge k\right)$,

• (3) $x=x^{* *} \wedge \left( x \vee k\right)$,

• (4) $x=x^{++} \vee \left( x \wedge k\right)$.

Definition 2.19

(Badawy 2018) A binary relation $\theta$ on a lattice L is called a lattice congruence on L if

• (1) $\theta$ is an equivalent relation on L,

• (2) $(a, b) \in \theta$ and $(c, d) \in \theta$ implies $(a \vee c, b \vee d) \in \theta$ and $(a \wedge c, b \wedge d) \in \theta$.

Theorem 2.20

(Grätzer 1971) An equivalent relation on a lattice L is a lattice congruence on L if and only if $(a, b) \in \theta$ implies $(a \vee c, b \vee c) \in \theta$ and $(a \wedge c, b \wedge c) \in \theta$ for all $c \in L$.

Definition 2.21

A lattice congruence $\theta$ on a Stone (dual Stone) algebra L is called a $\{^*\}$-congruence ($\{^+\}$-congruence) if $(a, b) \in \theta$ implies $\left( a^*, b^*\right) \in \theta$ ($(a, b) \in \theta$ implies $\left( a^+, b^+\right) \in \theta$).

Definition 2.22

A lattice congruence $\theta$ on a double Stone algebra L is called a congruence(or $\{^*,^+\}$-congruence) if $(a, b) \in \theta$ implies $\left( a^*, b^*\right) \in \theta$ and $\left( a^+, b^+\right) \in \theta$.

A binary relation $\Psi ^*$ defined on a double Stone algebra L by $(x, y) \in \Psi ^* \Leftrightarrow x^*=y^*$ is a $\{^*\}$-congruence relation which is called the Glivenko congruence relation on L. It is known that the quotient lattice $L / \Psi =\{[x] \Psi : x \in L\}$ is a Boolean algebra and $L / \Psi \cong B(L)$, where $[x] \Psi =\left\{ y \in L: y^*=x^*\right\}$ is the congruence class of x modulo $\Psi$. Moreover, the element $x^{**}$ is the greatest element of the congruence class $[x] \Psi$, $[1] \Psi =D(L)$ and $[0] \Psi =\{1\}$.

For a double Stone algebra L, we use Con(L) to denote the lattice of all congruences of L and $Con^*(L)$ to denote the lattice of all $\{^*\}$-congruences of a Stone algebra $(L, ^*)$. Also, we use $\nabla _L$ and $\Delta _L$ for the universal congruence $L\times L$ and equality congruence $\{(x,y): x\in L\}$ of L, respectively.

Definition 2.23

(Blyth 2005) A lattice congruence $\theta$ on a lattice L is called a principal congruence and denoted by $\theta (a,b)$ if $\theta$ is the smallest congruence on L containing a, b on the same class.

Theorem 2.24

(Blyth 2005) If L is a distributive lattice and $a, b\in L$ then the principal congruence $\theta (a,b)$ of L is given by

• (1) $(x,y)\in \theta (a,b)$ $\Leftrightarrow$ $x\vee a\vee b=y\vee a\vee b$ and $x\wedge a\wedge b=y\wedge a\wedge b$,

• (2) If $a\le b$, then $(x,y)\in \theta (a,b)$ $\Leftrightarrow$ $x\vee b=y\vee b$ and $x\wedge a=y\wedge a$,

• (3) If $b=1$, then $\theta (a,1)$ $\Leftrightarrow$ $x\wedge a=y\wedge a$.

Throughout the paper, we will use k for the core element of a CRD-Stone algebra L. For more information, we refer the reader to Kumar and Kumari (2021) and Varlet (1966) for Stone algebras, Zhang et al. (2018) for double Stone algebras, Comer (1995) for regular double Stone algebras, and Clouse (2016), Ravi Kumar et al. (2015), and Srikanth and Ravi Kumar (2017, 2018) for core regular double Stone algebra Burris and Sankappanavar (1981).

3 Properties of k-filters of a CRD-Stone algebra

In this section, we introduce the concept of k-filter of a CRD-Stone algebra L and study their related properties. Also, we prove that the class $F_{k}(L)$ of all k-filters of L is a bounded distributive lattice.

Definition 3.1

A filter F of a CRD-Stone algebra L with core k is called k-filter if $k\in F$.

Let A be a non-empty subset of a CRD-Stone algebra L. Define $A^{\bigtriangleup }$ by:

$$\begin{aligned} A^{\bigtriangleup }=\{x\in L:x^{**}\ge a^{**}\wedge k, ~\text { for some }~ a\in A\}. \end{aligned}$$

Lemma 3.2

Let A be a non-empty subset of a CRD-Stone algebra L, which is closed under $\wedge$. Then, $A^{\bigtriangleup }$ is a k-filter of L containing A.

Proof

Clearly, $k\in A^{\bigtriangleup }$. Let $x,y\in A^{\bigtriangleup }$. Thus, $x^{**}\ge a^{**}\wedge k$, $y^{**}\ge b^{**}\wedge k$ for some $a,b\in A$. Then, $(x\wedge y)^{**}\ge (a\wedge b)^{**}\wedge k$ and $a\wedge b \in A$, imply $x\wedge y \in A^{\bigtriangleup }$. Now, let $x\in L, y\in A^{\bigtriangleup }$ and $x\ge y$. Then $x^{**}\ge y^{**}\ge a^{**}\wedge k$. So $x \in A^{\bigtriangleup }$. Thus, $A^{\bigtriangleup }$ is a filter of L. Since $1=k^{**}\ge a^{**}\wedge k$ for $a\in A$, then $k\in A^{\bigtriangleup }$ and hence $A^{\bigtriangleup }$ is a k-filter of L containing A. $\square$

Lemma 3.3

Let A and B be two non empty subsets of a CRD-Stone algebra L, which are closed under $\wedge$. Then,

• (1) $[A)^{\bigtriangleup }=A^{\bigtriangleup }$,

• (2) $A\subseteq B \Longrightarrow A^{\bigtriangleup }\subseteq B^{\bigtriangleup }$,

• (3) $A^{\bigtriangleup }=[A)\vee D(L)$,

• (4) $A^{\bigtriangleup \bigtriangleup }=A^{\bigtriangleup }$.

Proof

• (1) Let $x\in [A)$. Then, $x^{**}\ge a^{**}\wedge k,~ for~ some~ a\in [A)$. Since $a\in [A)$, then by Definition 2.5, $a\ge a_{1}\wedge a_{2}\wedge \ldots a_{n},~ a_{i}\in A,~ i=1,2,\ldots ,n$ and hence $a^{**}\ge (a_{1}\wedge a_{2}\wedge \ldots a_{n})^{**}$. Thus, $x^{**}\ge a^{**}\wedge k\ge (a_{1}\wedge a_{2}\wedge \ldots a_{n})^{**}\wedge k,~ a_{1}\wedge a_{2}\wedge \ldots a_{n}\in A$ as A is closed under $\wedge$. This gives $x\in A^{\bigtriangleup }$ and hence $[A)^{\bigtriangleup }\subseteq A^{\bigtriangleup }$. Conversely, let $a\in A^{\bigtriangleup }$. Then, $x^{**}\ge a^{**}\wedge k~for~some~ a\in A$. Since $A \subseteq [A)$ and $x^{**}\ge a^{**}\wedge k~for~some~ a\in [A)$, then $x\in [A)^{\bigtriangleup }$. Therefore, $A^{\bigtriangleup } \subseteq [A)^{\bigtriangleup }$. Consequently, $[A)^{\bigtriangleup }=A^{\bigtriangleup }$.

• (2) Suppose $A\subseteq B$ and $x\in A^{\bigtriangleup }$. Then, $x^{**}\ge a^{**}\wedge k$ for some $a\in A\subseteq B$. Then, $x\in B^{\bigtriangleup }$. Thus, $A^{\bigtriangleup }\subseteq B^{\bigtriangleup }$.

• (3) Since $[A) \subseteq [A)^{\bigtriangleup }=A^{\bigtriangleup }$ by (1) and $D(L) =[k)\subseteq A^{\bigtriangleup }$, then $[A)^{\bigtriangleup }\vee D(L)\subseteq A^{\bigtriangleup }$. Conversely, let $x\in A^{\bigtriangleup }$. Then, $x^{**}\ge a^{**}\wedge k$ $~for ~some ~a\in A$. We get

  $$\begin{aligned}&x=x^{**}\wedge (x\vee k)\ge (a^{**}\wedge k)\wedge (x\vee k) ~~~~\\&\quad (\text {by Lemma }2.8.(3))\\&x\ge a^{**}\wedge (k\wedge (x\vee k))~~(\text {by Definition } 2.1(4))\\&x\ge a^{**}\wedge k\ge a\wedge k ~~~~~~\\&\quad (\text {by the obsorpation identity and})~ a\le a^{**}\\ {}&x\in [a\wedge k)=[a)\vee [k)\subseteq [A)\vee D(L) ~~~~~\\&\quad as~[a)\subseteq [A). \end{aligned}$$

  Consequently, $A^{\bigtriangleup }=[A)\vee D(L)$.

