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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 Λ(MS) of all subclasses of the class MS of all MS-algebras by identities.
A. Badawy, D. Guffovˊa 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 dL-filters of a principal MS-algebra and characterized certain congruences in terms of dL-filters. Recently, A. Badawy, M. Haviar and M. Ploˇsˇcica[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∘∘,D(L),φL) assosiated with a certain principal MS-algebra L. In Section 3, we give equivalent conditions for a pair of congruence (θ1,θ2)∈Con(L∘∘)×Conlat(D(L)) to become an MS-congruence pair of a principal MS-algebra L, where θ1 is a congruence on a de Morgan algebra L∘∘ and θ2 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 Ψ which defined on a principal MS-algebra L by
(a,b)∈Ψ⇔a∘=b∘⇔a∘∘=b∘∘. |
We prove that Ψ permutes with each congruence relation θ 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.
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;∨,∧,−,0,1) of type (2, 2, 1, 0, 0), where (L;∨,∧,0,1) is a bounded distributive lattice and the unary operation − satisfying:
(1)¯¯a=a,(2)¯(a∨b)=¯a∧¯b,(3)¯1=0. |
Definition 2.2. [15] A Stone algebra is an algebra (L;∨,∧,∗,0,1) of type (2,2,1,0,0), where (L;∨,∧,0,1) is a bounded distributive lattice and the unary operation ∗ satisfying:
(1)(a∧b)∗=a∗∨b∗,(2)a∗∨a∗∗=1,(3)1∗=0. |
Definition 2.3. [15] An MS-algebra is an algebra (L;∨,∧,∘,0,1) of type (2, 2, 1, 0, 0), where (L;∨,∧,0,1) is a bounded distributive lattice and a unary operation ∘ satisfying:
(1)a≤a∘∘,(2)(x∧y)∘=x∘∨y∘,(3)1∘=0. |
The class MS of all MS-algebras is equational. A de Morgan algebra is an MS-algebra satisfying the identity, a=a∘∘. The class S of a Stone algebra is a subclass of MS satisfying a∧a∘=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∘∘=0 and 1∘∘=1,
(2) a≤b⇒b∘≤a∘,
(3) a∘∘∘=a∘,
(4) a∘∘∘∘=a∘∘,
(5) (a∨b)∘=a∘∧b∘,
(6) (a∨b)∘∘=a∘∘∨b∘∘,
(7) (a∧b)∘∘=a∘∘∧b∘∘.
An element a of an MS-algebra L is called a closed element of L if a=a∘∘ and an element d∈L is called a dense element of L if d∘=0.
Theorem 2.2. Let L be an MS-algebra. Then
(1) the set L∘∘={a∈L:a=a∘∘} of all closed elements of L is a de Morgan subalgebra of L, see [17],
(2) the set D(L)={a∈L:a∘=0} of all dense elements of L is a principal filter of L, see [10].
Definition 2.4. [10] An MS-algebra (L;∨,∧,∘,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 dL∈L such that D(L)=[dL),
(2) a=a∘∘∧(a∨dL), for all a∈L.
A principal MS-algebra L is a principal Stone algebra if x∘∨x∘∘=1, for all x∈L.
Definition 2.5. [8] A principal MS-triple is (M,D,φ), where
(1) M is a de Morgan algebra,
(2) D is a bounded distributive lattice,
(3) φ is a (0,1)-lattice homomorphism from M into D.
Lemma 2.1. [10] Let L be a principal MS-algebra. Define a map φL:L∘∘→[dL) by the following rule
φL(a)=a∨dL,for all a∈L∘∘. |
Then φL 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∘∘,D(L)=[z),φL) which associated with L is given in Figure 2.
Definition 2.6. [22] An equivalence relation θ on a lattice L is called a lattice congruence if (a,b)∈θ and (c,d)∈θ imply (a∨c,b∨d)∈θ and (a∧c,b∧d)∈θ.
Definition 2.7. An equivalence relation θ on an MS-algebra L is called a congruence on L if
(1) θ is a lattice congruence,
(2) (a,b)∈θ implies (a∘,b∘)∈θ.
We use Conlat(L) for the lattice of all lattice congruences of a lattice (L;∨,∧) and Con(L) for the lattice of all congruences of an MS-algebra (L;∨,∧,∘,0,1).
Definition 2.8. Let L be a lattice and θ∈Conlat(L). Then we define the principal congruence generated by (a,b) which denoted by θ(a,b) as follows:
θ(a,b)=⋀{θ∈Conlat(L):(a,b)∈θ}. |
If L is an MS-algebra and θ∈Con(L), then
θ(a,b)=⋀{θ∈Con(L):(a,b)∈θ}. |
Let L be an MS-algebra. Then (θL∘∘,θD(L))∈Con(L∘∘)×Conlat(D(L)), where θL∘∘ and θD(L) are the restrictions of θ∈Con(L) to L∘∘ and D(L), respectively. It is clear that θL∘∘ is a congruence relation on a de Morgan algebra L∘∘ and θD(L) is a lattice congruence on a lattice D(L).
The symbols ▽L and △L will be used for the universal congruence L×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 dL. A pair of congruences (θ1,θ2)∈Con(L∘∘)×Conlat(D(L)) will be called an MS-congruence pairs if
(a,b)∈θ1 implies (a∨dL,b∨dL)∈θ2. |
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 dL. For every congruence θ on L, the restrictions of θ to L∘∘ and D(L) determine the MS-congruence pair (θL∘∘,θD(L)).
