Sanaa El-ASSAR Abd El-Mohsen BADAWY

Abstract In this paper,the authors first introduce the concept of congruence pairs on the class of decomposable MS-algebras generalizing that for principal MS-algebras(see[13]).They show that every congruence relation θ on a decomposable MS-algebra L can be uniquely determined by a congruence pair(θ1,θ2),where θ1 is a congruence on the de Morgan subalgebra L◦◦of L and θ2 is a lattice congruence on the sublattice D(L)of L.They obtain certain congruence pairs of a decomposable MS-algebra L via central elements of L.Moreover,they characterize the permutability of congruences and the strong extensions of decomposable MS-algebras in terms of congruence pairs.

Keywords MS-Algebras,Decomposable MS-algebras,Congruence pairs,Strong extension,Permutability,Congruence lattices

1Introduction

Blyth and Varlet[18]studied Morgan Stone algebras(briefly MS-algebras)as a generalization of the classes of de Morgan and Stone algebras.Such algebras are bounded distributive lattices with additional unary operation.Blyth and Varlet[19]described the lattice of subvarieties of the variety MS of all MS-algebras.Badawy,Guffova and Haviar[12]introduced and characterized the class of decomposable MS-algebras by means of decomposable MS-triples.They observed that every decomposable MS-algebra L has two auxiliary substructures,namely,the de Morgan subalgebra L◦◦of all closed elements of L and the sublattice D(L)of all dense elements of L.Also,they introduced and characterized principal MS-algebras by means of principal MS-triples.They observed that the class of decomposable MS-algebras contains the class of principal MS-algebras.Badawy[1,7]investigated a relationship between congruences and special filters of a principal MS-algebra and a decomposable MS-algebra,respectively.Also,Badawy and El-Fawal[9]studied homomorphisms and subalgebras of decomposable MSalgebras in terms of decomposable MS-triples.Recently,Badawy and Atallah[8]introduced and characterized the set B(L)of all central elements of an MS-algebra L and established the relationship between its MS-intervals and congruences.For recent studies of(decomposable)MS-algebras and double MS-algebras see also[2,4–6,10–11,14–15,23,25,31].

In this paper,we introduce a suitable notion of congruence pairs of decomposable MSalgebras which is a generalization of the notion of congruence pairs of both Stone algebras and principal MS-algebras.We study many properties of congruence pairs of a decomposable MSalgebra.We derive that every congruence relation θ on a decomposable MS-algebra L can be represented by a pair of congruences(θ1,θ2),where θ1∈Con(L◦◦)and θ2∈Con(D(L)).We establish that there is a one to one correspondence between the lattice Con(L)of all congruences of L and the lattice A(L)of all congruence pairs of L.Also,we investigate the relationship between the central elements of a decomposable MS-algebra L and the congruence pairs of the form(θ[a↓],θ[aϕ(L)])for a∈L◦◦.Using the concept of congruence pairs,we prove that a decomposable MS-algebra L is congruence permutable if and only if both L◦◦and D(L)are congruence permutable.If L is a subalgebra of a decomposable MS-algebra L1,we show that L1is a strong extension of L if and only ifis a strong extension of L◦◦and D(L1)is a strong extension of D(L).

2 Preliminaries

In this section,we give the definitions and the main results which are needed through this work.We refer the readers to[8–9,12–13,18–20,29–31]for more details.

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−is the unary operation of involution satisfying:

A Stone algebra is a universal algebra(L;∨,∧,*,0,1)of type(2,2,1,0,0),where the unary operation*of pseudocomplementation has the properties that x∧a=0⇔x≤a*and x**∨x*=1.

An MS-algebra is an algebra(L;∨,∧,◦,0,1)of type(2,2,1,0,0),where a unary operation◦satisfies:

The class MS of all MS-algebras is equational.A de Morgan algebra is an MS-algebra satisfying the identity,x=x◦◦.The class S of Stone algebras is a subclass of MS and is characterized by the identity x∧x◦=0.

