Skip to main content

Complete decomposable MS-algebras

Abstract

According to the characterization of decomposable MS-algebras in terms of triples (M,D,φ), where M is a de Morgan algebra, D is a distributive lattice with 1 and φ is a (0,1)-homomorphism of M into F(D), the filter lattice of D, we characterize complete decomposable MS-algebras in terms of complete decomposable MS-triples. Also, we describe the complete homomorphisms of complete decomposable MS-algebras by means of complete decomposable MS-triples.

Introduction

Morgan Stone algebras (or simply MS-algebras) are introduced and characterized by T.S. Blyth and J.C. Varlet [1] as a generalization of both de Morgan algebras and Stone algebras. In [2], T.S. Blyth and J.C. Varlet described the lattice Λ(MS) of subclasses of the class MS of all MS-algebras. A. Badawy, D. Guffova, and M. Haviar [3] introduced and characterized decomposable MS-algebras by means of decomposable MS-triples. Moreover, they constructed a one-to-one correspondence between decomposable MS-algebras and decomposable MS-triples. A. Badawy and R. El-Fawal [4] studied many properties of decomposable MS-algebras in terms of decomposable MS-triples as homomorphisms and subalgebras. Also, they formulated and solved some fill in problems concerning homomorphisms and subalgebras of decomposable MS-algebras. A. Badawy [5] introduced the notion of dL-filters of principal MS-algebras. Recently, A. Badawy [6] studied the relationship between de Morgan filters and congruences of decomposable MS-algebras. Also, many properties of ideals of MS-algebras are given in [7] and [8].

Several authors studied complete p-algebras, like C.C. Chain and G. Gr\(\ddot {a}\)tzer [9] for Stone algebras, S. El-Assar, and M. Atallah [10] for distributive p-algebras and P. Mederly [11] for modular p-algebras.

In this paper, we introduce complete decomposable MS-algebras and complete decomposable MS-triples. We show that a decomposable MS-algebra L constructed from the decomposable MS-triple (M,D,φ) is complete if and only if the triple (M,D,φ) is complete. Also, a description of complete homomorphisms of decomposable MS-algebras is given in terms of complete decomposable MS-triples.

Preliminaries

In this section, we present definitions and main results which are needed through this paper. We refer the reader to [14, 1215] 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 the unary operation of involution satisfies

\(\overline {\overline {x}}=x,\overline {(x\vee y)}=\overline {x}\wedge \overline {y},\overline {(x\wedge y)}=\overline {x}\vee \overline {y}.\)

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 the unary operation satisfies

xx,(xy)=xy,1=0.

The following Theorem gives the basic properties of MS-algebras.

Theorem 1

([1,12]). For any two elements a,b of an MS-algebra L, we have (1) 0=1, (2) abba, (3) a=a, (4) (ab)=ab, (5) (ab)=ab, (6) (ab)=ab.

Lemma 1

([1,3]). Let L be an MS-algebra. Then (1) L={xL:x=x} is a de Morgan subalgebra of L, (2) D(L)={xL:x=0} is a filter (filter of dense elements) of L.

For any lattice L, let F(L) denotes the set of all filters of L. It is known that, (F(L);,) is a distributive lattice if and only if L is a distributive lattice, where the operation and are given by

FG=FG and FG={xL:xfg,fF,gG},respectively for everyF,GF(L).

Also, [a)={xL:xa} is a principal filter of L generated by a.

Definition 1

[9]. Let \(\phantom {\dot {i}\!}L=(L;\vee,\wedge,0_{L},1_{L})\) and \(\phantom {\dot {i}\!}L_{1}=(L_{1};\vee,\wedge,0_{L_{1}},1_{L_{1}})\) be bounded lattices. The map h:LL1 is called (0,1)-lattice homomorphism if (1) \(\phantom {\dot {i}\!}0_{L}h=0_{L_{1}}\) and \(\phantom {\dot {i}\!}1_{L}h=1_{L_{1}}\), (2) h preserves joins, that is, (xy)h=xhyh for every x,yL, (3) h preserves meets, that is, (xy)h=xhyh for every x,yL.