Conversely, every MS-congruence pair (θ1,θ2) uniquely determines a congruence θ on L satisfying θL∘∘=θ1 and θD(L)=θ2. Such congruence can be defined by the rule
(x,y)∈θ if and only if (x∘,y∘)∈θ1 and (x∨dL,y∨dL)∈θ2. |
Throughout this paper, dL 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].
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 (θ1,θ2) be an MS-congruence pair. Then
(a,b)∈θ1 and (x,y)∈θ2 imply (a∨x,b∨y)∈θ2. |
Proof. Let (a,b)∈θ1 and (x,y)∈θ2. Then (a∨dL,b∨dL)∈θ2 (by Definition 2.9) and (x,y)∈θ2 imply that (a∨x∨dL,b∨y∨dL)∈θ2. Therefore (a∨x,b∨y)∈θ2 as x,y≥dL.
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 dL. Then the following statements are equivalent:
(1) (θ1,θ2) is an MS-congruence pair of L,
(2) ConφL(L)⊆θ2, where
ConφL(θ1)={(φL(a),φL(b)):(a,b)∈θ1} and φL(a)=a∨dL,∀a∈L∘∘. |
Proof. (1⇒2) Let (θ1,θ2) be an MS-congruence pair of L and (x,y)∈ConφL(θ1). Then (x,y)=(φL(a),φL(b))=(a∨dL,b∨dL), where (a,b)∈θ1. Since (a,b)∈θ1, then (a∨dL,b∨dL)∈θ2, by (1). Thus (x,y)∈θ2. So ConφL(θ1)⊆θ2.
(2⇒1) Let Con(θ1)⊆θ2. We prove that (θ1,θ2) is an MS-congruence pair of L. Let (a,b)∈θ1. Then by (2), (a∨dL,b∨dL)=(φL(a),φL(b))∈θ2.
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 (θ1,θ2)∈Con(L∘∘)×Conlat(D(L)). Then the following statements are equivalent:
(1) (θ1,θ2) is an MS-congruence pair of L,
(2) ConφL(θ1)⊆θ2,
(3) (a,1)∈θ1 and u≥a,u∈D(L) imply (u,1)∈θ2.
Proof. In Theorem 3.1, we proved that (1) and (2) are equivalent.
(2⇒3) Let ConφL(θ1)⊆θ2. Let (a,1)∈θ1 and u≥a,u∈D(L). Then (φL(a),φL(1))∈θ2 by (2). Thus (a∨dL,1)∈θ2.
Now (a∨dL,1)∈θ2 and (u,u)∈θ2,u≥a,u∈D(L) imply (a∨u∨dL,1∨u)∈θ2. Thus (u,1)∈θ2 as u≥a,dL.
(3⇒1) Since L is a Stone algebra then L∘∘ is a Boolean subalgebra of L and hence a∨a∘=1 for each a∈L∘∘. Suppose (3) holds and (a,b)∈θ1. Then (b∘,b∘)∈θ1 implies (a∨b∘,b∨b∘)=(a∨b∘,1)∈θ1 and (a∨a∘,b∨a∘)=(b∨a∘,1)∈θ1. Therefore (β,1)∈θ1, where β=(a∨b∘)∧(a∘∨b). It is clear that β∈L∘∘ and β∧a=β∧b=a∧b. Since β≤β∨dL∈D(L) and (β,1)∈θ1 then (β∨dL,1)∈θ2 by (3). Thus (a∨dL,a∨dL)∧(1,β∨dL)∈θ2 implies (a∨dL,(a∧β)∨dL)=(a∨dL,(a∧b)∨dL)∈θ2. Similarly, we can get (b∨dL,(b∧β)∨dL)=(b∨dL,(a∧b)∨dL)∈θ2. Then (b∨dL,b∨dL)∈θ2.
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,
△L={(x,x):x∈L},α={{0,c,a},{x,y,z},{b,d,1}},β={{0,c,a},{x,b,y,d,z,1}},γ={{0,x,b},{c,y,d},{a,z,1}},δ={{0},{c},{a},{x,b},{y,d},{z,1}},▽L=L×L. |
And
A(L)={(△L∘∘,△D(L)),(αL∘∘,αD(L)),(βL∘∘,βD(L)),(γL∘∘,γD(L)),(δL∘∘,δD(L)),(▽L∘∘,▽D(L))}={(△L∘∘,△D(L)),({{0,c,a},{b,d,1}},△D(L)),({{0,c,a},{b,d,1}},▽D(L)),({{a,1},{c,d},{0,b}},▽D(L)),(△L∘∘,▽D(L)),(▽L∘∘,▽D(L))}. |
It is clear that Con(L) isomorphic to A(L) under the isomorphism θ⟼(θL∘∘,θD(L)).