We recall some of the basic properties of MS-algebras which were proved in[18]or[20].

Theorem 2.1For any two elements a,b of an MS-algebra L,we have

(1)0◦=1,

(2)a≤b⇒b◦≤a◦,

(3)a◦◦◦=a◦,

(4)(a∨b)◦=a◦∧b◦,

(5)(a∨b)◦◦=a◦◦∨b◦◦,

(6)(a∧b)◦◦=a◦◦∧b◦◦.

We recall special subsets of an MS-algebra L which play an important role in the construction:

(1)L◦◦={x∈L:x=x◦◦}is the set of closed elements of L which is a de Morgan subalgebra of L(see[18]),

(2)D(L)={x∈L:x◦=0}is the set of dense elements of L which is a filter of L(see[12]),

(3)a↑={x∈L:x≥a}is the principal filter of L generated by the element a of L,

(4)a↓={x∈L:x≤a}is the principal ideal of L generated by the element a of L.

Now,we recall from[12]the definition of a decomposable MS-algebra and some related properties.

Definition 2.1(see[12])An MS-algebra(L;∨,∧,◦,0,1)is called a decomposable MSalgebra if for every x∈L there exists d∈D(L)such that x=x◦◦∧d.

The class of decomposable MS-algebras contains both the class M of all de Morgan algebras and the class S of all Stone algebras.

Let L be a decomposable MS-algebra.Define a map ϕ(L):L◦◦→F(D(L))(the lattice of all filters of D(L))by

It is known that ϕ(L)is a(0,1)-lattice homomorphism(see[12]).

An equivalence relation θ on a lattice L is called a lattice congruence on L if it is compatible with the lattice operations,that is,(a,b)∈θ and(c,d)∈θ imply(a∨c,b∨d)∈θ and(a∧c,b∧d)∈θ.

Let θ be a lattice congruence on a bounded lattice(a lattice with the smallest element 0 and the greatest element 1)L.Then the subset{x∈L:(x,0)∈θ}is called the Kernel of θ and is denoted by Ker θ.Also,the subset{x∈L:(x,1)∈θ}is called the Cokernel of θ and is denoted by Coker θ.It is clear that Ker θ and Coker θ are ideal and filter of L,respectively.

Theorem 2.2(see[26])An equivalence relation on a lattice L is a lattice congruence on L if and only if(a,b)∈θ implies(a∨c,b∨c)∈θ and(a∧c,b∧c)∈θ for all c∈L.

A lattice congruence θ on an MS-algebra(L;◦)is called a congruence on L if(a,b)∈θ implies(a◦,b◦)∈θ.

The symbols∇Land∆Lwill be used,as usual,for the universal congruence L×L and the equality congruence on L,respectively.

Let L be an MS-algebra.Then,we use Con(L)to denote the congruence lattice of L and we also use θL◦◦,θD(L)to denote the restrictions of a congruence θ∈Con(L)to L◦◦and D(L),respectively.Evidently,(θL◦◦,θD(L))∈Con(L◦◦)×Con(D(L)).

Now,we restrict the definition of a congruence pair of quasi-modular p-algebras(see[29,Definition 7])to Stone algebras.

Definition 2.2Let L be a Stone algebra.Then the pair(θ1,θ2)∈Con(L◦◦)×Con(D(L))is called a congruence pair if a∈L◦◦,u∈D(L),u≥a and a≡1(θ1)imply u≡1(θ2).

Definition 2.3(see[12])An MS-algebra(L;∨,∧,◦,0,1)is called a principal MS-algebra if it satisfies the following conditions:

(i)The filter D(L)is principal,i.e.,there exists an element dL∈L such that D(L)=[dL),

(ii)x=x◦◦∧(x∨dL)for any x∈L.

It is known that any principal MS-algebra is a decomposable MS-algebra(see[12]).From[13],we recall the definition of a congruence pair of a principal MS-algebra.