• (4) Using the definition of $A^{\bigtriangleup }$, we get

  $$\begin{aligned}&A^{\bigtriangleup \bigtriangleup } \\ {}&=\{x\in L: x^{**}\ge a_{1}^{**}\wedge k, \text {for~ some }~ a_{1}\in A^{\bigtriangleup }\}\\ {}&=\{x\in L: x^{**}\ge a_{1}^{**}\wedge k, a_{1}^{**}\ge a^{**}\wedge k ~\\&\quad \text {for ~some}~ a\in A\}\\ {}&=\{x\in L: x^{**}\ge a^{**}\wedge k,~\text {for~ some}~ a\in A\}=A^{\bigtriangleup }. \end{aligned}$$

$\square$

Now, a set of equivalent conditions for a filter F to become a k-filter of a CRD-Stone algebra L are given.

Theorem 3.4

Let F be a filter of a CRD-Stone algebra L. Then, the following propositions are equivalent:

• (1) F is a k-filter of L,

• (2) $D(L)\subseteq F$,

• (3) $x\vee x^* \in F$, for all $x\in L$,

• (4) $F=F^{\bigtriangleup }$.

Proof

• (1)–(2): Let F is a k-filter of L. Then $k\in F$ implies $D(L)=[k)\subseteq F$.

• (2)–(3): Let $D(L)\subseteq F$. For all $x\in L$, we have $x\vee x^{*}\in D(L)$ and hence $x\vee x^*\in F$.

• (3)–(4): We known that $F\subseteq F^{\bigtriangleup }$. Conversely, let $y\in F^{\bigtriangleup }$. Then, $y^{**}\ge f^{**}\wedge k$, for some $f\in F$. Thus, $y^{**}\ge f^{**}$. By Theorem 2.18.(3), $y=y^{**}\wedge (y\vee k)\ge f^{**}\wedge (y\vee k)$. By (3) $k=k\vee k^*\in F$. Now, $f^{**},~ y \vee k\in F$ imply that $f^{**}\wedge (y\vee k)\in F$ and hence $y \in F$. Therefore, $F=F^{\bigtriangleup }$.

• (4)–(1): Assume (4). Since $k\in F^{\bigtriangleup }=F$, it follows that F is a k-filter of L.

$\square$

As a consequent of Lemma 3.3 and Theorem 3.4, we have the following Corollary and Lemma, respectively.

Corollary 3.5

Let F, G be filters of a CRD-Stone algebra. Then,

• (1) $F\subseteq G \Rightarrow F^{\bigtriangleup }\subseteq G^{\bigtriangleup }$,

• (2) $F^{\bigtriangleup \bigtriangleup }=F^{\bigtriangleup }$.

Lemma 3.6

Let L be a CRD-Stone algebra L. Then,

• (1) $F^{\bigtriangleup }=F\vee D(L)$, for all $F\in F(L)$,

• (2) D(L) is the smallest and L is the greatest k-filter of L,

• (3) Every k-filter of L can be expressed in the form $F^{\bigtriangleup }$ for some $F\in F(L)$.

Theorem 3.7

Let F,  G be two filters of a CRD-Stone algebra L. Then,

• (1) $(F\cap G)^{\bigtriangleup }=F^{\bigtriangleup }\cap G^{\bigtriangleup }$,

• (2) $(F\vee G)^{\bigtriangleup }=F^{\bigtriangleup }\vee G^{\bigtriangleup }$.

Proof

• (1) Clearly, $(F\cap G)^{\bigtriangleup }\subseteq F^{\bigtriangleup }\cap G^{\bigtriangleup }$. Conversely, let $x\in F^{\bigtriangleup }\cap G^{\bigtriangleup }$. Then, $x^{**}\ge f^{**}\wedge k~and~ x^{**}\ge g^{**}\wedge k~for ~some~f\in F~ and~g\in G$. Hence, $x^{**}\ge (f^{**}\wedge k)\vee (g^{**}\wedge k)=(f^{**}\vee g^{**})\wedge k=(f\vee g)^{**}\wedge k$. Since $f\vee g\in F\cap G$. Then $x\in (F\cap G)^{\bigtriangleup }$. Thus $(F\cap G)^{\bigtriangleup }=F^{\bigtriangleup }\cap G^{\bigtriangleup }$.

• (2) Since $F,G \subseteq F\vee G$, Then by Corollary 3.5.(1), $F^{\bigtriangleup },G^{\bigtriangleup }\subseteq (F\vee G)^{\bigtriangleup }$. Thus, $(F\vee G)^{\bigtriangleup }$ is an upper bound of $F^{\bigtriangleup } ~and~G^{\bigtriangleup }$. Let $H^{\bigtriangleup }$ be an upper bound of both $F^{\bigtriangleup } ~and~G^{\bigtriangleup }$ in $F_{k}(L)$. Then, $F^{\bigtriangleup },G^{\bigtriangleup } \subseteq H^{\bigtriangleup }$ implies $F,G\subseteq H^{\bigtriangleup }$. Hence, $F\vee G\subseteq H^{\bigtriangleup }$. Therefore, by Corollary 3.5.(1) and (2), we get $(F\vee G)^{\bigtriangleup }\subseteq H^{\bigtriangleup \bigtriangleup }=H^\bigtriangleup$. This deduces that $(F\vee G)^{\bigtriangleup }$ is the least upper bound of both $F^{\bigtriangleup } ~and~G^{\bigtriangleup }$ in $F_{k}(L)$. Then, $(F\vee G)^{\bigtriangleup }=F^{\bigtriangleup }\vee G^{\bigtriangleup }$.

$\square$

Let $F_{k}(L)$=$\{F:F~ is~ a ~k- filter~ of~ L\}$=$\{F^{\bigtriangleup }:F\in F(L)\}$ be the set of all k-filters of L.

Theorem 3.8

The set $F_{k}(L)$ of all k-filters of a CRD-Stone algebra L is a bounded distributive lattice and $\{1\}$-sublattice of F(L).

Proof

We observe that the operations $\vee$ and $\wedge$ are defined on $F_{k}(L)$ (see Theorem 3.7) as follows:

$$\begin{aligned} (F\vee G)^{\bigtriangleup }= & {} F^{\bigtriangleup }\vee G^{\bigtriangleup } \text { and }(F\cap G)^{\bigtriangleup }\\= & {} F^{\bigtriangleup }\cap G^{\bigtriangleup } \quad \text {for all }F,G \in F(L). \end{aligned}$$

Then, $(F_{k}(L); \vee , \wedge )$ is sublattice of the lattice F(L). Since F(L) is a distributive lattice, then $F_{k}(L)$ is also distributive. Since D(L) and L are the smallest and the greatest k-filters of L, respectively. Then, $(F_{k}(L); \vee , \wedge ,D(L),L)$ is a bounded distributive lattice on its own. It is clear that $F_{k}(L)$ is $\{1\}$-sublattice of F(L). $\square$

4 Principal k-filters of a CRD-Stone algebra

In this section, we introduce the notion of principal k-filters of a CRD-Stone algebra L and consider many interesting related properties. A characterization of a k-filter of L is given in terms of principal k-filters. It is observed the set of all principal k-filters of a CRD-Stone algebra L forms a Boolean algebra and so a Boolean ring.

Definition 4.1

A principal filter of a CRD-Stone algebra, which containing the core k is called a principal k-filter of L.

Now, let $A=\{a\}$ be a subset of a CRD-Stone L. We realize that

$$\begin{aligned} \{a\}^{\bigtriangleup }=\{x\in L: x^{**}\ge a^{**}\wedge k\}. \end{aligned}$$

We will write $(a)^{\bigtriangleup }$ instead of $\{a\}^{\bigtriangleup }$. It is clear that $(1)^{\bigtriangleup }=D(L)$ and $(0)^{\bigtriangleup }=L$ are the least and largest principal k-filters of L, respectively.

Theorem 4.2

Let x, y be elements of a CRD-Stone algebra of L. Then, we have

• (1) $y\in (x)^{\bigtriangleup }$ $\Leftrightarrow$ $y^*\wedge x=0$,

• (2) $(x)^{\bigtriangleup }$ is a principal k-filter of L, more precisely $(x)^{\bigtriangleup }=[x^{**}\wedge k)=[x^{**})\vee D(L)$,

• (3) $x\in D(L)$ $\Leftrightarrow$ $(x)^{\bigtriangleup }=D(L)$.

Proof

• (1) Let $y\in (x)^{\bigtriangleup }$.Then,

  $$\begin{aligned}&y^{**}\ge x^{**}\wedge k\Leftrightarrow y^*\le x^*\\&\Leftrightarrow y^*\wedge x=0 ~\qquad \qquad (\text {by Definition }2.6). \end{aligned}$$

• (2) For all $x\in L$, we get

  $$\begin{aligned} (x)^{\bigtriangleup }&=\{y\in L: y^{**}\ge x^{**}\wedge k \}\\ {}&=\{y\in L: y^{**}\wedge (y\vee k)\ge (x^{**}\wedge k)\wedge (y\vee k)\}\\&=\{y\in L: y\ge x^{**}\wedge k\}\\&\quad (\text { by Theorem 2.18} (3) \text {and Definition } 2.1(4))\\&=[x^{**}\wedge k)\\ {}&=[x^{**})\vee [k)=[x^{**}) \vee D(L), ~\\&\quad where~ D(L)=[k). \end{aligned}$$

  It is follows that $(x)^{\bigtriangleup }$ is a principal k-filter of L.