A subset I of a lattice L with 0 is called an ideal of L if
(1)0∈I,(2)a∨b∈I,∀a,b∈I,(3)I is an down-set, that is, ifx≤y,y∈Iandx∈L,thenx∈I. |
A principal ideal of L generated by a∈L is defined by
(a]={x∈L:x≤a}. |
A subset F of a lattice L with 1 is called a filter of L if
(1)1∈F,(2)a∧b∈F,∀a,b∈F,(3)F is an up-set, that is, ifx≥y,y∈Fandx∈L,thenx∈F. |
A principal filter of L generated by a∈L is defined by
[a)={x∈L:x≥a}. |
Definition 3.1. Let θ be a lattice congruence on a bounded lattice L. Then we have the following important subsets
(i) The Kernel of θ (Kerθ) is the set {x∈L:(x,0)∈θ}, which is an ideal of L,
(ii) The Cokernel of θ (Cokerθ) is the set {x∈L:(x,1)∈θ}, 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∨c∘=1.
It is ready seen that the set B(L)={c:c∨c∘=1} of all Boolean elements of L forms a Boolean subalgebra of L∘∘.
Lemma 3.2. Let c be a Boolean element of a principal MS-algebra L. Then
(1) θ(0,c) is a principal congruence relation on L∘∘ with Kerθ(0,c)=(c], where
(a,b)∈θ(0,c)⇔a∧c∘=b∧c∘, |
(2) θ(φL(c),1) is a principal congruence relation on D(L) with Cokerθ(φL(c),1)=[φL(c)), where
(x,y)∈θ(φL(c),1)⇔x∧(c∨dL)=y∧(c∨dL). |
Proof.
(1) It is easy to check that θ(0,c) is a principal congruence relation on L∘∘ with Kerθ(0,c)=(c].
(2) It is easy to show that θ(φL(c),1) is an equivalence relation on D(L). Let (x,y),(m,n)∈θ(φL(c),1). Thus x∧(c∨dL)=y∧(c∨dL) and m∧(c∨dL)=n∧(c∨dL). Then, we get (x∧m,y∧n)∈θ(φL(c),1) and (x∨m,y∨n)∈θ(φL(c),1). Therefore θ(φL(c),1) is a principal congruence on the lattice D(L). Also, we have
Coker(θ(φL(c),1))={x∈D(L):(x,1)∈θ(φL(c),1)}={x∈D(L):x∧(c∨dL)=1∧(c∨dL)}={x∈D(L):x∧(c∨dL)=c∨dL}={x∈D(L):x≥c∨dL}=[c∨dL)=[φL(c)). |
Now, we observe that every Boolean element c of a principal MS-algebra L associated with the MS-congruence pair of the form (θ(0,c),θ(φL(c∘),1)).
Theorem 3.3. Let L be a principal MS-algebra and c∈L∘∘. Then c is a Boolean element of L if and only if (θ(0,c),θ(φL(c∘),1)) is an MS-congruence pair of L.
Proof. Let c be a Boolean element of L. We proved that θ(0,c) and θ(φL(c∘),1) are MS-congruence on L∘∘ and lattice congruence on D(L), respectively (Lemma 3.2). To show that (θ(0,c),θ(φL(c∘),1)) is an MS-congruence pair, let (x,y)∈θ(0,c). Then
(x,y)∈θ(0,c)⇒x∧c∘=y∧c∘⇒(x∧c∘)∨dL=(y∧c∘)∨dL⇒(x∨dL)∧(c∘∨dL)=(y∨dL)∧(c∘∨dL)⇒(x∨dL,y∨dL)∈θ(φL(c∘),1). |
Conversely, let (θ(0,c),θ(φL(c∘),1)) be an MS-congruence pair. Since (0,c)∈θ(0,c) then c∧c∘ = 0∧c∘=0. Now, c∨c∘=(c∘∧c)∘=1. Therefore c is a Boolean element.
The basic properties of principal congruence relations θ(0,a) and θ(φL(a),1),∀a∈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≤b if and only if θ(0,a)⊆θ(0,b),
(2) a=b if and only if θ(0,a)=θ(0,b),
(3) θ(0,0)=△L∘∘ and θ(0,1)=▽L∘∘,
(4) θ(0,a)∨θ(0,b)=θ(0,a∨b),
(5) θ(0,a)∩θ(0,b)=θ(0,a∧b).
Lemma 3.4. Let L be a principal MS-algebra. Then for every a,b∈B(L), we have
(1) a≤b implies θ(φL(a∘),1)⊆θ(φL(b∘),1),
(2) θ(φL(a∘),1)∨θ(φL(b∘),1)=θ(φL(a∨b)∘,1),
(3) θ(φL(a∘),1)∧θ(φL(b∘),1)=θ(φL(a∧b)∘,1),
(4) θ(φL(0),1)=▽D(L) and θ(φL(1),1)=△D(L).
Proof.
(1) Let a≤b. Then a∘≥b∘. Let (x,y)∈θ(φL(a∘),1) then x∧(a∘∨dL)=y∧(a∘∨dL). Now
x∧(b∘∨dL)=x∧((a∘∧b∘)∨dL)=x∧{(a∘∨dL)∧(b∘∨dL)}=x∧(a∘∨dL)∧(b∘∨dL)=y∧(a∘∨dL)∧(b∘∨dL)=y∧{(a∘∧b∘)∨dL}=y∧(b∘∨dL). |
Then (x,y)∈θ(φL(b∘),1). Therefore θ(φL(a∘),1)⊆θ(φL(b∘),1).