Definition 2.4(see[13])Let L be a principal MS-algebra with a smallest dense element dL.A pair of congruences(θ1,θ2)∈Con(L◦◦)×Con(D(L))will be called a congruence pair if

3 Congruence Pairs of a Decomposable MS-Algebra

The notion of a congruence pair was studied on various classes of algebras containing the class S of all Stone algebras.Katrik[27,29]studied the congruence pairs and the lattices of congruence pairs of certain p-algebras,El-Assar[21]characterized the congruence lattices of quasi-modular p-algebras,Badawy and Shume[16]considered the congruence pairs and related properties of principal p-algebras.Also,Badawy[3]presented a characterization of the congruence lattices of principal p-algebras.Beazear[17]introduced the notion of congruence pairs on MS-algebras from the subvariety K2(K2-algebras).Recently,Badawy,Haviar and Ploica[13]studied the concept of congruence pairs of principal MS-algebras.Also,they characterized the congruence lattices of principal MS-algebras in terms of congruence pairs.

In this section we introduce the concept of congruence pairs on decomposable MS-algebras generalizing that for principal MS-algebras.Some properties of congruence pairs of a decomposable MS-algebra L will be investigated.

Definition 3.1Let L be a decomposable MS-algebra.An arbitrary pair(θ1,θ2)in Con(L◦◦)×Con(D(L))is called a congruence pair if a≡b(θ1)implies a∨d≡b∨d(θ2)for all d∈D(L).

It is clear that if L is a principal MS-algebra with a smallest dense element dL,then Definition 2.6 implies Definition 3.1.

Lemma 3.1Let L be a decomposable MS-algebra and(θ1,θ2)be a congruence pair.Then we have the following property:

ProofLet a≡b(θ1).Thus by Definition 3.1,we get a∨c≡b∨c(θ2),a∨d≡b∨d(θ2)and hence a∨c∨d≡b∨c∨d(θ2)as c,d,c∨d∈D(L).Then a∨c≡b∨c(θ2)and c≡d(θ2)imply a∨c≡b∨c∨d.Also a∨d≡b∨d(θ2)and c≡d(θ2)imply a∨c∨d≡b∨d(θ2).Consequently a∨c≡b∨d(θ2).

For a Stone algebra,the following lemma shows that Definitions 2.4 and 3.1 are equivalent.

Lemma 3.2Let L be a Stone algebra.Then(θ1,θ2)∈Con(L◦◦)×Con(D(L))is a congruence pair according to Definition 2.4 if and only if it is a congruence pair by Definition 3.1.

ProofLet L be a Stone algebra.Then L◦◦is a Boolean subalgebra of L.Thus a∨a◦=1 for all a∈L◦◦.Let(θ1,θ2)∈Con(L◦◦)×Con(D(L))be a congruence pair by Definition 2.4.Suppose that a≡b(θ1).Let α=(a∨b◦)∧(a◦∨b).Then α∈L◦◦and α∧a=α∧b=a∧b.Since a∨b◦≡b∨b◦(θ1)=1 and a◦∨b≡a◦∨a(θ1)=1,we have α≡1(θ1)and by Definition 2.4,α≤α∨d∈D(L)implies α∨d≡1(θ2)for all d∈D(L).Since L is a distributive lattice,we have

In a similar way,we get b∨d≡(a∧b)∨d(θ2).Thus a∨d≡b∨d(θ2).For the converse,let a∈L◦◦,a≤u∈D(L)and a≡1(θ1).Then we have a∨d≡1∨d(θ2)for all d∈D(L)by Definition 3.1.Without loss of generality we can take u≥d.Then u=a∨d∨u≡1∨u∨d(θ2)=1.Therefore(θ1,θ2)is a congruence pair according to Definition 2.4.

The following theorem gives one of the main results of this paper.We give a characterization of congruence pairs of a decomposable MS-algebra.