• (3) Let $x\in D(L)$. Then, $x^{*}=0$. Now,

  $$\begin{aligned} (x)^{\bigtriangleup }&=[x^{**}\wedge k)\qquad \qquad \qquad (\text {by }(2))\\ {}&=[1\wedge k)=[k)=D(L). \end{aligned}$$

The converse implication is obvious. $\square$

More important properties of principal k-filters are given in the following two lemmas.

Lemma 4.3

Let x, y be elements of a CRD-Stone algebra L. Then,

• (1) $(x)^{\bigtriangleup \bigtriangleup }=(x)^{\bigtriangleup }$,

• (2) $[x)^{\bigtriangleup }=(x)^{\bigtriangleup }$,

• (3) $x\in (y)^{\bigtriangleup }\Leftrightarrow (x)^{\bigtriangleup }\subseteq (y)^{\bigtriangleup }$,

• (4) $x\le y \Rightarrow (x)^{\bigtriangleup }\subseteq (y)^{\bigtriangleup }$.

Lemma 4.4

Let L be a CRD-Stone algebra L. For any $x,y \in L$, we have.

• (1) $(x)^{\bigtriangleup }=(x^{**})^{\bigtriangleup }$,

• (2) $(x\wedge y)^{\bigtriangleup }$=$(x)^{\bigtriangleup }\vee (y)^{\bigtriangleup }$,

• (3) $(x\vee y)^{\bigtriangleup }=(x)^{\bigtriangleup }\cap (y)^{\bigtriangleup }$,

• (4) $(x\wedge x^*)^{\bigtriangleup }= L$,

• (5) $(x\vee x^{*})^{\bigtriangleup }=D(L)$.

Proof

• (1) $(x)^{\bigtriangleup }=\{y\in L:y^{**}\ge x^{**}\wedge k=(x^{**})^{**}\wedge k \}=(x^{**})^{\bigtriangleup }$, as $x^{****}=x^{**}$.

• (2) By Theorem 4.2.(2), we get

  $$\begin{aligned} (x\wedge y)^{\bigtriangleup }&=[(x\wedge y)^{**}) \vee D(L)\\&=[(x^{**}\wedge y^{**}))\vee D(L)\\ {}&=([x^{**})\vee [y^{**}))\vee D(L)\\&= ([x^{**})\vee D(L))\vee ( [y^{**})\vee D(L))\\ {}&=(x)^{\bigtriangleup }\vee (y)^{\bigtriangleup }. \end{aligned}$$

• (3) By Theorem 4.2.(2), we get

  $$\begin{aligned}&(x\vee y)^{\bigtriangleup }\\ {}&=[(x\vee y)^{**}) \vee D(L)\\ {}&= [x^{**}\vee y^{**})\vee D(L)\\ {}&=([x^{**})\cap [y^{**}))\vee D(L)\\ {}&= ([x^{**})\vee D(L))\cap ([y^{**})\vee D(L))\\&\quad (\text {by distributivity of F(L)})\\ {}&=(x)^{\bigtriangleup }\cap (y)^{\bigtriangleup }. \end{aligned}$$

• (4), (5) are obvious. As a consequent of the above results, we can prove the following.

$\square$

Lemma 4.5

Let x, y be elements of a CRD-Stone algebra. Then,

• (1) $(x)^{\bigtriangleup }=(y)^{\bigtriangleup }$ $\Leftrightarrow$ $x^{**}=y^{**}$ $\Leftrightarrow x^{*}=y^{*}$,

• (2) $(x)^{\bigtriangleup }=(y)^{\bigtriangleup }\Rightarrow (x\wedge z)^{\bigtriangleup }=(y\wedge z)^{\bigtriangleup }$, for all $z \in L$,

• (3) $(x)^{\bigtriangleup }=(y)^{\bigtriangleup }\Rightarrow (x\vee z)^{\bigtriangleup }=(y\vee z)^{\bigtriangleup }$, for all $z \in L$.

In the following, we observe that any principal k-filter [x) of L is identical with $(x)^{\bigtriangleup }$.

Theorem 4.6

Every principal k-filter of L can be written as $(x)^{\bigtriangleup }$ for some $x\in L$.

Proof

Let $F=[x)$ be a principal k-filter of L. We claim that $F=(x)^{\bigtriangleup }$. Since $x \in (x)^{\bigtriangleup }~then~ [x) \subseteq (x)^{\bigtriangleup }$. Conversely, let $y\in (x)^{\bigtriangleup }$. Then,

$$\begin{aligned}&y\in (x)^{\bigtriangleup } \\ {}&\Rightarrow y^{**}\ge x^{**}\wedge k\\ {}&\Rightarrow y^{**}\wedge (y \vee k)\ge (x^{**}\wedge k)\wedge (y\vee k)\\&= x^{**}\wedge ( k \wedge (y\vee k))\\&=x^{**}\wedge k\\ {}&\Rightarrow y \ge x \wedge k~~~~as~y=y^{**}\wedge (y \vee k) ~and~ a\le a^{**}\\ {}&\Rightarrow y\in [x\wedge k)\subseteq [x)~~~ as~x\wedge k\le x. \end{aligned}$$

Then, $(x)^{\bigtriangleup }\subseteq [x)$ and hence $(x)^{\bigtriangleup }=[x)$. $\square$

Now, a characterization of a k-filter via principal k-filters of a CRD-Stone algebra of L is given.

Theorem 4.7

Let F be a filter of a CRD-Stone algebra L, Then, the following conditions are equivalent:

• (1) F is a k-filter,

• (2) $x^{**}\in F$ implies $x\in F$,

• (3) for all $x,y \in L,(x)^{\bigtriangleup }=(y)^{\bigtriangleup }$ and $y\in F$ imply $x\in F$,

• (4) $F=\bigcup _{x\in F}(x)^{\bigtriangleup }$,

• (5) $x\in F$ implies $(x)^{\bigtriangleup }\subseteq F$.

Proof

$(1)\Rightarrow (2)$: Let F be a k-filter of L and $x^{**}\in F$. Then, $k\in F$ implies $x\vee k\in F$. Now, $x^{**},~x\vee k\in F$ imply that $x=x^{**}\wedge ( x\vee k) \in F$.

$(2)\Rightarrow (3)$: Let $(x)^{\bigtriangleup }=(y)^{\bigtriangleup }$ and $y\in F$. We have $x \in (y)^{\bigtriangleup }$. Then, $x^{**}\ge y^{**}\wedge k$ implies $x^{**}\ge y^{**}\ge y \in F$. Then, $x^{**}\in F$ implies $x\in F$, by(2).

$(3)\Rightarrow (4)$: For any $x \in F$, we have $x \in (x)^{\bigtriangleup }\subseteq \bigcup _{x\in F}(x)^{\bigtriangleup }$. Then, $F\subseteq \bigcup _{x\in F}(x)^{\bigtriangledown }$. Conversely, let $y\in \bigcup _{x\in F}(x)^{\bigtriangleup }$. Then, $y\in (z)^{\bigtriangleup }$ for some $z\in F$. Hence, $(y)^{\bigtriangleup }\subseteq (z)^{\bigtriangleup }$, by Lemma 4.3.(3). It follows that $(z)^{\bigtriangleup }=(y)^{\bigtriangleup }\vee (z)^{\bigtriangleup }=(y\wedge z)^{\bigtriangleup }$. Since $z \in F$, then by (3), we get $y\wedge z\in F$. Thus $y\in F$. Therefore, $\bigcup _{x\in F}(x)^{\bigtriangleup }\subseteq F$ and hence $\bigcup _{x\in F}(x)^{\bigtriangleup }=F$.

$(4)\Rightarrow (5)$: Assume (4). Let $x\in F$. Then by (4), we get $x\in (f)^{\bigtriangleup }$ for some $f\in F$. Suppose $t\in (x)^{\bigtriangleup }$. Then, it concludes $t \in (x)^{\bigtriangleup }\subseteq (f)^{\bigtriangleup }$. Then, $t\in \bigcup _{f\in F}(f)^{\bigtriangleup }=F$. Therefore, $(x)^{\bigtriangleup }\subseteq F$.

$(5)\Rightarrow (1)$: Assume (5). Since $k\in (x)^{\bigtriangleup }$, $\forall x\in F$, then by (5), $k\in (x)^{\bigtriangleup }\subseteq F$. This proves that F is a k-filter of L. $\square$

Consider $F^{p}_{k}(L)=\{(x)^{\bigtriangleup }:x\in L\}$, the set of all principal k-filters of a CRD-Stone algebra L.