(2) Since a,b≤a∨b then by (1), we have
θ(φL(a∘),1),θ(φL(b∘),1)⊆θ(φL(a∨b)∘,1). |
Then θ(φL(a∨b)∘,1) is an upper bound of θ(φL(a∘),1) and θ(φL(b∘),1). Let θ(φL(c∘),1) be an upper bound of θ(φL(a∘),1) and θ(φL(b∘),1).
Then θ(φL(a∘),1),θ(φL(b∘),1)⊆θ(φL(c∘),1) implies φL(c∘)≤φL(a∘), φL(b∘). Thus φL(c∘)≤φL(a∘∧b∘)=φL(a∨b)∘ and hence θ(φL(a∨b)∘,1)⊆θ(φL(c∘),1). Therefore θ(φL(a∨b)∘,1) is the least upper bound of θ(φL(a∘),1) and θ(φL(b∘),1).
(3) Since a∧b≤a,b then by (1) θ(φL(a∧b)∘,1)⊆θ(φL(a∘),1),θ(φL(b∘),1). Then θ(φL(a∧b)∘,1) is a lower bound of θ(φL(a∘),1) and θ(φL(b∘),1). Let θ(φL(c∘),1) be a lower bound of θ(φL(a∘),1) and θ(φL(b∘),1). Then θ(φL(c∘),1)⊆θ(φL(a∘),1),θ(φL(b∘),1) implies φL(a∘),φL(b∘)≤φL(c∘). Thus φL(a∧b)∘=φL(a∘)∨φL(b∘)≤φL(c∘) and hence θ(φL(c∘),1)⊆θ(φ(a∧b)∘,1). Therefore θ(φL(a∧b)∘,1) is the greatest lower bound of θ(φL(a∘),1) and θ(φL(b∘),1).
(4) Since dL,1∈θ(dL,1), then θ(dL,1)=D(L)×D(L)=▽D(L). Also, let (x,y)∈θ(φL(1),1) then x∧(1∨dL)=y∧(1∨dL). It follows, x=y and hence θ(φL(1),1)=△D(L).
Theorem 3.4. Let L be a principal MS-algebra. Consider the subsets H and G of Con(L∘∘) and Con(D(L)), respectively as follows:
H={θ(0,a):a∈B(L)},G={θ(φL(a),1):a∈B(L)}. |
Then we have
(1) (H;∨,∧,′,△L∘∘,▽L∘∘) is a Boolean algebra, where (θ(0,a))′=θ(a∘,0),
(2) (G;∨,∧,′,△D(L),▽DL) is a Boolean algebra, where (θ(φL(a),1))′=θ(φL(a∘),1).
Proof.
(1) Since θ(0,0)=△L∘∘ and θ(0,1)=▽L∘∘. Then ▽L∘∘,△L∘∘∈H. Let θ(0,a),θ(0,b)∈H. Then we get
θ(0,a)∨θ(0,b)=θ(0,a∨b), |
and
θ(0,a)∧θ(0,b)=θ(0,a∧b). |
Therefore H is a bounded lattice. Since B(L) is a Boolean algebra, then a∘∈B(L) for all a∈B(L). Thus θ(0,a∘)∈H. Then we have
θ(0,a)∨[θ(0,a)]′=θ(0,a)∨θ(0,a∘)=θ(0,a∨a∘)=θ(0,1)=▽L∘∘, |
and
θ(0,a)∧[(0,a)]′=θ(0,a)∧θ(0,a∘)=θ(φL(0,a∧a∘)=θ(0,0)=△L∘∘. |
Therefore (H;∨,∧,′,△L∘∘,▽L∘∘) is a Boolean algebra.
(2) We have △D(L)=θ(φL(1),1)∈G and ▽D(L)=θ(φL(0),1)=▽D(L)∈G. Let θ(φL(a),1),θ(φL(b),1)∈G. Then we get
θ(φL(a),1)∨θ(φL(b),1)=θ(φL(a∧b),1), |
and
θ(φL(a),1)∧θ(φL(b),1)=θ(φL(a∨b),1). |
Therefore G is a bounded lattice. Since B(L) is a Boolean algebra, then a∘∈B(L) for all a∈B(L). Thus θ(φL(a∘),1)∈G. Then we have
θ(φL(a),1)∨[θ(φL(a),1)]′=θ(φL(a),1)∨θ(φL(a∘),1)=θ(φL(a∧a∘),1)=θ(φL(0),1)=▽D(L), |
and
θ(φL(a),1)∧[θ(φL(a),1)]′=θ(φL(a),1)∧θ(φL(a∘),1)=θ(φL(a∨a∘),1)=θ(φL(1),1)=△D(L). |
Therefore (G;∨,∧,′,△DL,▽DL) 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′(L) of A(L) as follows:
A′(L)={(θ(0,a),θ(φL(a∘),1)):a∈B(L)}. |
From the above results, we observe that the set A′(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′(L) is a Boolean algebra on its own.