Theorem 3.1Let L be a decomposable MS-algebra.Then every congruence relation θ of L determines a congruence pair(θL◦◦,θD(L)).Conversely,every congruence pair(θ1,θ2)uniquely determines a congruence relation θ on L satisfying θL◦◦=θ1and θD(L)=θ2by the rule

ProofLet θ∈Con(L)and a≡b(θL◦◦)for a,b∈L◦◦.Then a≡b(θ).This result implies that a∨d≡b∨d(θ).Hence,a∨d≡b∨d(θD(L)),where a∨d,b∨d∈D(L).This shows that(θL◦◦,θD(L))is a congruence pair.Conversely,let(θ1,θ2)be a congruence pair and let θ be defined as above.It is clear that θ is an equivalence relation.We now proceed to show that θ is a congruence on L.Let a≡b(θ)and c≡f(θ).Then we get a◦◦≡b◦◦(θ1),c◦◦≡f◦◦(θ1)and a∨d≡b∨d(θ2),c∨d≡f∨d(θ2)for all d∈D(L).Now,we have

Then a∧c≡b∧f(θ),and therefore θ preserves the meet operation of L.Also,θ preserves the join operation of L since the following equalities hold on L:

In order to show that θ preserves the unary operation◦,we let a≡b(θ),then a◦◦≡b◦◦(θ1).Hence,a◦=a◦◦◦≡b◦◦◦(θ1)=b◦.Thus by Definition 3.1,we have shown that a◦∨d≡b◦∨d(θ2)for all d∈D(L).Therefore,a◦≡b◦(θ).

Corollary 3.1Let L be a decomposable MS-algebra.Then the set A(L)of congruence pairs of L is a bounded sublattice of Con(L◦◦)×Con(D(L))and θ(θL◦◦,θD(L))is an isomorphism of Con(L)and A(L).

ProofIt is clear that(△L◦◦,△D(L)),(▽L◦◦,▽D(L))∈A(L).Let(θ1,θ2),(ψ1,ψ2)∈A(L).Then,it is easy to verify that(θ1∧ψ1,θ2∧ψ2)∈A(L).Now,we proceed to show that(θ1∨ψ1,θ2∨ψ2)∈A(L).Let a≡b(θ1∨ψ1).Then there is a finite sequence a=a0,a1,···,an=b in L◦◦such that,for each i with 0≤i≤n−1,either ai−1≡ai(θ1)or ai≡ai+1(ψ1).Then ai−1∨d≡ai∨d(θ2)or ai∨d≡ai+1∨d(ψ2),for every d∈D(L)by Definition 3.1.Thus we have the sequence

The above result leads to a∨d≡b∨d(θ2∨ψ2)and hence(θ1∨ψ1,θ2∨ψ2)∈A(L).Thus we conclude that A(L)is a bounded sublattice of Con(L◦◦)×Con(D(L)).It is clear that the map θ(θL◦◦,θD(L))of Con(L)into A(L)is an isomorphism.

The next corollary follows immediately.

Corollary 3.2Let L be a decomposable MS-algebra.Then the following statements hold:

(1)(∀Φ∈Con(D(L)))(△L◦◦,Φ)∈A(L),

(2)(∀Ψ∈Con(L◦◦)(Ψ,▽D(L))∈A(L).

4 Congruence Pairs via Central Elements of a Decomposable MSalgebra

In this section,we investigate the relationship between the central elements of a decomposable MS-algebra L and the congruence pairs of L.

From[8],we recall the following.

Definition 4.1(see[8])An element a of an MS-algebra L is called a central element of L if a∨a◦=1.The set of all central elements of L is denoted by B(L).

Theorem 4.1(see[8])Let L be an MS-algebra.Then B(L)is a Boolean subalgebra of L◦◦.

For each central element a of an MS-algebra L,we define a relation θ[a↓]on L◦◦as follows:

For each central element a of a decomposable MS-algebra L,we define a relation θ[aϕ(L)]on D(L)as follows:

The properties of the above two relations are given in the following two lemmas,respectively.