Definition 2

[14] A (0,1)-lattice homomorphism h:LL1 of an MS-algebra L into an MS-algebra L1 is called a homomorphism if xh=xh for all xL. If L and L1 are de Morgan algebras, then h is called a de Morgan homomorphism.

Definition 3

[3] An MS-algebra L is called decomposable MS-algebra if for every xL there exists dD(L) such that x=xd.

Definition 4

[3] A decomposable MS-triple is (M,D,φ), where (i) \((M;\vee,\wedge,\bar {},0,1)\) is a de Morgan algebra, (ii) (D;,,1) is a distributive lattice with 1, (iii) φ is a (0,1)-homomorphism from M into F(D) such that for every element aM and for every yD there exists an element tD with aφ∩[y)=[t).

Theorem 2

[3] (Construction Theorem) Let (M,D,φ) be a decomposable MS-triple. Then

$$L=\left\{(a,\bar{a}\varphi \vee \left[x\right)):a\in M,x\in D\right\} $$

is a decomposable MS-algebra, if we define

$$(a,\bar{a}\varphi \vee \left[x\right))\vee (b,\bar{b}\varphi \vee \left[y\right))=\left(a\vee b,\overline{(a\vee b)}\varphi \vee \left[t\right)\right)\text{for some}t\in D, $$
$$(a,\bar{a}\varphi \vee \left[x\right))\wedge (b,\bar{b}\varphi \vee \left[y\right))=\left(a\wedge b,\overline{(a\wedge b)}\varphi \vee \left[x\wedge y\right)\right), $$
$$\left(a,\bar{a}\varphi \vee \left[x\right)\right)^{\circ }=(\bar{a},a\varphi), $$
$$1_{L}=(1,\left[1\right)), $$
$$0_{L}=(0,D). $$

Conversely, every decomposable MS-algebra L can be associated with the decomposable MS-triple (L,D(L),φ(L)), where

aφ(L)=[a)(L),aL.

The decomposable MS-algebra L constructed in Theorem 2 is called the decomposable MS-algebra associated with the decomposable MS-triple (M,D,φ) and the construction of L described in Theorem 2 is called a decomposable MS-construction.

Corollary 1

[3] Let L be a decomposable MS-algebra associated with the decomposable MS-triple (M,D,φ). Then (1) \(L^{\circ \circ }=\left \{(a,\bar {a}\varphi):a\in M\right \}\), (2) D(L)={(1,[x)):xD}, (3) DD(L) and ML, (4) The order of L is given as follows: \((a,\bar {a}\varphi \vee [x))\leq (b,\bar {b}\varphi \vee [y))\) iff ab and \(\bar {a}\varphi \vee [x)\supseteq \bar {b}\varphi \vee [y)\).

Definition 5

[14] A lattice L is called complete if infLH and supLH exist for each ϕHL.

Definition 6

[14] A lattice L is called conditionally complete if every upper bounded subset of L has a supermum in L and every lower bounded subset of L has an infimum in L.

An MS-algebra L is called complete if it is complete as a lattice.

Definition 7

[14] A lattice homomorphism h:LL1 of a complete lattice L into a complete lattice L1 is called complete if

\((\inf _{L}H)h=\inf _{L_{1}}Hh\) and \((\sup _{L}H)h=\sup _{L_{1}}Hh\) for each ϕHL.

A homomorphism h:LL1 of a complete MS-algebra L into a complete MS-algebra L1 is called complete if it is complete as a lattice homomorphism.

Characterization of complete decomposable MS-algebras via triples

In this section, we introduce and characterize complete decomposable MS-triples of complete decomposable MS-algebras.

Let L be a decomposable MS-algebra L. For ϕNL, define N as follows:

N={n:nN}.