Theorem 3.5. Let L be a principal MS-algebra. Then (A′(L);∨,∧,′,0A′(L),1A′(L)) is a Boolean algebra, where
(θ(0,a),θ(φL(a∘),1))∨(θ(0,b),θ(φL(b∘),1))=(θ(0,a∨b),θ(φL(a∨b)∘,1)),(θ(0,a),θ(φL(a∘),1)∧(θ(0,b),θ(φL(b∘),1))=(θ(0,a∧b),θ(φL(a∧b)∘,1)),[(θ(0,a),θ(φL(a∘),1))]′=(θ(0,a∘),θ(φL(a),1)),1A′(L)=(▽L∘∘,△D(L)),0A′(L)=(△L∘∘,△D(L)). |
Example 3.2. Consider the principal MS-algebra L in Example 2.1. The lattices B(L) and A′(L) of all Boolean elements of L and all MS-congruence pairs of L of the form (θ(0,a),θ(φL(a∘),1)),a∈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⟼(θ(0,a),θ(a∘∨z,1)). Also, A′(L) is a bounded sublattice of A(L).
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 θ,Φ∈Con(L) are 2-permutable if θ∘Φ=Φ∘θ, that is, (x,a)∈θ and (a,y)∈Φ, x,y,a∈L imply (x,b)∈Φ and (b,y)∈θ for some b∈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 θ and Φ be arbitrary congruences on L and (a,b)∈(θ∘Φ),a,b∈L. Then (a,t)∈θ and (t,b)∈Φ for some t∈L. Then (a,t)∈θ(a,t) and (t,b)∈θ(t,b) imply that (a,b)∈θ(a,t)∘θ(t,b). Thus (a,b)∈θ(t,b)∘θ(a,t). Since θ(a,t)⊆θ and θ(t,b)⊆Φ, then (a,b)∈(Φ∘θ). Therefore θ and Φ permute. The second implication is obvious.
Define a relation Ψ on a principal MS-algebra L as follows:
(a,b)∈Ψ⇔a∘=b∘⇔a∘∘=b∘∘. |
Theorem 4.1. Let L be a principal MS-algebra with the smallest dense element dL. Then
(1) Ψ is a congruence relation on L with KerΨ={0} and CokerΨ=D(L),
(2) Ψ is closed congruence on L, that is, (x,x∘∘)∈Ψ,∀x∈L,
(3) max [x]Ψ=x∘∘,x∈L, where [x]Ψ={y∈L:y∘∘=x∘∘} is the congruence class of x modulo Ψ.
Proof.
(1) One can check that Ψ is a congruence relation on L. Now, we have
KerΨ={a∈L:(a,0)∈Ψ}={a∈L:a∘=1}={0}, |
and
CokerΨ={z∈L:(z,1)∈Ψ}={z∈L:z∘=1}=D(L)=[dL). |
(2) Since x∘∘∘∘=x∘∘ then x∘∘∈[x]Ψ. Thus (x∘∘,x)∈Ψ.
(3) Let y∈[x]Ψ. Then y≤y∘∘=x∘∘ and x∘∘∈[x]Ψ. Thus x∘∘ is the greatest member of the congruence class [x]Ψ.
Theorem 4.2. Let L be a principal MS-algebra. Then
(1) L/Ψ is a de Morgan algebra,
(2) L/Ψ and L∘∘ are isomorphic de Morgan algebras.
Proof.
(1) It is ready seen that (L/Ψ;∨,∧,{0},D(L)) is a bounded distributive lattice, where L/Ψ={[a]Ψ:a∈L} is the set of all congruence classes module Ψ and
[a]Ψ∨[b]Ψ=[a∨b]Ψ,[a]Ψ∧[b]Ψ=[a∧b]Ψ. |
Define ◻ on L/Ψ by ([a]Ψ)◻=[a∘]Ψ,∀a∈L. We have
([0]Ψ)◻=[1]Ψ,([a]Ψ)◻◻=[a∘∘]Ψ=[a]Ψ,([a]Ψ∧[b]Ψ)◻=([a]Ψ)◻∨([b]Ψ)◻. |
Then L/Ψ is a de Morgan algebra.
(2) Define g:L∘∘→L/Ψ by
g(a)=[a]Ψ,∀a∈L∘∘. |
Let a,b∈L∘∘ and a=b. Then a∘∘=b∘∘ implies [a]Ψ=[b]Ψ. It follows that g is well defined map of L∘∘ into L/Ψ. Let a,b∈L∘∘, we get
g(a∨b)=[a∨b]Ψ=[a]Ψ∨[b]Ψ=g(a)∨g(b), |
g(a∧b)=[a∧b]Ψ=[a]Ψ∧[b]Ψ=g(a)∧g(b), |
and
g(a∘)=[a∘]Ψ=([a]Ψ)◻=(g(a))◻. |
Let [a]Ψ=[b]Ψ. Then a∘∘=b∘∘ implies a=b. Therefore g is an injective map. Let [a]Ψ∈L/Ψ. Then [x]Ψ=[x∘∘]Ψ=g(x). Then g is a surjective map. This deduce that g is an isomorphism of de Morgan algebras.
Now, we observe that Ψ satisfies the following property,
Ψ∘θ=θ∘Ψ forall θ∈Con(L). |
Theorem 4.3. Let L be a principal MS-algebra. Then Ψ permutes with each congruence of L.
