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Research article

Forming localized waves of the nonlinearity of the DNA dynamics arising in oscillator-chain of Peyrard-Bishop model

  • Received: 28 October 2019 Accepted: 24 February 2020 Published: 09 March 2020
  • MSC : 65D19, 65H10, 35A20, 35A24, 35C08, 35G50

  • In this article, the mathematical modeling of DNA vibration dynamics has been considered that describes the nonlinear interaction between adjacent displacements along with the Hydrogen bonds with utilizing five techniques, namely, the improved tan(φ/2)-expansion method (ITEM), the exp(-Ω(η))-expansion method (EEM), the improved exp(-Ω(η))-expansion method (IEEM), the generalized (G'/G)-expansion method (GGM), and the exp-function method (EFM) to get the new exact solutions. This model of the equation is analyzed using the aforementioned schemes. The different kinds of traveling wave solutions: solitary, topological, periodic and rational, are fall out as a by-product of these schemes. Finally, the existence of the solutions for the constraint conditions is also shown.

    Citation: Jalil Manafian, Onur Alp Ilhan, Sizar Abid Mohammed. Forming localized waves of the nonlinearity of the DNA dynamics arising in oscillator-chain of Peyrard-Bishop model[J]. AIMS Mathematics, 2020, 5(3): 2461-2483. doi: 10.3934/math.2020163

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  • In this article, the mathematical modeling of DNA vibration dynamics has been considered that describes the nonlinear interaction between adjacent displacements along with the Hydrogen bonds with utilizing five techniques, namely, the improved tan(φ/2)-expansion method (ITEM), the exp(-Ω(η))-expansion method (EEM), the improved exp(-Ω(η))-expansion method (IEEM), the generalized (G'/G)-expansion method (GGM), and the exp-function method (EFM) to get the new exact solutions. This model of the equation is analyzed using the aforementioned schemes. The different kinds of traveling wave solutions: solitary, topological, periodic and rational, are fall out as a by-product of these schemes. Finally, the existence of the solutions for the constraint conditions is also shown.


    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)Ψa=ba=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)¯(ab)=¯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)(ab)=ab,(2)aa=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)aa,(2)(xy)=xy,(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 aa=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) abba,

    (3) a=a,

    (4) a=a,

    (5) (ab)=ab,

    (6) (ab)=ab,

    (7) (ab)=ab.

    An element a of an MS-algebra L is called a closed element of L if a=a and an element dL is called a dense element of L if d=0.

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

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

    (2) the set D(L)={aL: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 dLL such that D(L)=[dL),

    (2) a=a(adL), for all aL.

    A principal MS-algebra L is a principal Stone algebra if xx=1, for all xL.

    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)=adL,for  all  aL.

    Then φL is a (0,1)-lattice homomorphism.

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

    Figure 1.  L is a principal MS-algebra.

    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.

    Figure 2.  (L,D(L),φL) is a principal MS-triple.

    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 (ac,bd)θ and (ac,bd)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 θConlat(L). Then we define the principal congruence generated by (a,b) which denoted by θ(a,b) as follows:

    θ(a,b)={θConlat(L):(a,b)θ}.

    If L is an MS-algebra and θCon(L), then

    θ(a,b)={θCon(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 θCon(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  (adL,bdL)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  (xdL,ydL)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  (ax,by)θ2.

    Proof. Let (a,b)θ1 and (x,y)θ2. Then (adL,bdL)θ2 (by Definition 2.9) and (x,y)θ2 imply that (axdL,bydL)θ2. Therefore (ax,by)θ2 as x,ydL.

    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)=adL,aL.

    Proof. (12) Let (θ1,θ2) be an MS-congruence pair of L and (x,y)ConφL(θ1). Then (x,y)=(φL(a),φL(b))=(adL,bdL), where (a,b)θ1. Since (a,b)θ1, then (adL,bdL)θ2, by (1). Thus (x,y)θ2. So ConφL(θ1)θ2.

    (21) Let Con(θ1)θ2. We prove that (θ1,θ2) is an MS-congruence pair of L. Let (a,b)θ1. Then by (2), (adL,bdL)=(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 ua,uD(L) imply (u,1)θ2.

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

    (23) Let ConφL(θ1)θ2. Let (a,1)θ1 and ua,uD(L). Then (φL(a),φL(1))θ2 by (2). Thus (adL,1)θ2.

    Now (adL,1)θ2 and (u,u)θ2,ua,uD(L) imply (audL,1u)θ2. Thus (u,1)θ2 as ua,dL.