Theorem 4.8

Let L be a CRD-Stone algebra. Then,

$(F^{p}_{k}(L); \vee , \cap , ^{\prime },(0)^{\bigtriangleup }, (1)^{\bigtriangleup })$ is a Boolean algebra, where

$$\begin{aligned} (x)^{\bigtriangleup }\vee (y)^{\bigtriangleup }= & {} (x\wedge y)^{\bigtriangleup },\\ (x)^{\bigtriangleup }\cap (y)^{\bigtriangleup }= & {} (x\vee y)^{\bigtriangleup },\\ {(x)^{\bigtriangleup }}^{\prime }= & {} (x^{*})^{\bigtriangleup }. \end{aligned}$$

Proof

It is clear that $(F^{p}_{k}(L); \vee , \cap , ^{\prime },(0)^{\bigtriangleup }, (1)^{\bigtriangleup })$ is a bounded lattice. For all $(x)^{\Delta },(y)^{\Delta }$ and $(z)^{\Delta }$ in $F^{p}_{k}(L)$, we have

$$\begin{aligned} (x)^{\bigtriangleup } \cap ((y)^{\bigtriangleup } \vee (z)^{\bigtriangleup })= & {} (x)^{\bigtriangleup } \cap (y \wedge z)^{\bigtriangleup } \\= & {} (x \vee (y \wedge z))^{\bigtriangleup } \\= & {} ((x \vee y) \wedge (x \vee z))^{\bigtriangleup }~~~~~~~\\{} & {} \text { by~ the~ distributivity ~of~ L }\\= & {} (x \vee y)^{\bigtriangleup } \vee (x \vee z)^{\bigtriangleup } \\= & {} ((x)^{\bigtriangleup } \cap (y)^{\bigtriangleup }) \vee ((x)^{\bigtriangleup } \cap (z)^{\bigtriangleup }) \end{aligned}$$

Therefore, $F^{p}_{k}(L)$ is a bounded distributive lattice. Now, for each $(x)^{\bigtriangleup }\in F^{p}_{k}(L)$, we have

$$\begin{aligned}{} & {} (x)^{\bigtriangleup } \cap {(x)^{\bigtriangleup }}^{\prime }=(x)^{\bigtriangleup } \cap \left( x^*\right) ^{\bigtriangleup }=(x \vee x^*)^{\bigtriangleup }=(1)^{\bigtriangleup }\\{} & {} \quad =D(L),~ as~ x\vee x^*\in D(L),\\{} & {} \quad (x)^{\bigtriangleup } \vee {(x)^{\bigtriangleup }}^{\prime }=(x)^{\bigtriangleup } \vee (x^*)^{\bigtriangleup }=(x \wedge x^*)^{\bigtriangleup }=(0)^{\bigtriangleup }=L. \end{aligned}$$

Then, $F^{p}_{k}(L)$ is a complemented lattice. Therefore,

$\left( F^{p}_{k}(L), \cap , \vee ,^{\prime },(0)^{\triangle },(1)^{\triangle }\right)$ is a Boolean algebra. $\square$

Theorem 4.9

Let L be a CRD-Stone algebra. Then,

• (1) $(F^{p}_{k}(L);\vee ,\wedge ,D(L),L)$ is a $\{1\}$-sublattice of F(L),

• (2) $(F^{p}_{k}(L);\vee , \wedge , (0)^{\bigtriangleup }, (1)^{\bigtriangleup })$ is a bounded sublattice of $F_{k}(L)$,

• (3) B(L) is isomorphic to $F^{p}_{k}(L)$ as Boolean algebras.

Proof

• (1) and (2) are clear.

• (3) Define a map**: $f: B(L) \longrightarrow F^{p}_{k}(L)$ by $f(x)=(x^*)^{\bigtriangleup }, \forall x \in B(L)$. To prove that f is a homomorphism, let $x,y\in B(L)$,

  $$\begin{aligned} f(x\vee y)&=((x\vee y)^*)^{\bigtriangleup }\\ {}&=(x^*\wedge y^*)^{\bigtriangleup } \qquad \qquad (\text {by Lemma } 4.4(2))\\ {}&=(x^*)^{\bigtriangleup }\vee (y^*)^{\bigtriangleup }\\ {}&=f(x)\vee f(y). \end{aligned}$$

  Thus, $f(x\vee y)=f(x)\vee f(y)$. Similarly, we can get $f(x\wedge y)=f(x)\cap f(y)$ since $f(x^*)=(x^{**})^{\bigtriangleup }=(x^{*{\bigtriangleup }})^{\prime }=(f(x))^{\prime }$, that f is a homomorphism of Boolean algebras $f(0)=(1)^{\bigtriangleup }$ and $f(1)=(0)^{\bigtriangleup }$. This f is a $\{0,1\}$-lattice homomorphism. Let $f(x)=f(y)$. Then, $(x)^{\bigtriangleup }=(y)^{\bigtriangleup }$. Hence, $x^{**}=y^{**}$. Therefore, $x=y$. Then, f is an injective map. $\forall (x)^{\bigtriangleup }\in F^{p}_{k}(L)$, then $x^*\in B(L)$ such that $f(x^*)=(x^{**})^{\bigtriangleup }=(x)^{\bigtriangleup }$. Then f is a surjective homomorphism. Therefore f is an isomorphism and $B(L)\cong F^{p}_{k}(L)$.

$\square$

Corollary 4.10

Let L be a CRD-Stone algebra. Then,

$(I^{p}_{k}(L);+, \bullet , (0)^{\bigtriangleup }, (1)^{\bigtriangleup })$ can be made into a Boolean ring, where $+$ the addition operation and $\bullet$ the multiplication operation are defined as follows:

$$\begin{aligned} (x)^{\bigtriangleup }+(y)^{\bigtriangleup }= & {} ((x\vee y^{*})\wedge (y\vee x^{*}))^{\bigtriangleup },\\ (x)^{\bigtriangleup }\bullet (y)^{\bigtriangleup }= & {} (x\wedge y)^{\bigtriangleup }. \end{aligned}$$

5 k-$\{^*\}$-congruences on a CRD-Stone algebra

In this section, we investigate the relationship between k-filters and k-$\{^*\}$-congruences of a CRD-Stone algebra L. Also, we describe the lattice $Con^*_{k}(L)$ of all k-$\{^*\}$-congruences of L.

Definition 5.1

A $\{^*\}$-congruence relation $\theta$ on a CRD-Stone algebra L is called a k-$\{^*\}$-congruence if $k \in Coker~{\theta }$.

It is known that $Coker~\theta$=$[1]\theta$. Using the properties of the principal k-filters of L, one can verify the following.

Proposition 5.2

Define a binary relation $\theta$ on a core regular double Stone L as follows:

$$\begin{aligned} (x,y)\in \theta \Leftrightarrow (x)^{\bigtriangleup }=(y)^{\bigtriangleup }. \end{aligned}$$

Then, $\theta$ is a k-$\{^*\}$-congruence on L. Moreover, $\theta =\psi ^*$

Let F be a k-filter of CRD-Stone algebra L. Define a binary relation $\theta _F$ on L as follows:

$$\begin{aligned}{} & {} \theta _F=\left\{ (a, b) \in L \times L: a \wedge f\wedge k =b \wedge f\wedge k,\right. \\{} & {} \quad \left. \text { for some ~f }\in F \right\} . \end{aligned}$$

Now, we investigate the k-$\{^*\}$-congerences via k-filters of L.

Theorem 5.3

Let F be a k-filter of CRD-Stone algebra L. Then $\theta _F$ is a k-$\{^*\}$-congerence on L such that $Coker~\theta _F=F$.

Proof

Clearly, $\theta _F$ is an equivalent relation on L. Let $(a, b) \in \theta _F$. Then, $a \wedge f\wedge k=b \wedge f\wedge k$ for some $f \in F$. Now for all $c \in L$, then by distributivity of L, we have

$$\begin{aligned} \begin{aligned}&(a \wedge c)\wedge f\wedge k =(b \wedge c) \wedge i\wedge k, \\&(a \vee c)\wedge f\wedge k=(b \wedge c) \wedge f\wedge k. \end{aligned} \end{aligned}$$

Therefore, $(a \wedge c, b \wedge c),(a \vee c, b \vee c) \in \theta _F$. Therefore, by Definition 2.19, $\theta _F$ is a lattice congruence on L. It remains to show that $(a, b) \in \theta _F$ implies $\left( a^*, b^*\right) \in \theta _F$.

$$\begin{aligned}&(a, b) \in \theta _F \\ {}&\Rightarrow a\wedge f\wedge k=b \wedge f\wedge k \\&\Rightarrow a^* \vee f^*\vee k^*= b^* \vee f^*\vee k^* \\&\Rightarrow a^* \vee f^*=b^* \vee f^* ~ as~ k^*=0 \\&\Rightarrow \left( a^* \vee f^*\right) \wedge f=\left( b^* \vee f^*\right) \wedge f \\&\Rightarrow \left( a^* \wedge f\right) \vee \left( f^* \wedge f\right) =\left( b^* \wedge f\right) \vee \left( f^* \wedge f\right) \\&(\text {by distributivity of L }) \\&\Rightarrow \left( a^*\wedge f\right) \vee 0=\left( b^* \wedge f\right) \vee 0 \qquad (\text {by Theorem }2.9(2)) \\&\Rightarrow a^* \wedge f=b^* \wedge f \\&\Rightarrow a^* \wedge f\wedge k=b^* \wedge f\wedge k \\&\Rightarrow \left( a^*, b^*\right) \in \theta _F. \end{aligned}$$

Then, $\theta _F$ is a $\{^*\}$-congruence on L. Now, we prove that $Coker~\theta _ F=F$.

$$\begin{aligned} Coker\theta _F&=\{x\in L: (1,x)\in \theta _{F}\} \\&=\{x\in L: 1\wedge f\wedge k=x\wedge f\wedge k , f\in F\} \\&=\{x\in L: f\wedge k=x\wedge f\wedge k\}\\&=\{x\in L: x\ge f\wedge k\}\\&=\{x\in L: x^{**}\ge f^{**}\ge f^{**}\wedge k \}\\&=\{x: x\in F^{\bigtriangleup }=F\}=F. \end{aligned}$$

Since $k\in F=Coker~\theta _F$, then $\theta _F$ is a a k-$\{^*\}$-congruence on L. $\square$

Theorem 5.4

For any k-filters F, G of a CRD-Stone algebra L, we have

• (1) $F \subseteq G \Leftrightarrow \theta _F \subseteq \theta _G$,

• (2) $\psi ^* \subseteq \theta _F$, where $\psi ^*$ is the Glivenko congruence on $(L,^{*})$,

• (3) $\theta _{D(L)}=\psi ^*$,

• (4) $\theta _{L}=\nabla _L$,

• (5) $L / \theta _F$ is a Boolean algebra.