Lemma 2

If L is a complete decomposable MS-algebra, then for ϕNL,ϕCL and ϕED(L), we have (1) (supLN)= infLN, (2) \(\sup _{L^{\circ \circ }}C=(\sup _{L}C)^{\circ \circ }=(\inf _{L} C^{\circ })^{\circ }\), (3) \(\inf _{L^{\circ \circ }} C=\inf _{L}C\), (4) infD(L)E= infLE and supD(L)E= supLE.

Proof

(1). Let x= supLN. Then xn for all nN implies xn. Hence x is a lower bound of N. Let y be a lower bound of N. Then yn for all nN implies ynn. So, y is an upper bound of N. Thus xy as x= supLN. This gives xyy. Therefore x= infLN=(supLN). (2) Let supLC=x. Then x=(supLC). We have to show that \(x^{\circ \circ }=\sup _{L^{\circ \circ }}C\). Since supLC=x, then xc for all cC. so, xc=c for all cC. Therefore x is an upper bound of C. Let y be another upper bound of C in L. Then yc for all cC. Thus yc=c. Hence y is an upper bound of C. So yx as x= supLC. It follows that y=yx. Hence x is the least upper bound of C. Since xL, then \(x^{\circ \circ }=\sup _{L^{\circ \circ }}C\). By (1) we have (supLC)=(infLC). (3) Let x= infLC. Then xc for all cC. Then xc=c. Hence x is a lower bound of C. Thus xx as x= infLC. But xx. Then x=x and xL. Thus \(\inf _{L^{\circ \circ }}C=x\). (4) Let x= infLE and y= infD(L)E. Then xe and ye for all eE imply that x=y. Now we prove supD(L)E= supLE. Let y= supLE. Then ye for all eE. It follows that ye=0. Then yD(L) implies y= supD(L)E. □

Let (M,D,φ) be a decomposable MS-triple. For any ED, consider the set ME as follows:

\(M_{E}=\left \{a\in M : \bar {a}\varphi \vee [z)\supset E \ \text {for some} \ z\in D\right \}\).

Lemma 3

Let (M,D,φ) be a decomposable MS-triple. For any ED, we have (1) ME is an ideal of M, (2) [E)={[t):tE}, (2) ME=M[E).

Proof

(1). Let a,bME. Then \(\bar {a}\varphi \vee [z_{1})\supset E\) and \(\bar {b}\varphi \vee [z_{2})\supset E\) for some z1,z2D. Hence \(E\subset (\bar {a}\varphi \vee [z_{1})) \cap (\bar {b}\varphi \vee [z_{2}))=\overline {(a\vee b)}\varphi \vee [t)\) for some tD (see Theorem 2). It follows that abME. Now, let aME and cM. Then, zD such that \(\bar {a}\varphi \vee [z)\supset E\). Since aca, then \(\overline {a\wedge c}\geq \bar {a}\). This gives \(\overline {(a\wedge c)}\varphi \supseteq \bar {a}\varphi \). It follows that \(\overline {(a\wedge c)}\varphi \vee [z)\supseteq \bar {a}\varphi \vee [z)\supset E\). Then acME. Consequently, ME is an ideal of M. (2) Obvious. (3) Clearly, M[E)ME. Let aME. Then, zD such that \(\bar {a}\varphi \vee [z)\supset E\). Since \(\bar {a}\varphi \vee [z)\) is a filter of D and [E) is the smallest filter of D containing E, then \(\bar {a}\varphi \vee [z)\supset [E)\). Hence, aM[E) and MEM[E). Therefore, ME=M[E). □

Definition 8

A complete decomposable MS-triple is a decomposable MS-triple (M,D,φ) satisfying the following conditions: (i) M is complete, (ii) D is conditionally complete, (iii) For each ED, the set ME has the greatest element in M.

Theorem 3

Let L be a complete decomposable MS-algebra constructed from the decomposable MS-triple (M,D,φ). Then, the triple (M,D,φ) is complete.