Proof. We prove that Ψ∘θ=θ∘Ψ for all θ∈Con(L). Let (a,b)∈Ψ∘θ. Then (a,z)∈Ψ and (z,b)∈θ for some z∈L. It follows that a∘∘=z∘∘ and (z,b)∈θ. Now
(z,b)∈θ⇒(z∘∘,b∘∘)∈θand(a∨dL,a∨dL)∈θ⇒(a∘∘∧(a∨dL),b∘∘∧(a∨dL))∈θ⇒(a,b∘∘∧(a∨dL))∈θ as a=a∘∘∧(a∨dL) |
Since [b∘∘∧(a∨dL)]∘∘=b∘∘, then (b∘∘∧(a∨dL),b)∈Ψ. Therefore (a,b∘∘∧(a∨dL))∈θ and (b∘∘∧(a∨dL),b)∈Ψ. imply (a,b)∈θ∘Ψ. Then Ψ∘θ=θ∘Ψ,∀θ∈Con(L).
Lemma 4.2. Let L be a principal MS-algebra. Then
(1) △L permutes with every congruence on L,
(2) ▽L permutes with every congruence on L.
Proof.
(1) Let (a,b)∈θ∘△L. Then (a,t)∈θ and (t,b)∈△L for some t∈L. Thus (a,t)∈θ and t=b. Then (a,a)∈△L and (a,b)∈θ imply (a,b)∈△L∘θ. Therefore θ∘△L=△L∘θ.
(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∘∘ and D(L) have 2-permutable congruences.
Proof. To prove the conditions (1) and (2) are equivalent, we show that the two congruences θ,Φ∈Con(L) are 2-permutable if and only if their restrictions θL∘∘,ΦL∘∘ and θD(L),ΦD(L) have 2-permutable congruences on L∘∘ and D(L), respectively. Firstly, suppose that θ,Φ have 2-permutable congruences on L. Let (x,z)∈θL∘∘∘ΦL∘∘. Then there exist y∈L∘∘ such that (x,y)∈θL∘∘ and (y,z)∈ΦL∘∘. Then we have (x,y)∈θ and (y,z)∈Φ. Since θ,Φ are 2-permutable, then there exists a∈L such that (x,a)∈Φ and (a,y)∈θ. We get (x,a∘∘)=(x∘∘,a∘∘)∈ΦL∘∘ and (a∘∘,z∘∘)∈θL∘∘. Therefore θL∘∘,ΦL∘∘ are 2-permutable. Also, to prove that θD(L),ΦD(L) are 2-permutable, let (x,z)∈θD(L)∘ΦD(L). Then there exist y∈D(L) such that (x,y)∈θD(L) and (y,z)∈ΦD(L). Then we have (x,y)∈θ and (y,z)∈Φ. Since θ,Φ are 2-permutable, there exists a∈L such that
(x,a)∈Φand(a,z)∈θ⇒(x,a∨dL)=(x∨dL,a∨dL)∈Φand(a∨dL,z∨dL)∈θ⇒(x,a∨dL)∈Φand(a∨dL,z)∈θ,⇒(x,a∨dL)∈ΦD(L)and(a∨dL,z)∈θD(L),asx,a∨dL,z∈D(L)⇒(x,z)∈ΦD(L)∘θD(L). |
Thus θD(L),ΦD(L) have 2-permutable congruences on D(L).
Conversely, let θ,Φ∈Con(L). Assume that θL∘∘,ΦL∘∘ and θD(L),ΦD(L), have 2-permutable congruences on L∘∘ and D(L), respectively. Let (x,z)∈θ∘Φ. Suppose that x,y,z∈L with (x,y)∈θ and (y,z)∈Φ. Then, by using Theorem 2.3, we get the following statements.
(x∘∘,y∘∘),(y∘∘,z∘∘)∈ΦL∘∘,(x∨dL,y∨dL),(y∨dL,z∨dL)ΦD(L). |
Since θL∘∘,ΦL∘∘ have 2-permutable congruences on L∘∘, there exists an element a∈L∘∘ such that
(x∘∘,a)∈ΦL∘∘ and (a,z∘∘)∈θL∘∘. |
Since θD(L),ΦD(L) have 2-permutable congruences on D(L), there exists e∈D(L) such that (x∨dL,e)∈ΦDL and (e,z∨dL)∈θDL). It follows that (x∘∘,a)∈Φ,(a,z∘∘)∈θ and (x∨dL,e)∈Φ,(e,z∨dL)∈θ. Since L is a principal MS-algebra we have
x=x∘∘∧(x∨dL) and z=z∘∘∧(z∨dL). |
Since θ,Φ are compatible with the ∧ operation, then (x∘∘,a)∈Φ and (x∨dL,e)∈Φ imply (x,a∧e)=(x∘∘∧(x∨dL),a∧e)∈Φ. Also (a,z∘∘)∈θ and (e,z∨dL)∈θ imply (a∧e,z)=(a∧e,z∘∘∧(z∨dL))∈θ. Consequently, we deduce that (x,a∧e)∈Φ and (a∧e,z)∈θ,a∧e∈L. Then (x,z)∈Φ∘θ, and hence θ,Φ 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∘∘andD(L) have 2-permutable congruences, respectively. Where αL∘∘=βL∘∘ and δL∘∘=△L∘∘.