Proof

• (1) The first implication is obvious. For the converse implication, let $\theta _F \subseteq \theta _G$. Then, by the above Theorem 5.3. $F=Coker~\theta _F\subseteq Coker~\theta _G=G$.

• (2) Let $(a,b)\in \psi ^*$. Then $a^*=b^* ~implies~ a^{**}=b^{**}$. Now, we have

  $$\begin{aligned} a\wedge f\wedge k&=(a^{**}\wedge (a\vee k))\wedge f\wedge k\\&(\text {by Lemma }2.18(3))\\&=a^{**}\wedge f\wedge ((a\vee k)\wedge k)\\&=a^{**}\wedge f\wedge k\qquad \qquad (\text {by Definition } 2.1(4))\\&=b^{**}\wedge f\wedge k\\ {}&=b^{**}\wedge f \wedge ((b\vee k)\wedge k)\\ {}&=(b^{**}\wedge (b\vee k))\wedge f\wedge k\\ {}&=b\wedge f\wedge k. \end{aligned}$$

  Thus, $(a,b)\in \theta _F$ and hence $\psi ^*\subseteq \theta _{F}$.

• (3) Since $f^*=0$, for all $f\in {D(L)}$, we get

  $$\begin{aligned} \begin{aligned}&\theta _{D(L)}\\ {}&=\{(a,b)\in L \times L: a\wedge f\wedge k=b\wedge f\wedge k,~\\&f\in {D(L)}\}\\ {}&=\{(a,b)\in L \times L: a^*\vee f^*\vee k^*=b^*\vee f^*\vee k^*\}\\ {}&=\{(a,b)\in L \times L: a^*=b^*\}=\\&\psi ^* ~~~~(as~ f^*=k^*=0) \end{aligned} \end{aligned}$$

• (4) Since $a\wedge 0\wedge k=b\wedge 0\wedge k$ for all $a,b\in L$, then $(a,b)\in \theta _{L}$ and hence $\theta _{L}=\nabla _L$.

• (5) The quotient set $L / \theta _F$ is $\left\{ [a] \theta _F: a \in L\right\}$, where $[a] \theta _F$ is the congruence class of an element $a \in L$ modulo $\theta _F$. It is clear that $L / \theta _F=\left( L / \theta _F; \vee , \wedge ,[1]\theta _F,[0] \theta _F\right)$ is a bounded distributive lattice, where $[1]_F=F,[0] \theta _F$ are the bounds of $L / \theta _F$ and $[a] \theta _F \wedge [b] \theta _F=[a \wedge b] \theta _F,[a] \theta _F \vee [b] \theta _F=[a \vee b] \theta _F$. Define a unary operation on $L / \theta _F$ by $[a]^{\prime } \theta _F=[a^*] \theta _F$, since $[a] \theta _F \wedge \left[ a^*\right] \theta _F=[a\wedge a^*] \theta _F=[0]\theta _{F}$, $[a] \theta _F \vee [a^*] \theta _F=[a\vee a^*] \theta _F=[1]\theta _{F}$ and $[a]^{\prime \prime } \theta _F=[a^*]^{\prime } \theta _F=[a^{**}] \theta _F=[a] \theta _F$. Then, $(L / \theta _F; \vee , \wedge , ^{\prime }, [0]\theta _{F}, [1]\theta _{F})$ is a Boolean algebra.

$\square$

Consider $Con^{*}_{k}(L)$ =$\{\theta _{F}: F\in F_{k}(L)\}$, the set of all k-$\{^*\}$-congruences which induced by k-filters of L.

Theorem 5.5

For any $\theta _{F}$ and $\theta _{F}$ of $Con^{*}_{k}(L)$, we have

• (1) $\theta _{F}\cap \theta _{G}=\theta _{(F\cap G)}$,

• (2) $\theta _{F}\vee \theta _{G}=\theta _{(F\vee G)}$,

• (3) $(Con^{*}_{k}(L); \vee , \cap ,\theta _{D(L)}, \theta _{L} )$ forms a bounded lattice and a $\{1\}$-sublattice of $Con^{*}(L)$.

Proof

• (1) Since $F\cap G \subseteq F, G$, by Theorem 5.4. $\theta _{(F \cap G)}\subseteq \theta _{F}, \theta _{G}$ implies $\theta _{(F\cap G)}\subseteq \theta _{F} \cap \theta _{G}$. Conversely,

  $$\begin{aligned} \begin{aligned}&Let~ (a,b)\in \theta _F\cap \theta _G\Rightarrow (a,b)\in \theta _F~\text { and}~(a,b)\in \theta _G\\&{\Rightarrow } a\wedge f\wedge k=b\wedge f\wedge k \text {~for~some~}f\in F~and~ a\wedge g \wedge \\&k=b\wedge g\wedge k~\text { for~some~}g\in G\\&\Rightarrow (a\wedge f\wedge k)\vee (a\wedge g\wedge k)\\&=(b\wedge f\wedge k)\vee (a\wedge g\wedge k)\\&\Rightarrow (a\wedge k\wedge f)\vee (a\wedge k\vee g)\\&=(b\wedge k\wedge f)\vee (a\wedge k\wedge g)\\&\Rightarrow a\wedge k\wedge (f\vee g)=b\wedge k \wedge (f\vee g) \\&\Rightarrow (a,b)\in \theta _{(F\cap G)}~ as~ f\vee g\ge f,g\\&\Rightarrow f\vee g\in F\cap G. \end{aligned} \end{aligned}$$

  Then, $\theta _F\cap \theta _G \subseteq \theta _{(F\cap G)}$. Therefore, $\theta _{(F\cap G)}$ is the greatest lower bound of $\theta _F$ and $\theta _G$ on $Con^{*}_{k}(L)$. Therefore, $\theta _{(F\cap G)}= \theta _{F} \cap \theta _{G}$.

• (2) Since $F,G \subseteq F\vee G$, then by $\theta _F, \theta _G\subseteq \theta _{(F\vee G)}$. Thus, $\theta _{(F\vee G)}$ is an upper bound of $\theta _F$,$\theta _G$ on $Con^{*}_{k}(L)$. Let $\theta _{H}$ be an upper bound of $\theta _F$ and $\theta _G$, for $\theta _{H}\in Con^{*}_{k}(L)$ and $H\in F_{k}(L)$. Then $\theta _F$, $\theta _G$ $\subseteq$ $\theta _{H}$. Hence, $F,~G\subseteq H$ implies $F\vee G\subseteq H$ is the least upper bound of F,  G on $F_{k}(L)$. By Theorem 5.4. $\theta _{(F\vee G)}$ $\subseteq$ $\theta _{H}$. Therefore, $\theta _{(F\vee G)}$ is the least upper bound of $\theta _F$, $\theta _G$ on $Con^{*}_{k}(L)$. This proves that $\theta _{F}\vee \theta _{G}=\theta _{(F\vee G)}$.

• (3) From (1) and (2), it is clear that $(Con^{*}_{k}(L); \vee , \wedge )$ forms a sublattice of $Con^*(L)$. Since $\theta _{D(L)}$ and $\theta _{L}$ are the smallest and the greatest members of $Con^{*}_{k}(L)$, respectively. Then, $(Con^{*}_{k}(L); \vee , \wedge ,\theta _{D(L)}, \theta _{L} )$ is a bounded lattice. Since $\nabla _L=\theta _{L}$ is the largest element of $Con^{*}_{k}(L)$ then $Con^{*}_{k}(L)$ is $\{1\}$-sublatttice of $Con^{*}(L)$.

$\square$

Now, we introduce the following interesting results.

Theorem 5.6

For every k-$\{^*\}$-congruence $\alpha$ on a CRD-Stone algebra L, we have

• (1) $[1] \alpha$ is a k-filter of L,

• (2) $\alpha$ can be expressed as $\theta _F$ for some k-filter F of L.

Proof

• (1) It is known that $[1]\alpha =\{x\in L:(x,1)\in \alpha )\}=Coker~\alpha$. Which is a filter of L. Since $\alpha$ is a k-$\{^*\}$-congruence, then $k\in Coker~\alpha$. Therefore, $[1]\alpha$ is a k-filter of L.