Proof

Since L is associated with the decomposable MS-triple (M,D,φ), then by Theorem 2, we have

\(L=\left \{(a,\bar {a}\varphi \vee [x)): a\in M,x\in D\right \}\).

Corollary 1(1)-(3), gives

\(L^{\circ \circ }=\left \{(a,\bar {a}\varphi): a\in M\right \}\cong M\) and D(L)={(1,[x)):xD}D.

We have to prove that a decomposable MS-triple (M,D,φ) is complete. So we proceed to prove (i)–(iii) of Definition 8. For (i), let CM. Consider a subset \(\acute {C}=\{(c,\bar {c}\varphi):c\in C\}\) of L corresponding to C. Since L is complete, then \(\inf _{L}\acute {C}=(a,\bar {a}\varphi \vee [x))\) for some \((a,\bar {a}\varphi \vee [x))\in L\). Thus, \((a,\bar {a}\varphi \vee [x))\leq (c,c\varphi)\) for all cC. Then ac for all cC implies that a is a lower bound of C. We verify that a is the greatest lower bound of C in M. Let b be a lower bound of C. Then bc for all cC. This gives \(\bar {b}\varphi \supseteq \bar {c}\varphi \). Therefore, \((b,\bar {b}\varphi)\leq (c,\bar {c}\varphi)\) for all cC and (b,bφ) is a lower bound of \(\acute {C}\). Then \((a,\bar {a}\varphi \vee [x))\geq (b,b\varphi)\) as \(\inf _{L}C=(a,\bar {a}\varphi \vee [x))\). Consequently, ab and a= infMC. Since a= infMC and M is bounded above by 1, then, M is complete.Now we prove (ii). Let ϕED. Consider \(\acute {E}\subseteq D(L)\) corresponding to E. Then

\(\acute {E}=\left \{(1,[e)): e\in D\right \}\).

Let z be a lower bound of E. Since L is complete, then \(\inf _{L}\acute {E}\) exists. Let \(\inf _{L}\acute {E}=(a,\bar {a}\varphi \vee [x))\). Since ze for all eE as z is a lower bound of E. Then, [z)[e) and (1,[z))≤(1,[e)). Thus, (1,z) is a lower bound of \(\acute {E}\). Then, \((a,\bar {a}\varphi \vee [x))\geq (1,[z))\) because of \(\inf _{L}\acute {E}=(a,\bar {a}\varphi \vee [x))\). This implies that a≥1 and \(\bar {a}\varphi \vee [x)\subseteq [z)\). Consequently, a=1 and \(\bar {a}\varphi \vee [x)=0\varphi \vee [x)=[x)\). Thus [x)[z) implies xz. This shows that x is the greatest lower bound of E in D and x= infDE. Using a similar way, we can show that, if E has an upper bound, then supDE exists. Therefore, D is a conditionally complete lattice as required.

Now we prove (iii). Let ED. Consider \(\acute {E}\subseteq D(L)\) corresponding to E. Then

$$\begin{array}{@{}rcl@{}} \acute{E}=\left\{(1,[x)): x\in E\right\}. \end{array} $$

Since L is complete, then \(\inf _{L} \acute {E}\) exists. Let \((b,\bar {b}\varphi \vee [z))=\inf _{L} \acute {E}\). We show that b is the largest element of ME. Since \((b,\bar {b}\varphi \vee [z))=\inf _{L} \acute {E}\), then \((b,\bar {b}\varphi \vee [z))\leq (1,[x)), \ \forall x\in E\). This gives b≤1 and \(\bar {b}\varphi \vee [z)\supseteq [x), \ \forall x\in E\). Therefore, \(\bar {b}\varphi \vee [z)\supseteq \cup _{x\in E}[x)=[E)\supset E\). Thus, bME. Now, let cME. Then \(\bar {c}\varphi \vee [y)\supset E\) for some yD. It follows that \(\bar {c}\varphi \vee [y)\supseteq [E)\supseteq [x)\) for all xE. Hence, \((1,[x))\leq (c,\bar {c}\varphi \vee [y))\) for all xE. Thus, \((c,\bar {c}\varphi \vee [y))\) is a lower bound of \(\acute {E}\) and therefore \((c,\bar {c}\varphi \vee [y))\leq (b,\bar {b}\varphi \vee [z))\). Then, cb. This deduce that b is the largest element of ME in M. Therefore, (M,D,φ) is a complete decomposable MS-triple. □