Lemma 4.1Let L be an MS-algebra.Then for every a,b of B(L),we have

(1)θ[a↓]is a congruence on L◦◦with Ker(θ[a↓])=a↓,

(2)a≤b if and only if θ[a↓]⊆θ[b↓],

(3)a=b if and only if θ[a↓]=θ[b↓],

(4)θ[0↓]=△L◦◦and θ[1↓]=▽L◦◦,

(5)θ[a↓]∨θ[b↓]=θ[(a∨b)↓],

(6)θ[a↓]∩θ[b↓]=θ[(a∧b)↓].

Proof(1)It is clear that θ[a↓]is an equivalence relation on L◦◦for every a∈B(L).Now let(x,y)∈θ[a↓]and c∈L◦◦.Then x∧a◦=y∧a◦and hence

Therefore(x∨c,y∨c)∈θ[a↓]for all c∈L◦◦.Also,we can deduce that(x∧c,y∧c)∈θ[a↓].Then by Theorem 2.6,θ[a↓]is a lattice congruence on L◦◦.To show that θ[a↓]is preserved by a unary operation◦on L◦◦,let(x,y)∈θ[a↓].Then we have:

Further,

as a=a∨0=a∨(x∧a◦)=a∨x implies x≤a.

(2)Let a≤b and(x,y)∈θ[a↓].Then x∧a◦=y∧a◦.Thus x∧a◦∧b◦=y∧a◦∧b◦and b◦≤a◦imply x∧b◦=y∧b◦.So(x,y)∈θ[b↓]and hence θ[a↓]⊆θ[b↓].Conversely,let θ[a↓]⊆θ[b↓].As a is a central element of L,then(a∧b)∧a◦=0=a∧a◦.Hence(a∧b,a)∈θ[a↓].By hypotheses,(a∧b,a)∈θ[b↓].Since b is a central element of L,then(a∧b)∧b◦=a∧b◦implies a∧b◦=0.Now,since a∧b◦=0 and a,b belong to the Boolean algebra B(L)then a≤b◦◦=b.

(3)It is obvious.

(4)Let(x,y)∈θ[0↓].Then x=x∧0◦=y∧0◦=y.Therefore θ[0↓]=△L◦◦.For all x,y∈L,we have x∧1◦=0=y∧1◦and hence(x,y)∈θ[1↓].Then θ[1↓]=▽L◦◦.

(5)Since a,b≤a∨b,then by(2),θ[a↓],θ[b↓]⊆θ[(a∨b)↓].Therefore θ[(a∨b)↓]is an upper bound of both θ[a↓]and θ[b↓].Suppose that θ[c↓]is an upper bound of θ[a↓]and θ[b↓].Then θ[a↓],θ[b↓]⊆θ[c↓].Thus by(2)we get a,b≤c.Then a∨b≤c.Again by(2),θ[(a∨b)↓]⊆θ[(c)↓].Therefore θ[(a∨b)↓]is the least upper bound of both θ[a↓]and θ[b↓].This deduces that θ[a↓]∨θ[b↓]=θ[(a∨b)↓].

(6)Since a∧b≤a,b,then by(2),θ[(a∧b)↓]⊆θ[a↓],θ[a↓].Thus θ[(a∧b)↓]⊆θ[a↓]∩θ[a↓].Conversely,let(x,y)∈θ[a↓]∩θ[b↓].Then

Therefore θ[(a↓]∩θ[b↓]⊆θ[(a∧b)↓]and hence θ[(a∧b)↓]=θ[a↓]∩θ[b↓].

Lemma 4.2Let L be a decomposable MS-algebra.Then for every a,b of B(L),we have

(1)θ[aϕ(L)]is a congruence on D(L)with Coker(θ[aϕ(L)])=aϕ(L),

(2)a≤b implies θ[aϕ(L)]⊆θ[bϕ(L)],

(3)θ[(0ϕ(L)]=△D(L)and θ[1ϕ(L)]=▽D(L),

(4)θ[aϕ(L)]∨θ[bϕ(L)]=θ[(a∨b)ϕ(L)],

(5)θ[aϕ(L)]∧θ[bϕ(L)]=θ[(a∧b)ϕ(L)].