• (2) We claim that $\alpha =\alpha _{[1]\theta }$. Let $(x, y) \in \alpha$. Then, $(k,k) \in \alpha$ hence $\left( x \wedge k, y \wedge k\right) \in \alpha$. Since $[1]\alpha$ is a k-ideal of L, then $x \vee k,~ y \vee k \in [1]\alpha$. Hence, $\left( x \wedge k,~ y \wedge k\right) \in \alpha _{[1]\alpha } \text{. } \text{ Now }$, we prove that $\left( x^{**}, y^{**}\right) \in \alpha _{[1] \alpha }$.

  $$\begin{aligned}&\left( x^*, y^*\right) \in \alpha \quad \Rightarrow \quad \left( x^* \vee x^{**}, y^* \vee x^{**}\right) \in \alpha \text{ and } \\&\left( x^* \vee y^{**}, y^* \vee y^{**}\right) \in \alpha \\&\Rightarrow \quad \left( 1, y^* \vee x^{**}\right) \in \alpha \text{ and } \\&\left( x^* \vee y^{**}, 1\right) \in \alpha \qquad \qquad \text {(by Definition} 2.8) \\&\Rightarrow \quad x^* \vee y^{**}, y^* \vee x^{**} \in [1]\alpha \\&\Rightarrow \quad \left( x^* \vee y^{**}, y^* \vee x^{**}\right) \in \alpha _{[1]\alpha } \\&\Rightarrow \quad \left( x^* \wedge \left( x^* \vee y^{**}\right) , x^* \wedge \left( y^* \vee x^{**}\right) \right) \\&=\left( x^*, x^* \vee y^*\right) \alpha _{[1] \alpha }\qquad \qquad \text{(by } \text{ Definition } 2.1(4)) \\&\text{ and } \quad \left( y^* \wedge \left( x^* \vee y^{**}\right) , y^* \wedge \left( y^* \vee x^{**}\right) \right) \\&=\left( x^* \wedge y^*, y^*\right) \in \alpha _{[1] \alpha } \\&\Rightarrow \quad \left( x^*, y^*\right) \in \alpha _{[1]\alpha } \\&\Rightarrow \quad \left( x^{**}, y^{**}\right) \in \alpha _{[1]\alpha } \text{. } \end{aligned}$$

  Now, $\left( x^{**}, y^{**}\right) \in \alpha _{[1] \alpha }$ and $\left( x \vee k, y \vee k\right) \in \alpha _{[1] \alpha }$ imply that $(x, y)=\left( x^{**} \wedge \left( x \vee k\right) , y^{**} \wedge \left( y \vee k\right) \right) =\left( x^{**}, y^{**}\right) \wedge \left( x \vee k, y \vee k\right) \in \alpha _{[1] \alpha } \text{. } \text{ Then }$ $\alpha \subseteq \alpha _{[1] \alpha }$. For the converse let $(x, y) \in \alpha _{[1] \alpha }$. Then $\left( x \vee k, y \vee k\right) \in \alpha _{[1] \alpha }$. Since $x \vee k, y \vee k \in [1] \alpha$, then $\left( x \vee k, y \vee k\right) \in \alpha$. Now, we prove that $\left( x^{**}, y^{**}\right) \in \alpha$ for all $(x, y) \in \alpha _{[1] \alpha }$

  $$\begin{aligned} \begin{aligned}&(x, y) \in \alpha _{[1] \alpha } \Rightarrow \left( x^*, y^*\right) \in \alpha _{[1] \alpha } \\&\Rightarrow \left( x^* \vee x^{**}, y^* \vee x^{**}\right) ,\left( x^* \vee y^{**}, y^* \vee y^{**}\right) \in \alpha _{[1] \alpha } \\&\Rightarrow \left( 1, y^* \vee x^{**}\right) ,\left( x^* \vee y^{**}, 1\right) \in \alpha _{[1] \alpha } \text{ as } \\&x^* \vee x^{**}=1, y^* \vee y^{**}=1 \\&\Rightarrow x^* \vee y^{**}, y^* \vee x^{**} \in [1] \alpha \\&\Rightarrow \left( x^* \vee y^{**}, y^* \vee x^{**}\right) \in [1] \alpha \\&\Rightarrow \left( x^* \wedge \left( x^* \vee y^{**}\right) , x^* \wedge \left( y^* \vee x^{**}\right) \right) ,\\&\left( y^* \wedge \left( x^* \vee y^{**}\right) , y^* \wedge \left( y^* \vee x^{**}\right) \right) \in \alpha \\&\Rightarrow \left( x^*,\left( x^* \wedge y^*\right) \vee \left( x^* \wedge x^{**}\right) \right) ,\\&\left( \left( y^* \wedge x^*\right) \vee \left( y^* \wedge y^{**}\right) , y^*\right) \in \alpha \qquad \qquad \\&\text{(by } \text{ Definition } 2.1(4)) \\&\Rightarrow \left( x^*, x^* \wedge y^*\right) ,\\&*\left( x^* \wedge y^*, y^*\right) \in \alpha \qquad \qquad \text{(by } \text{ Definition } 2.8) \\&\Rightarrow \left( x^*, y^*\right) \in \alpha \\&\Rightarrow \left( x^{**}, y^{**}\right) \in [1]\alpha . \end{aligned} \end{aligned}$$

  Now, $\left( x^{**}, y^{**}\right) \in \alpha$ and $\left( x \vee k, y \vee k\right) \in [1] \alpha$ imply that $(x, y)=\left( x^{**}, y^{**}\right) \wedge \left( x \vee k, y \vee k \right) \in \alpha$. Therefore, $\alpha _{[1] \alpha } \subseteq \alpha$ and $\alpha =\alpha _{[1] \alpha }$.

$\square$

As a consequent of Theorem 5.5 and Theorem 5.6, we can show that there is a one to one correspondence between the elements of the lattice $F_{k}(L)$ of all k-filters of a CRD-Stone algebra L and the elements of the lattice $Con^{*}_{k}(L)$ of all k-$\{^*\}$-Congruences of L. This proves that the lattices $F_{k}(L)$ and $Con^{*}_{k}(L)$ are isomorphic and hence the lattice $Con^{*}_{k}(L)$ is a bounded distributive lattice.

Theorem 5.7

Let L be a CRD-Stone algebra. Then, the lattices $F_{k}(L)$ and $Con^{*}_{k}(L)$ are isomorphic and hence $Con^{*}_{k}(L)$ is a bounded distributive lattice.

Proof

Define a map h:$F_{k}(L)$ $\longrightarrow$ $Con^{*}_{k}(L)$ by $h(F)=\theta _{F}$, for all $F \in F_{k}(L)$. From Theorem 5.5, for $F,G \in F_{k}(L)$, we have

$$\begin{aligned}{} & {} h(F\vee G)=\theta _{(F\vee G)}=\theta _{F}\vee \theta _{G}=h(F) \vee h(G),\\{} & {} h(F\cap G)=\theta _{(F\cap G)}=\theta _{F}\cap \theta _{G}=h(F) \cap h(G),\\{} & {} h(D(L))=\theta _{D(L)}=\psi ^*,\\{} & {} h(L)=\theta _{L}=\nabla _L. \end{aligned}$$

Then, h is (0,1)-lattice homomorphism. Let $h(F)=h(G)$. Then, $\theta _{F}=\theta _{G}$ implies $F=Corker\theta _{F}=Coker\theta _{G}=G$. Thus, h is an injective map. For each $\alpha \in Con^{*}_{k}(L)$, by Theorem 5.6.(2), we have $\alpha =\theta _{F}$ for some $F\in F_{k}(L)$. Then $h(F)=\theta _{F}=\alpha$ implies that h is a surjective. Therefore, h is a lattice isomorphism and hence $F_{k}(L)$ and $Con^{*}_{k}(L)$ are isomorphic lattices. Since $F_{k}(L)$ is a bounded distributive lattice (see Theorem 3.8), then also, $Con^{*}_{k}(L)$ a bounded distributive lattice. $\square$

6 Principal k-$\{^*\}$-congruences on a CRD-Stone algebra

In this section, we characterize principal k-$\{^*\}$-Congruences on a CRD-Stone algebra L via principal k-filters of L and establish the algebraic structure of the class of all principal k-$\{^*\}$-congruences of L.

Proposition 6.1

Let L be a CRD-Stone algebra L and $F=(x)^{\bigtriangleup }$. Then, $\theta _{(x)^{\bigtriangleup }}$ is described as follows:

$$\begin{aligned} \theta _{(x)^{\bigtriangleup }}=\left\{ (a, b) \in L \times L: a \wedge x \wedge k=b \wedge x\wedge k\right\} . \end{aligned}$$

and $Coker~\theta _{(x)^{\bigtriangleup }}=(x)^{\bigtriangleup }$.

Proof

Let $F=(x)^{\bigtriangleup }$. Then,

$\theta _F=\theta _{(x)^{\bigtriangleup }}=\left\{ (a, b) \in L \times L: a \wedge f \wedge k=b \wedge f\wedge k,\right.$ $\left. ~\textrm{for}~ \textrm{some}~ f\in (x)^{\bigtriangleup }\right\}$.