∘ | △L | α | β | γ | δ | ▽L |
△L | △L | α | β | γ | δ | ▽L |
α | α | α | β | ▽L | β | ▽L |
β | β | β | β | ▽L | β | ▽L |
γ | γ | ▽L | ▽L | γ | ▽L | ▽L |
δ | δ | β | β | ▽L | δ | ▽L |
▽L | ▽L | ▽L | ▽L | ▽L | ▽L | ▽L |
∘ | △L∘∘ | αL∘∘ | γL∘∘ | ▽L∘∘ |
△L∘∘ | △L∘∘ | αL∘∘ | γL∘∘ | ▽L∘∘ |
αL∘∘ | αL∘∘ | αL∘∘ | ▽L∘∘ | ▽L∘∘ |
γL∘∘ | γL∘∘ | ▽L∘∘ | γL∘∘ | ▽L∘∘ |
▽L∘∘ | ▽L∘∘ | ▽L∘∘ | ▽L∘∘ | ▽L∘∘ |
∘ | △D(L) | ▽D(L) |
△D(L) | △D(L) | ▽D(L) |
▽D(L) | ▽D(L) | ▽D(L) |
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,z1,1} and L∘∘={0,c,a,b,d,1}. Consider θ,Φ∈Con(L) as follows:
θ=△L∪{{x1,b},{y1,d},{z1,1}},Φ=△L∪{{x,x1},{y,y1},{z,z1}}. |
We observe that θ∘Φ≠Φ∘θ as (z,1)∈Φ∘θ but (z,1)∉θ∘Φ.
Then L has not 2-permutable congruences and D(L) has not 2-permutable congruences as θD(L)∘ΦD(L)≠ΦD(L)∘θD(L).
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 θ,Φ of L are n-permutable, that is,
θ∘Φ∘θ∘...=Φ∘θ∘Φ∘...(n times). |
At first, we need the following two lemmas.
Lemma 5.1. Let L be a principal MS- algebra with the smallest dense element dL. Let θ,Φ be two congruences on L. Then we have
(i) (θ∘Φ∘θ∘...)L∘∘=θL∘∘∘ΦL∘∘∘...(n times),
(ii) (θ∘Φ∘θ∘...)D(L)=θD(L)∘ΦD(L)∘...(n times).
Proof.
(i) Recall that θL∘∘,ΦL∘∘ are the restrictions of θ,Φ on L∘∘, respectively. Let a,b∈L∘∘ and (a,b)∈(θL∘∘∘ΦL∘∘∘...). Then there exist c1,c2,...,cn−1∈L∘∘ such that
(a,c1)∈θL∘∘,(c1,c2)∈ΦL∘∘,...,(cn−1,b)∈Ψ,whereΨ={θL∘∘,ifnodd,ΦL∘∘,ifneven. |
Then
(a,c1)∈θ,(c1,c2)∈Φ,...,(cn−1,b)∈Ψ,whereΨ={θ,ifnodd,Φ,ifneven. |
Then (a,b)∈(θ∘Φ∘θ∘...) and hence (a,b)∈(θ∘Φ∘θ∘...)L∘∘. Therefore
θL∘∘∘ΦL∘∘∘θL∘∘...⊆(θ∘Φ∘θ∘...)L∘∘. |
Conversely, let a,b∈L∘∘ such that (a,b)∈(θ∘Φ∘θ∘...)L∘∘. Then (a,b)∈(θ∘Φ∘θ∘...). Then there exist c1,c2,...,cn−1∈L such that
(a,c1)∈θL∘∘,(c1,c2)∈ΦL∘∘,...,(cn−1,b)∈Ψ,whereΨ={θL∘∘,ifnodd,ΦL∘∘,ifneven. |
Then we get
(a,c1∘∘)∈θ,(c1∘∘,c2∘∘)∈Φ,...,(cn−1∘∘,b)∈Ψ,whereΨ={θ,ifnodd,Φ,ifneven. |
Since c1∘∘,c2∘∘,...,cn−1∘∘∈L∘∘, we have
(a,c1∘∘)∈θL∘∘,(c1∘∘,c2∘∘)∈ΦL∘∘,...,(cn−1∘∘,b)∈Ψ,whereΨ={θL∘∘,ifnodd,ΦL∘∘,ifneven. |
Therefore
(a,b)∈(θL∘∘∘ΦL∘∘∘θL∘∘∘...) |
and hence
(θ∘Φ∘θ∘...)L∘∘⊆θL∘∘∘ΦL∘∘∘θL∘∘∘...(ntimes). |
Then we get
(θ∘Φ∘θ∘...)L∘∘=θL∘∘∘ΦL∘∘∘θL∘∘∘...(ntimes). |
(ii) Take x,y∈D(L), let (x,y)∈(θD(L)∘ΦD(L)∘ΦD(L)∘θD(L)∘...). Then (x,y)∈(θ∘Φ∘θ∘...) and hence (a,b)∈(θ∘Φ∘θ∘...)D(L). Then (θ∘Φ∘θ∘...)⊆(θ∘Φ∘θ∘...)D(L).
Conversely, let (x,y)∈(θ∘Φ∘θ∘...)D(L), then (x,y)∈(θ∘Φ∘...). There exist d1,d2,...dn−1∈L, such that
(x,d1)∈θ,(d1,d2)∈Φ,...,(dn−1,y)∈ψ,whereψ={θ,ifnodd,Φ,ifneven. |
Since x,y≥dL, we get,
(x,d1)=(x∨dL,d1∨dL)∈θD(L),(d1∨dL,d2∨dL)∈ΦD(L),...,(dn−1,b∨dL1)=(dn−1,y)∈ψ,whereψ={θD(L),ifnodd,ΦD(L),ifneven. |
Hence (x,y)∈(θD(L)∘ΦD(L)∘θD(L)∘...). Then the required equality is proved.