Proof(1)We know that aϕ(L)=a◦↑∩D(L)is a filter of D(L).Obviously,θ[aϕ(L)]is an equivalence relation on D(L).Let(x,y),(x′,y′)∈θ[aϕ(L)].Thus x∧d=y∧d and x′∧e=y′∧e for some d,e∈aϕ(L).Then

Hence(x∨x′,y∨y′)∈θ[aϕ(L)].Using a similar way,we get(x∧x′,y∧y′)∈θ[aϕ(L)],so θ[aϕ(L)]is lattice congruence on D(L).Also,we have

(2)Let a≤b.Then aϕ(L)⊆bϕ(L).Let(x,y)∈θ[aϕ(L)].Then x∧d=y∧d for some d∈aϕ(L).Since d∈aϕ(L)and aϕ(L)⊆bϕ(L),then d∈bϕ(L).So,(x,y)∈θ[bϕ(L)].Therefore θ[aϕ(L)]⊆θ[abϕ(L)].

(3)Let(x,y)∈θ[0ϕ(L)].Since 0ϕ(L)=(1],then x=y and hence θ[0ϕ(L)]=△D(L).Since 1ϕ(L)=D(L),then θ[1ϕ(L)]=θ[D(L)]=D(L)×D(L)=▽D(L).

(4)Since a,b≤a∨b,then aϕ(L),bϕ(L)⊆(a∨b)ϕ(L).Hence by(2),we have

Then θ[(a∨b)ϕ(L)]is an upper bound of θ[aϕ(L)]and θ[bϕ(L)].Let θ[cϕ(L)]be an upper bound of θ[aϕ(L)]and θ[bϕ(L)].Then θ[aϕ(L)],θ[bϕ(L)]⊆θ[cϕ(L)]implies aϕ(L),bϕ(L)⊆cϕ(L).Thus(a∨b)ϕ(L)=aϕ(L)∨bϕ(L)⊆cϕ(L)and hence θ[(a∨b)ϕ(L)]⊆θ[cϕ(L)].Therefore θ[(a∨b)ϕ(L)]is the least upper bound of both θ[aϕ(L)]and θ[bϕ(L)].

(5)Since a∧b≤a,b,then by(2),θ[(a∧b)ϕ(L)]⊆θ[aϕ(L)],θ[bϕ(L)]and hence θ[(a∧b)ϕ(L)]⊆θ[aϕ(L)]∩θ[bϕ(L)].Conversely,let(x,y)∈θ[aϕ(L)]∩θ[bϕ(L)].Then(x,y)∈θ[aϕ(L)]and(x,y)∈θ[bϕ(L)].Thus x∧d=y∧d for some d∈aϕ(L)and x∧e=y∧e for some e∈bϕ(L).Since d∨e≥d,e and d∈aϕ(L),b∈bϕ(L),then d∨e∈aϕ(L)∩bϕ(L)=(a∧b)ϕ(L).Now

Therefore(x,y)∈θ[(a∧b)ϕ(L)]and hence θ[aϕ(L)]∩θ[bϕ(L)]⊆θ[(a∧b)ϕ(L)].

Let L be a decomposable MS-algebra.Consider the subsets B and D of Con(L◦◦)and Con(D(L)),respectively as follows:

The proof of the following theorem is a consequence of Lemmas 4.3–4.4.

Theorem 4.2Let L be a decomposable MS-algebra.Then

(1)(B,∨,∧,′,△L◦◦,▽L◦◦)is a Boolean algebra,where(θ[a↓])′=θ[a◦↓],

(2)(D,∨,∧,′,△D(L),▽D(L))is a Boolean algebra,where(θ[aϕ(L)])′=θ[a◦ϕ(L)].

Now,we observe that every central element a of a decomposable MS-algebra L associated with the congruence pair(θ[a↓],θ[aϕ(L)]).