Let $(a,b)\in \theta _F$. Since $F=(x)^{\bigtriangleup }$, thus $a \wedge f \wedge k=b \wedge f\wedge k$, for some $f\in (x)^{\bigtriangleup }$ and hence $a^{**}\wedge f^{**}=b^{**}\wedge f^{**}$. Since $f\in (x)^{\bigtriangleup }$, then $f^{**}\ge x^{**}\wedge k$ and we have $f^{**}\ge x^{**}$. Now, we get

$$\begin{aligned}&a\wedge x\wedge k \\ {}&=[a^{**}\wedge (a\vee k)]\wedge [x^{**}\wedge (x\vee k)]\wedge k \qquad \qquad \\&(\text {by Lemma }2.18(3)) \\ {}&=[a^{**}\wedge (a\vee k)]\wedge x^{**}\wedge [(x\vee k)\wedge k]\\ {}&=[a^{**}\wedge (a\vee k)]\wedge x^{**}\wedge k \qquad \qquad (\text {by Definition }2.1(4) )\\ {}&=a^{**}\wedge x^{**}\wedge [(a\vee k)\wedge k]\\ {}&=a^{**}\wedge x^{**}\wedge k\qquad \qquad (\text {by Definition } 2.1(4))\\ {}&=b^{**}\wedge x^{**}\wedge k\\ {}&=b^{**}\wedge x^{**}\wedge (x\vee k)\wedge (b\vee k)\wedge k \\&as~(x\vee k)\wedge (b\vee k)\wedge k=k \\ {}&=[b^{**}\wedge (b\vee k)]\wedge [x^{**}\wedge (x\vee k)]\wedge k\\ {}&=b\wedge x\wedge k. \end{aligned}$$

Thus, $(a,b)\in \theta _{(x)^{\bigtriangleup }}$  if and only if  $a~\wedge ~ x~\wedge ~k=b~\wedge ~x~\wedge ~k$ and hence

$$\begin{aligned} \theta _{(x)^{\bigtriangleup }}=\left\{ (a, b) \in L \times L: a \wedge x \wedge k=b \wedge x\wedge k\right\} . \end{aligned}$$

From Theorem 5.3. $Coker~\theta _{(x)^{\bigtriangleup }}=(x)^{\bigtriangleup }$. $\square$

Definition 6.2

A k-$\{^*\}$-congruence $\theta$ is called a principal k-$\{^*\}$-congruence if $\theta$ is a principal $\{^*\}$-congruence on L.

Proposition 6.3

For any element x of a CRD-Stone algebra L, define $\theta {(x^{**}\wedge k,1)}$ by

$$\begin{aligned} \theta (x^{**}\wedge k,1){=}\{(a,b)\in L\times L: a \wedge x^{**}\wedge k{=}b\wedge x^{**}\wedge k\} \end{aligned}$$

Then, $\theta (x^{**}\wedge k,1)$ is a principal k-$\{^*\}$-congruence on L and $Coker~ \theta (x^{**}\wedge k,1)=(x)^{\bigtriangleup }$.

Proof

It is clear that $\theta (x^{**}\wedge k,1)$ is a lattice congruence on L.

Let $(a,b)\in \theta (x^{**}\wedge k,1)$. Then,

$$\begin{aligned}&a\wedge x^{**}\wedge k=b\wedge x^{**}\wedge k \\ {}&\Rightarrow a^*\vee x^*\vee k^*=b^*\vee x^*\vee k^*\\ {}&\Rightarrow a^*\vee x^*=b^*\vee x^* ~as~k^*=0\\ {}&\Rightarrow (a^*\vee x^*)\wedge (x^{**}\wedge k)=(b^*\vee x^*)\wedge (x^{**}\wedge k)\\ {}&\Rightarrow (a^*\wedge x^{**}\wedge k)\vee (x^*\wedge x^{**}\wedge k)\\&=(b^*\wedge x^{**}\wedge k)\vee (x^*\wedge x^{**}\wedge k)\\ {}&\Rightarrow a^*\wedge x^{**}\wedge k=b^*\wedge x^{**}\wedge k~ as~ x^*\wedge x^{**}=0. \end{aligned}$$

Then, $(a^*,b^*)\in \theta (x^{**}\wedge k,1)$. Thus, $\theta (x^{**}\wedge k,1)$ a principal $\{^*\}$-congruence on L.

Since $1\wedge x^{**}\wedge k=k\wedge x^{**}\wedge k$, then $(1,k)\in \theta (x^{**}\wedge k,1)$. Then, $k\in [1]\theta (x^{**}\wedge k,1)$ and hence $\theta$ is a principal k-$\{^*\}$-congruence on L.

Now, for every $x\in L$, we prove $Coker~\theta (x^{**}\wedge k,1)=[x^{**}\wedge k)$.

$$\begin{aligned} \begin{aligned}&Coker~\theta (x^{**}\wedge k,1) \\&=\left\{ y \in L:(1, y) \in \theta {(x^{**}\wedge k,1)}\right\} \\&=\left\{ y \in L: 1\wedge x^{**}\wedge k=y \wedge x^{**}\wedge k\right\} \\&=\left\{ y \in L: y\ge x ^{**}\wedge k\right\} \\&=[x^{**}\wedge k)\\&=(x)^{\bigtriangleup }. \end{aligned} \end{aligned}$$

$\square$

Theorem 6.4

Let x be an element of a CRD-Stone algebra L. Then, $\theta _{(x)^{\bigtriangleup }} ~and ~\theta (x^{**}\wedge k,1)$ are identical, that is, $\theta (x^{**}\wedge k,1)=\theta _{(x)^{\bigtriangleup }}$.

Proof

Let $(a,b)\in \theta {(x^{**}\wedge k,1)}$. Then,

$$\begin{aligned} \begin{aligned}&a\wedge x^{**}\wedge k=b\wedge x^{**}\wedge k \\ {}&\Rightarrow a\wedge x^{**}\wedge x\wedge k=b\wedge x^{**}\wedge x\wedge k\\&\Rightarrow a\wedge x\wedge k=b\wedge x\wedge k\\&\Rightarrow (a,b)\in \theta _{(x)^{\bigtriangleup }}. \end{aligned} \end{aligned}$$

Thus, $\theta {( x^{**}\wedge k,1)}\subseteq \theta _{(x)^{\bigtriangleup }}$. Conversely, let $(a,b)\in \theta _{(x)^{\bigtriangleup }}$. Then, we get

$$\begin{aligned}&a\wedge x\wedge k=b\wedge x\wedge k \Rightarrow a\wedge (x^{**}\wedge (x\vee k))\wedge x\wedge k\\&=b\wedge (x^{**})\wedge (x\vee k)\wedge x\wedge k ~~~~by~ Lemma ~2.18(3)\\&\Rightarrow a\wedge x^{**}\wedge ( (x\vee k)\wedge k)=b\wedge x^{**}\wedge ( (x\vee k)\wedge k) \\&\Rightarrow a\wedge x^{**}\wedge k =b\wedge x^{**}\wedge k\\&\Rightarrow (a,b)\in \theta {( x^{**}\wedge k,1)}. \end{aligned}$$

Then, $\theta _{(x)^{\bigtriangleup }}\subseteq \theta {( x^{**}\wedge k,1)}$ and hence $\theta _{(x)^{\bigtriangleup }}=\theta {( x^{**}\wedge k,1)}$. $\square$

Corollary 6.5

Let L be a CRD-Stone algebra. Then, Coker $\theta _{(x)^{\bigtriangleup }}=Coker\theta {(x^{**}\wedge k,1)}=(x)^{\bigtriangleup }$.

Lemma 6.6

Let $\theta (a,1)$ is a principle $\{^*\}$-congruence of L. Then $\theta (a,1)$ is principle k-$\{^*\}$-congruence if and only if $k\ge a$.

Proof

If $\theta (a,1)$ is a principle k-$\{^*\}$-congruence, then $k\in Coker\theta (a,1)$ implies $(k,1)\in \theta (a,1)$ and hence $k\wedge a=1\wedge a=a$. Thus, $k\ge a$. For the converse, let $k\ge a$ and $\theta (a,1)$ is a principal $\{^*\}$-congruence. Then, $(k,1)\in \theta (a,1)$. Since $k\in Coker~\theta (a,1)$, thus $\theta (a,1)$ is a principle k-$\{^*\}$-congruence. $\square$

Lemma 6.7

Let $\theta (a,1)$ be principle k-$\{^*\}$-congruence on L. Then, $\theta (a,1)= \theta _{(a)^{\bigtriangleup }}$ if and only if $k\ge a$.