    (31) Since L is a Stone algebra then L is a Boolean subalgebra of L and hence aa=1 for each aL. Suppose (3) holds and (a,b)θ1. Then (b,b)θ1 implies (ab,bb)=(ab,1)θ1 and (aa,ba)=(ba,1)θ1. Therefore (β,1)θ1, where β=(ab)(ab). It is clear that βL and βa=βb=ab. Since ββdLD(L) and (β,1)θ1 then (βdL,1)θ2 by (3). Thus (adL,adL)(1,βdL)θ2 implies (adL,(aβ)dL)=(adL,(ab)dL)θ2. Similarly, we can get (bdL,(bβ)dL)=(bdL,(ab)dL)θ2. Then (bdL,bdL)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.

    Figure 3.  The congruence lattices Con(L) and A(L).

    Where,

    L={(x,x):xL},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)0I,(2)abI,a,bI,(3)I is an down-set, that is, ifxy,yIandxL,thenxI.

    A principal ideal of L generated by aL is defined by

    (a]={xL:xa}.

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

    (1)1F,(2)abF,a,bF,(3)F is an up-set, that is, ifxy,yFandxL,thenxF.

    A principal filter of L generated by aL is defined by

    [a)={xL:xa}.

    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 {xL:(x,0)θ}, which is an ideal of L,

    (ii) The Cokernel of θ (Cokerθ) is the set {xL:(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 cc=1.

    It is ready seen that the set B(L)={c:cc=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)ac=bc,

    (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(cdL)=y(cdL).

    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(cdL)=y(cdL) and m(cdL)=n(cdL). Then, we get (xm,yn)θ(φL(c),1) and (xm,yn)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))={xD(L):(x,1)θ(φL(c),1)}={xD(L):x(cdL)=1(cdL)}={xD(L):x(cdL)=cdL}={xD(L):xcdL}=[cdL)=[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 cL. 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)xc=yc(xc)dL=(yc)dL(xdL)(cdL)=(ydL)(cdL)(xdL,ydL)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 cc = 0c=0. Now, cc=(cc)=1. Therefore c is a Boolean element.

    The basic properties of principal congruence relations θ(0,a) and θ(φL(a),1),aB(L) are given in the following:

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

    (1) ab 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,ab),

    (5) θ(0,a)θ(0,b)=θ(0,ab).

    Lemma 3.4. Let L be a principal MS-algebra. Then for every a,bB(L), we have

    (1) ab implies θ(φL(a),1)θ(φL(b),1),

    (2) θ(φL(a),1)θ(φL(b),1)=θ(φL(ab),1),

    (3) θ(φL(a),1)θ(φL(b),1)=θ(φL(ab),1),

    (4) θ(φL(0),1)=D(L) and θ(φL(1),1)=D(L).

    Proof.

    (1) Let ab. Then ab. Let (x,y)θ(φL(a),1) then x(adL)=y(adL). Now

    x(bdL)=x((ab)dL)=x{(adL)(bdL)}=x(adL)(bdL)=y(adL)(bdL)=y{(ab)dL}=y(bdL).

    Then (x,y)θ(φL(b),1). Therefore θ(φL(a),1)θ(φL(b),1).

    (2) Since a,bab then by (1), we have

    θ(φL(a),1),θ(φL(b),1)θ(φL(ab),1).

    Then θ(φL(ab),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(ab)=φL(ab) and hence θ(φL(ab),1)θ(φL(c),1). Therefore θ(φL(ab),1) is the least upper bound of θ(φL(a),1) and θ(φL(b),1).

    (3) Since aba,b then by (1) θ(φL(ab),1)θ(φL(a),1),θ(φL(b),1). Then θ(φL(ab),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(ab)=φL(a)φL(b)φL(c) and hence θ(φL(c),1)θ(φ(ab),1). Therefore θ(φL(ab),1) is the greatest lower bound of θ(φL(a),1) and θ(φL(b),1).

    (4) Since dL,1θ(dL,1), then θ(dL,1)=D(L)×D(L)=D(L). Also, let (x,y)θ(φL(1),1) then x(1dL)=y(1dL). 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):aB(L)},G={θ(φL(a),1):aB(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,LH. Let θ(0,a),θ(0,b)H. Then we get

    θ(0,a)θ(0,b)=θ(0,ab),

    and

    θ(0,a)θ(0,b)=θ(0,ab).

    Therefore H is a bounded lattice. Since B(L) is a Boolean algebra, then aB(L) for all aB(L). Thus θ(0,a)H. Then we have

    θ(0,a)[θ(0,a)]=θ(0,a)θ(0,a)=θ(0,aa)=θ(0,1)=L,

    and

    θ(0,a)[(0,a)]=θ(0,a)θ(0,a)=θ(φL(0,aa)=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(ab),1),

    and

    θ(φL(a),1)θ(φL(b),1)=θ(φL(ab),1).