Theorem 4.3Let L be a decomposable MS-algebra and a∈L◦◦.Then a is a central element of L if and only if(θ[a↓],θ[aϕ(L)])is a congruence pair of L.

ProofLet a be a central element of L.By Lemmas 4.3(1)and 4.4(1),θ[a↓]and θ[aϕ(L)]are congruences on L◦◦and D(L),respectively.To show that(θ[a↓],θ[aϕ(L)])is a congruence pair,let(b,c)∈θ[a↓].Then

Thus(θ[a↓],θ[aϕ(L)])∈A(L).Conversely,let(θ[a↓],θ[aϕ(L)])∈A(L).Since(a,0)∈θ[a↓],then a∧a◦=0∧a◦=0.Now,a∨a◦=(a◦∧a)◦=0◦=1.Therfore a∈B(L).

Let L be a decomposable MS-algebra.Consider the set

From Theorems 4.5–4.6,we observe the following important results.

Theorem 4.4Let L be a decomposable MS-algebra.Then(A′(L);∨,∧,′,0A′(L),1A′(L))is a Boolean algebra,where

Theorem 4.5Let L be a decomposable MS-algebra.Then B(L)is isomorphic to A′(L)under the isomorphism a(θ[a↓],θ[aϕ(L)]).

5 Congruence Permutable of Decomposable MS-Algebras

El-Assar[21]studied the notion of n-permutability of congruences of p-algebras satisfying certain condition.Also,El-Assar and Abd El-Hakim[24]characterized the permutability of congruences of modular p-algebras.Badawy and Shume[16]characterized the permutability of congruences of the class of principal p-algebras.

Let L be an algebra.We say that θ,ψ∈Con(L)permute if for any a,b,c∈L with(a,b)∈θ and(b,c)∈ψ,there exists h∈L such that(a,h)∈ψ and(h,c)∈θ,that is θ◦ψ=ψ◦θ,where θ◦ψ is the relational product of θ and ψ.

An algebra L is said to be congruence permutable(briefly,permutable)if every pair of congruences on it is permutable.

We characterize the congruence permutable of a decomposable MS-algebra in the following theorem.

Theorem 5.1Let L be a decomposable MS-algebra.Then the following conditions are equivalent:

(1)L has congruence permutable,

(2)L◦◦and D(L)both are congruence permutable.

ProofTo show the equivalence of the conditions(1)and(2),we have to show that two congruences θ,ψ∈Con(L)are permutable if and only if their restrictions θL◦◦,ψL◦◦and θD(L),ψD(L)both are congruence permutable on L◦◦and D(L),respectively.Let θ,ψ be permutable on L.Firstly,we will prove that θL◦◦,ψL◦◦are permutable on L◦◦.Let a,b,c∈L◦◦be such that(a,b)∈θL◦◦and(b,c)∈ψL◦◦.Then(a,b)∈θ and(b,c)∈ψ.Since θ,ψ are permutable,then there exists x∈L such that(a,x)∈ψ and(x,c)∈θ.Thus(a,x◦◦)∈ψ and(x◦◦,c)∈θ.Then(a,x◦◦)∈ψL◦◦and(x◦◦,c)∈θL◦◦as x◦◦∈L◦◦.Therefore θL◦◦,ψL◦◦are permutable on L◦◦.Now we prove that permutability of θ and ψ implies permutability of θD(L)and ψD(L).Let x,y,z∈D(L)be such that(x,y)∈θD(L)and(y,z)∈ψD(L).Then(x,y)∈θ and(y,z)∈ψ.Since θ,ψ are permutable,then there exists a∈L such that(x,a)∈ψ and(a,z)∈θ.Then for every d∈D(L),we have(x∨d,a∨d)∈ψ and(a∨d,z∨d)∈θ.We can choose d≤x,z.Then(x,a∨d)∈ψD(L)and(a∨d,z)∈θD(L)with a∨d∈D(L).Therefore θD(L)and θD(L)both are congruence permutable on D(L).