Lemma 5.2. Let θ and Φ be congruences on a principal MS-algebra L. Let ψ denoted to the relation θ∘Φ∘θ∘...(n times). Then (a,b)∈ψ and (c,d)∈ψ imply (a∧c,b∧d)∈ψ.
Proof. Let (a,b)∈ψ and (c,d)∈ψ. Then there exist a1,a2,...,an−1,c1,c2,...,cn−1∈L such that
(a,a1)∈θ,(a1,a2)∈Φ,...,(an−1,b)∈Ψ,whereΨ={θ,ifnodd,Φ,ifneven. |
And
(c,c1)∈θ,(c1,c2)∈Φ,...,(cn−1,d)∈Ψ,whereΨ={θ,ifnodd,Φ,ifneven. |
Since θ,Φ are congruence on L then we get,
(a∧c,a1∧c1)∈θ,(a1∧c1,a2∧c2)∈Φ,...,(an−1∧cn−1,b∧d)∈Ψ,whereΨ={θ,ifnodd,Φ,ifneven. |
Thus (a∧c,b∧d)∈ψ.
Now, a characterization of n-permutability of principal MS-algebras is given.
Theorem 5.1. Two congruences θ,Φ on a principal MS-algebra L are n-permutable if and only if their restrictions θL∘∘,ΦL∘∘ and θD(L),ΦD(L) with respect to L∘∘ and D(L) respectively are n-permutable.
Proof. Let L has n-permutable congruences. Let θ,Φ be any two congruences on L. Then by (i) and (ii) of Lemma 5.1, respectively, we have
θL∘∘∘ΦL∘∘∘θL∘∘∘...=(θ∘Φ∘θ∘...)L∘∘=(Φ∘θ∘Φ∘...)L∘∘=ΦL∘∘∘θL∘∘∘ΦL∘∘∘..., |
and
θD(L)∘ΦD(L)∘θD(L)∘...=(θ∘Φ∘θ∘...)D(L)=(Φ∘θ∘Φ∘...)D(L)=ΦD(L)∘θD(L)∘ΦD(L)∘.... |
Therefore L∘∘ and D(L) have n-permutable congruences.
Conversely, let L∘∘ and D(L) have n-permutable congruences. If (a,b)∈(θ∘Φ∘θ∘...), using Theorem 2.3, we get
(a∘∘,b∘∘)∈(θ∘Φ∘θ∘...)L∘∘, |
and
(a∨dL,b∨dL)∈(θ∘Φ∘θ∘...)DL. |
By lemma 5.1, we get
(a∘∘,b∘∘)∈(θL∘∘∘ΦL∘∘∘θL∘∘∘...), |
and
(a∨dL,b∨dL)∈(θD(L)∘ΦD(L)∘ΦD(L)∘...). |
Since θL∘∘,ΦL∘∘ and θD(L),ΦD(L) are n-permutable on L∘∘,D(L), respectively, we have
(a∘∘,b∘∘)∈(ΦL∘∘∘θL∘∘∘ΦL∘∘∘...), |
and
(a∨dL≡b∨dL)∈(ΦD(L)∘θD(L)∘ΦD(L)∘...). |
It follows that, (a∘∘,b∘∘)∈(Φ∘θ∘Φ∘...) and (a∨dL,b∨dL)∈(Φ∘θ∘Φ∘...). Since L is a principal MS-algebra, then a=a∘∘∧(a∨dL) and b=b∘∘∧(b∨dL). By Lemma 5.2, we get
(a∘∘∧(a∨dL),b)=(a,b)∈(Φ∘θ∘Φ∘...). |
Thus (θ∘Φ∘θ∘...)⊆(Φ∘θ∘Φ∘...). Similarly we can show that (Φ∘θ∘Φ∘...)⊆(θ∘Φ∘θ∘...). Then L has n-permutable congruences.
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.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
The authors declare no conflict of interest.
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∘ | △L | α | β | γ | δ | ▽L |
△L | △L | α | β | γ | δ | ▽L |
α | α | α | β | ▽L | β | ▽L |
β | β | β | β | ▽L | β | ▽L |
γ | γ | ▽L | ▽L | γ | ▽L | ▽L |
δ | δ | β | β | ▽L | δ | ▽L |
▽L | ▽L | ▽L | ▽L | ▽L | ▽L | ▽L |
∘ | △L∘∘ | αL∘∘ | γL∘∘ | ▽L∘∘ |
△L∘∘ | △L∘∘ | αL∘∘ | γL∘∘ | ▽L∘∘ |
αL∘∘ | αL∘∘ | αL∘∘ | ▽L∘∘ | ▽L∘∘ |
γL∘∘ | γL∘∘ | ▽L∘∘ | γL∘∘ | ▽L∘∘ |
▽L∘∘ | ▽L∘∘ | ▽L∘∘ | ▽L∘∘ | ▽L∘∘ |
∘ | △D(L) | ▽D(L) |
△D(L) | △D(L) | ▽D(L) |
▽D(L) | ▽D(L) | ▽D(L) |