    Therefore G is a bounded lattice. Since B(L) is a Boolean algebra, then aB(L) for all aB(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(aa),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(aa),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)):aB(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,ab),θ(φL(ab),1)),(θ(0,a),θ(φL(a),1)(θ(0,b),θ(φL(b),1))=(θ(0,ab),θ(φL(ab),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)),aB(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(az,1)). Also, A(L) is a bounded sublattice of A(L).

    Figure 4.  The lattices B(L) and 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,ΦCon(L) are 2-permutable if θΦ=Φθ, that is, (x,a)θ and (a,y)Φ, x,y,aL imply (x,b)Φ and (b,y)θ for some bL. 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)(θΦ),a,bL. Then (a,t)θ and (t,b)Φ for some tL. 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)(Φθ). Therefore θ and Φ permute. The second implication is obvious.

    Define a relation Ψ on a principal MS-algebra L as follows:

    (a,b)Ψa=ba=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,xL,

    (3) max [x]Ψ=x,xL, where [x]Ψ={yL: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={aL:(a,0)Ψ}={aL:a=1}={0},

    and

    CokerΨ={zL:(z,1)Ψ}={zL:z=1}=D(L)=[dL).

    (2) Since x=x then x[x]Ψ. Thus (x,x)Ψ.

    (3) Let y[x]Ψ. Then yy=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:aL} is the set of all congruence classes module Ψ and

    [a]Ψ[b]Ψ=[ab]Ψ,[a]Ψ[b]Ψ=[ab]Ψ.

    Define on L/Ψ by ([a]Ψ)=[a]Ψ,aL. 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:LL/Ψ by

    g(a)=[a]Ψ,aL.

    Let a,bL 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,bL, we get

    g(ab)=[ab]Ψ=[a]Ψ[b]Ψ=g(a)g(b),
    g(ab)=[ab]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]ΨL/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,

    Ψθ=θΨ  forall  θCon(L).

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

    Proof. We prove that Ψθ=θΨ for all θCon(L). Let (a,b)Ψθ. Then (a,z)Ψ and (z,b)θ for some zL. It follows that a=z and (z,b)θ. Now

    (z,b)θ(z,b)θand(adL,adL)θ(a(adL),b(adL))θ(a,b(adL))θ         as  a=a(adL)

    Since [b(adL)]=b, then (b(adL),b)Ψ. Therefore (a,b(adL))θ and (b(adL),b)Ψ. imply (a,b)θΨ. Then Ψθ=θΨ,θCon(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)θL. Then (a,t)θ and (t,b)L for some tL. Thus (a,t)θ and t=b. Then (a,a)L and (a,b)θ imply (a,b)Lθ. Therefore θL=Lθ.

    (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,ΦCon(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)θLΦL. Then there exist yL 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 aL 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 yD(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 aL such that

    (x,a)Φand(a,z)θ(x,adL)=(xdL,adL)Φand(adL,zdL)θ(x,adL)Φand(adL,z)θ,(x,adL)ΦD(L)and(adL,z)θD(L),asx,adL,zD(L)(x,z)ΦD(L)θD(L).

    Thus θD(L),ΦD(L) have 2-permutable congruences on D(L).

    Conversely, let θ,ΦCon(L). Assume that θL,ΦL and θD(L),ΦD(L), have 2-permutable congruences on L and D(L), respectively. Let (x,z)θΦ. Suppose that x,y,zL with (x,y)θ and (y,z)Φ. Then, by using Theorem 2.3, we get the following statements.

    (x,y),(y,z)ΦL,(xdL,ydL),(ydL,zdL)ΦD(L).

    Since θL,ΦL have 2-permutable congruences on L, there exists an element aL such that

    (x,a)ΦL  and  (a,z)θL.

    Since θD(L),ΦD(L) have 2-permutable congruences on D(L), there exists eD(L) such that (xdL,e)ΦDL and (e,zdL)θDL). It follows that (x,a)Φ,(a,z)θ and (xdL,e)Φ,(e,zdL)θ. Since L is a principal MS-algebra we have

    x=x(xdL)  and  z=z(zdL).

    Since θ,Φ are compatible with the operation, then (x,a)Φ and (xdL,e)Φ imply (x,ae)=(x(xdL),ae)Φ. Also (a,z)θ and (e,zdL)θ imply (ae,z)=(ae,z(zdL))θ. Consequently, we deduce that (x,ae)Φ and (ae,z)θ,aeL. Then (x,z)Φθ, 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 LandD(L) have 2-permutable congruences, respectively. Where αL=βL and δL=L.

    Table 1.  (Con(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

     | Show Table
    DownLoad: CSV
    Table 2.  (Con(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

     | Show Table
    DownLoad: CSV
    Table 3.  (Con(D(L));).
    D(L) D(L)
    D(L) D(L) D(L)
    D(L) D(L) D(L)

     | Show Table
    DownLoad: CSV

    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.