Conversely,let θ,ψ∈Con(L)such that θL◦◦,ψL◦◦and θD(L),ψD(L)are congruence permutable on L◦◦and D(L)respectively.Consider the elements x,y,z∈L with(x,y)∈θ and(y,z)∈ψ.By Theorem 3.4,we get(x◦◦,y◦◦)∈θL◦◦,(y◦◦,z◦◦)∈ψL◦◦and(x∨d,y∨d)∈θD(L),(y∨d,z∨d)∈ψD(L)for all d∈D(L).Since θL◦◦,ψL◦◦are permutable,then there exists a∈L◦◦with(x◦◦,a)∈ψL◦◦and(a,z◦◦)∈θL◦◦.Since θD(L),ψD(L)are permutable congruences on D(L),then there exists e∈D(L)such that(x∨d,e)∈ψD(L)and(e,z∨d)∈θD(L).It follows that

Since L is a decomposable MS-algebra,then there exist d1,d2∈D(L)such that x=x◦◦∧d1and z=z◦◦∧d2.Hence x≤d1and z≤d2.Since θ and ψ are compatible with the∧operation,then we have

and

Consequently,we deduce that(x,a∧e)∈ψ and(a∧e,z)∈θ.Therefore θ,ψ are permutable.

Let L be an MS-algebra.Define the relation Φ on L as follows:

It is known that Φ is a congruence relation on L(see[18]).Then Φ satisfies the following property.

Corollary 5.1Let L be a decomposable MS-algebra.Then the congruence relation Φ permutes with any element of Con(L),as ΦL◦◦=△L◦◦and ΦD(L)=∇D(L).

6 Strong Extensions of Decomposable MS-Algebras

It is known that the class of distributive lattices satisfies the Congruence Extension Property(CEP for short)briefly.Luo[30]proved that the class MS of all MS-algebras satisfies the CEP.The notion of a strong extension of algebras was first introduced by Varlet[32].EL-Assar and Abd El-Hakim[24]studied the strong extension for modular p-algebras.Also EL-Assar[22]introduced the strong extension for quasi-modular p-algebras.Now we recall the following two definitions.

Definition 6.1(see[28])An algebra A satisfies the CEP if for every subalgebra B of A and every θ of B,θ extends to a congruence of A.

Definition 6.2(see[28])An algebra L is said to be a strong extension of the algebra M,if M is a subalgebra of L and every congruence of M has at most one extension to L.

In the following theorem,we study strong extensions of decomposable MS-algebras using the congruence pairs technique.

Theorem 6.1Let L be a subalgebra of a decomposable MS-algebra L1.Then L1is a strong extension of L if and only if the following conditions hold:

(1)D(L1)is a strong extension of D(L),

Corollary 6.1Let L1and L be decomposable MS-algebras.If L1is a strong extension of L,then Con(L1)Con(L).

ProofSince the class of MS-algebras satisfies the CEP,then every congruence of L has an extension.By hypotheses this extension is unique.Then Con(L1)Con(L).

7 Conclusion

In this paper,we introduced the notion of congruence pairs of decomposable MS-algebras.It is proved that every congruence relation θ on a decomposable MS-algebra L can be represented by a unique congruence pair(θ1,θ2),where θ1is a congruence relation on the de Morgan algebra L◦◦and θ2is a lattice congruence relation on the lattice D(L).Also,it is observed that Con(L),the lattice of all congruences of a decomposable MS-algebra L,is isomorphic to A(L),the lattice of all congruence pairs of L.It is observed that there is a one to one correspondence between the set B(L)of central elements of a decomposable MS-algebra L and the set of congruence pairs of the form(θ[a↓],θ[aϕ(L)]),where a∈B(L).Permutability of congruences and strong extensions of decomposable MS-algebras are considered in terms of congruence pairs.In a future work,we will describe the congruence lattices of decomposable MS-algebras by means of congruence pairs.

AcknowledgementThe authors would like to thank the editors and referees for their valuable comments and suggestions to improve this presentation.