Proof

Let $\theta (a,1)$ be a k-$\{^*\}$-congruence on L and $\theta _{(a,1)}=\theta _{(a)^{\bigtriangleup }}$

$$\begin{aligned}&\theta (a,1)=\theta _{(a)^{\bigtriangleup }} \\ {}&\Rightarrow k\in Coker\theta (a,1) \\&=Coker\theta _{(a)^{\bigtriangleup }}\\&\Rightarrow (k,1)\in \theta (a,1)\\&\Rightarrow k\wedge a= 1\wedge a=a\\&\Rightarrow k\ge a. \end{aligned}$$

For the converse, let $k\ge a$ and $(x,y)\in \theta (a,1)$.

$$\begin{aligned}&(x,y)\in \theta (a,1) \\ {}&\Rightarrow x\wedge a=y\wedge a\\&\Rightarrow x\wedge a\wedge k=y\wedge a\wedge k\\&\Rightarrow (x,y)\in \theta _{(a)^{\bigtriangleup }}. \end{aligned}$$

Then, $\theta (a,1)\subseteq \theta _{(a)^{\bigtriangleup }}$. Let $(x,y)\in \theta _{(a)^{\bigtriangleup }}$. Then we have

$$\begin{aligned}&(x,y)\in \theta _{(a)^{\bigtriangleup }}\\ {}&\Rightarrow x\wedge a\wedge k=y\wedge a\wedge k\\&\Rightarrow x\wedge a= y\wedge a\\&\Rightarrow (x,y)\in \theta (a,1). \end{aligned}$$

Then, $\theta _{(a)^{\bigtriangleup }}\subseteq \theta (a,1)$ and hence $\theta _{(a)^{\bigtriangleup }}=\theta (a,1)$. $\square$

Corollary 6.8

Every principle k-$\{^*\}$-congruence $\theta (a,1)$ on CRD-Stone algebra L can be expressed as $\theta (a^{**}\wedge k,1)$.

Theorem 6.9

Let L be a CRD-Stone algebra. Then, for every $x, y \in L$, we have

(1) $\theta _{(x)^{\bigtriangleup }}\cap \theta _{(y)^{\bigtriangleup }}=\theta _{(x\vee y)^{\bigtriangleup }}$,

(2) $\theta _{(x)^{\bigtriangleup }}\vee \theta _{(y)^{\bigtriangleup }}=\theta _{(x\wedge y)^{\bigtriangleup }}$,

(3) $\theta _{(x)^{\bigtriangleup }} \cap \theta _{(x^*)^{\bigtriangleup }}=\nabla _L$,

(4) $\theta _{(x)^{\bigtriangleup }} \vee \theta _{(x^*)^{\bigtriangleup }}=\Psi ^*$.

Proof

• (1) Since $x\vee y\subseteq x, y$, then $(x \vee y)^{\bigtriangleup }\subseteq (x)^{\bigtriangleup }, (y)^{\bigtriangleup }$ and by Theorem 5.4. $\theta _{(x\vee y)^{\bigtriangleup }} \subseteq \theta _{(x)^{\bigtriangleup }},\theta _{(y)^{\bigtriangleup }}$. Thus $\theta _{(x\vee y)^{\bigtriangleup }}\subseteq \theta _{(x)^{\bigtriangleup }}\cap \theta _{(y)^{\bigtriangleup }}$. Conversely, let $(a,b)\in \theta _{(x)^{\bigtriangleup }}\cap \theta _{(y)^{\bigtriangleup }}$. Then, we have $(a,b)\in \theta _{(x)^{\bigtriangleup }}~and~(a,b)\in \theta _{(y)^{\bigtriangleup }}$

  $$\begin{aligned} \begin{aligned}&\Rightarrow a\wedge x\wedge k=b\wedge x\wedge k ~and~ a\wedge y\wedge k=b\wedge y\wedge k \\&\Rightarrow (a\wedge x\wedge k)\vee (a\wedge y\wedge k)\\&\quad =(b\wedge x\wedge k)\vee (a\wedge y\wedge k)\\&\Rightarrow a\wedge (x\vee y)\wedge k=b\wedge (x\vee y)\wedge k \\&\Rightarrow (a,b)\in \theta _{(x\vee y)^{\bigtriangleup }}. \end{aligned} \end{aligned}$$

  Then, $\theta _{(x)^{\bigtriangleup }}\cap \theta _{(y)^{\bigtriangleup }}\subseteq \theta _{(x\vee y)^{\bigtriangleup }}$.

• (2) Since $x\wedge y \subseteq x,y$, then by Theorem 5.4. $\theta _{(x\wedge y)^{\bigtriangleup }}\subseteq \theta _{(x)^{\bigtriangleup }}, ~\theta _{(y)^{\bigtriangleup }}$. Thus, $\theta _{(x\wedge y)^{\bigtriangleup }}$ is lower bound of $\theta _{(x)^{\bigtriangleup }}$, $\theta _{(y)^{\bigtriangleup }}$. Let $\theta _{(z)^{\bigtriangleup }}$ be lower bound of $\theta _{(x)^{\bigtriangleup }}$ and $\theta _{(y)^{\bigtriangleup }}$. Then $\theta _{(z)^{\bigtriangleup }}$ $\subseteq$ $\theta _{(x)^{\bigtriangleup }}$, $\theta _{(y)^{\bigtriangleup }}$. Hence $(z)^{\bigtriangleup }\subseteq (x)^{\bigtriangleup },~(y)^{\bigtriangleup }$ implies $(z)^{\bigtriangleup }\subseteq (x)^{\bigtriangleup }\vee (y)^{\bigtriangleup }=(x\wedge y)^{\bigtriangleup }$ is the greatest lower bound of $(x)^{\bigtriangleup },~(y)^{\bigtriangleup }$. By Theorem 5.4. $\theta _{(z)^{\bigtriangleup }}$ $\subseteq$ $\theta _{(x)^{\bigtriangleup }\vee (y)^{\bigtriangleup }}=\theta _{(x\wedge y) ^{\bigtriangleup }}$. Therefore, $\theta _{(x\wedge y)^{\bigtriangleup }}$ is the greatest lower bound of $\theta _{(x)^{\bigtriangleup }}$, $\theta _{(x)^{\bigtriangleup }}$. This proves that $\theta _{(x)^{\bigtriangleup }}\vee \theta _{(y)^{\bigtriangleup }}=\theta _ {(x\cap y)^{\bigtriangleup }}$.

• (3) $\theta _{(x)^{\bigtriangleup }} \cap \theta _{(x^*)^{\bigtriangleup }}=\theta _{(x\vee x^*)^{\bigtriangleup }}=\theta _{(1)^{\bigtriangleup }}=\theta _{D(L)}=\nabla _L$.

• (4) $\theta _{(x)^{\bigtriangleup }} \vee \theta _{(x^*)^{\bigtriangleup }}=\theta _{(x\wedge x^*)^{\bigtriangleup }}=\theta _{(0)^{\bigtriangleup }}=\theta _{L}=\Psi ^*$.

$\square$

Let ${\text {Con}}_k^{p}(L)=\left\{ \theta _{(x)^{\bigtriangleup }}: x \in L\right\}$ be the class of all principal k-$\{^*\}$-congerences induced by principal k-filters of L. Using Theorem 6.9, we deduce that the set ${\text {Con}}_k^{p}(L)$ forms a Boolean algebra which is isomorphic to the Boolean algebra $F^{p}_{k}(L)$.

Theorem 6.10

Let L be a CRD-Stone algebra. Then, $\left( {\text {Con}}_k^{p}(L); \vee , \cap ,{ }^{\prime }, \theta _{(1)^{\bigtriangleup }}, \theta _{(0)^{\bigtriangleup }}\right)$ is a Boolean algebra, whenever

$$\begin{aligned} \begin{aligned}&\theta _{(x)^{\bigtriangleup }} \vee \theta _{(y)^{\bigtriangleup }} =\theta _{(x \wedge y)^{\bigtriangleup }}, \\&\theta _{(x)^{\bigtriangleup }} \cap \theta _{(y)^{\bigtriangleup }} =\theta _{(x \vee y)^{\bigtriangleup }}, \\&\theta _{(x)^{\bigtriangleup }}^{\prime } =\theta _{\left( x^*\right) ^{\bigtriangleup }.} \end{aligned} \end{aligned}$$

Combining the above Theorem 6.10 and Corollary 4.10, we will investigate the following important result.

Corollary 6.11

Let L be a CRD-Stone algebra. Then

$\left( {\text {Con}}_k^{p}(L); \oplus , \odot , \theta _{(1)^{\bigtriangleup }}, \theta _{(0)^{\bigtriangleup }}\right)$ forms a Boolean ring, where

$$\begin{aligned} \begin{aligned}&\theta _{(x)^{\bigtriangleup }} \oplus \theta _{(y)^{\bigtriangleup }}=\theta _{((x\vee y^{*})\wedge (y\vee x^{*}))^{\bigtriangleup }}, \\&\theta _{(x)^{\bigtriangleup }} \odot \theta _{(y)^{\bigtriangleup }}=\theta _{(x\wedge y)^{\bigtriangleup }.} \end{aligned} \end{aligned}$$

7 Conclusion

In this article, the notion of k-filters of a core regular double Stone algebra is introduced and characterized. Many properties of principal k-filter of a CRD-Stone algebra are considered. It is observed that the class $F_{k}(L)$ of all k-filters of a CRD-Stone algebra L forms a bounded distributive lattice and the class $F^{p}_{k}(L)$ of all principal k-filters of L is a bounded sublattice of $F_{k}(L)$ and a Boolean algebra on its own. It is obtained the relationship between k-filters (principal k-filter) of a CRD-Stone algebra L and the k-$\{^*\}$-congruences (principal k-$\{^*\}$-congruences). This study leads us to consider many ideas in the future work, we would like to discuss the relationship between the prime k-filters and the maximal k-filters of a CRD-Stone algebras. Also, we shall study the effect of the core element k of a CRD-Stone algebra L in studying the atoms and coatoms of L and B(L).