    Figure 5.  L is a principal MS-algebra with D(L)=[z).

    It is clear that D(L)={z,z1,1} and L={0,c,a,b,d,1}. Consider θ,ΦCon(L) as follows:

    θ=L{{x1,b},{y1,d},{z1,1}},Φ=L{{x,x1},{y,y1},{z,z1}}.

    We observe that θΦΦθ as (z,1)Φθ but (z,1)θΦ.

    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,

    θΦθ...=ΦθΦ...(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) (θΦθ...)L=θLΦL...(n times),

    (ii) (θΦθ...)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,bL and (a,b)(θLΦL...). Then there exist c1,c2,...,cn1L such that

    (a,c1)θL,(c1,c2)ΦL,...,(cn1,b)Ψ,whereΨ={θL,ifnodd,ΦL,ifneven.

    Then

    (a,c1)θ,(c1,c2)Φ,...,(cn1,b)Ψ,whereΨ={θ,ifnodd,Φ,ifneven.

    Then (a,b)(θΦθ...) and hence (a,b)(θΦθ...)L. Therefore

    θLΦLθL...(θΦθ...)L.

    Conversely, let a,bL such that (a,b)(θΦθ...)L. Then (a,b)(θΦθ...). Then there exist c1,c2,...,cn1L such that

    (a,c1)θL,(c1,c2)ΦL,...,(cn1,b)Ψ,whereΨ={θL,ifnodd,ΦL,ifneven.

    Then we get

    (a,c1)θ,(c1,c2)Φ,...,(cn1,b)Ψ,whereΨ={θ,ifnodd,Φ,ifneven.

    Since c1,c2,...,cn1L, we have

    (a,c1)θL,(c1,c2)ΦL,...,(cn1,b)Ψ,whereΨ={θL,ifnodd,ΦL,ifneven.

    Therefore

    (a,b)(θLΦLθL...)

    and hence

    (θΦθ...)LθLΦLθL...(ntimes).

    Then we get

    (θΦθ...)L=θLΦLθL...(ntimes).

    (ii) Take x,yD(L), let (x,y)(θD(L)ΦD(L)ΦD(L)θD(L)...). Then (x,y)(θΦθ...) and hence (a,b)(θΦθ...)D(L). Then (θΦθ...)(θΦθ...)D(L).

    Conversely, let (x,y)(θΦθ...)D(L), then (x,y)(θΦ...). There exist d1,d2,...dn1L, such that

    (x,d1)θ,(d1,d2)Φ,...,(dn1,y)ψ,whereψ={θ,ifnodd,Φ,ifneven.

    Since x,ydL, we get,

    (x,d1)=(xdL,d1dL)θD(L),(d1dL,d2dL)ΦD(L),...,(dn1,bdL1)=(dn1,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 θΦθ...(n times). Then (a,b)ψ and (c,d)ψ imply (ac,bd)ψ.

    Proof. Let (a,b)ψ and (c,d)ψ. Then there exist a1,a2,...,an1,c1,c2,...,cn1L such that

    (a,a1)θ,(a1,a2)Φ,...,(an1,b)Ψ,whereΨ={θ,ifnodd,Φ,ifneven.

    And

    (c,c1)θ,(c1,c2)Φ,...,(cn1,d)Ψ,whereΨ={θ,ifnodd,Φ,ifneven.

    Since θ,Φ are congruence on L then we get,

    (ac,a1c1)θ,(a1c1,a2c2)Φ,...,(an1cn1,bd)Ψ,whereΨ={θ,ifnodd,Φ,ifneven.

    Thus (ac,bd)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

    θLΦLθL...=(θΦθ...)L=(ΦθΦ...)L=ΦLθLΦL...,

    and

    θD(L)ΦD(L)θD(L)...=(θΦθ...)D(L)=(ΦθΦ...)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)(θΦθ...), using Theorem 2.3, we get

    (a,b)(θΦθ...)L,

    and

    (adL,bdL)(θΦθ...)DL.

    By lemma 5.1, we get

    (a,b)(θLΦLθL...),

    and

    (adL,bdL)(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)(ΦLθLΦL...),

    and

    (adLbdL)(ΦD(L)θD(L)ΦD(L)...).

    It follows that, (a,b)(ΦθΦ...) and (adL,bdL)(ΦθΦ...). Since L is a principal MS-algebra, then a=a(adL) and b=b(bdL). By Lemma 5.2, we get

    (a(adL),b)=(a,b)(ΦθΦ...).

    Thus (θΦθ...)(ΦθΦ...). Similarly we can show that (ΦθΦ...)(θΦθ...). 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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