The converse of the above theorem is given in the following.

Theorem 4

Let L be a decomposable MS-algebra constructed from the complete decomposable MS-triple (M,D,φ). Then L is complete.

Proof

Let (M,D,φ) be a complete decomposable MS-triple. Then –(iii) of Definition 8 hold. Let NL, where L is constructed as in construction Theorem from the decomposable MS-triple (M,D,φ) as follows:

\(L=\left \{(a,\bar {a}\varphi \vee [x)):a\in M,x\in D\right \}\).

Since L is bounded, it is enough to show the existence of infLN. Denote a= infMN and \(F=\cup \left \{[t): (c,\bar {c}\varphi \vee [t))\in N \text {for some} \ c\in M\right \}\) ( means the union in F(D)). Let b= maxMF. Now, we prove that there exists an element zD such that \(\bar {b}\varphi \vee [z)\supset F\) and if \(\bar {b}\varphi \vee [y)\supset F\) for some yD then \(\bar {b}\varphi \vee [y) \supseteq \bar {b}\varphi \vee [z)\). For this purpose, consider the following set:

\(\left \{x_{\gamma }:\gamma \in \Gamma \text {for all}x_{\gamma }\text {with}\bar {b}\varphi \vee [x_{\gamma })\supset F\right \}\).

Thus, we have to find a zD with \(\bar {b}\varphi \vee [y)\supset F\) and \(\bar {b}\varphi \vee [y)\supseteq \bar {b}\varphi \vee [z)\) for all γΓ. The set \(\left \{x_{\gamma }:\gamma \in \Gamma \text {for all} x_{\gamma }\text {with}\bar {b}\varphi \vee [x_{\gamma })\supset F\right \}\) is bounded from above. Then, by (ii), there exists s= supD{xγ:γΓ}. We prove that ∩γΓ[xγ)=[s).

$$\begin{array}{@{}rcl@{}} y\in\cap_{\gamma\in\Gamma}[x_{\gamma})&\Leftrightarrow& y\in[x_{\gamma}),~ \ \forall\gamma\in\Gamma\\&\Leftrightarrow& y\geq x_{\gamma},~ \ \forall\gamma\in\Gamma\\ &\Leftrightarrow& y \ \text{is an upper bound of} \ \{x_{\gamma}: \gamma\in\Gamma\}\\ &\Leftrightarrow& y\geq s\ \text{as}\ s=\sup_{D}\{x_{\gamma} : \gamma\in\Gamma\}\\&\Leftrightarrow& y\in[s). \end{array} $$

Then it is sufficient to prove the following equality.

$$\begin{array}{@{}rcl@{}} \cap_{\gamma\in\Gamma}(\bar{b}\varphi\vee[x_{\gamma}))=\bar{b}\varphi\vee\cap_{\gamma\in\Gamma}[x_{\gamma})=\bar{b}\varphi\vee[s). \end{array} $$
(1)

Let \(t\in \bar {b}\varphi \vee [s)\). Then

$$\begin{array}{@{}rcl@{}} t\in \bar{b}\varphi\vee[s)&\Rightarrow& t\geq t_{1}\wedge s\ \text{where}\ t_{1}\in \bar{b}\varphi\\ &\Rightarrow& t\geq t_{1}\wedge (s\vee x_{\gamma})\ \text{as}\ s\geq x_{\gamma}\text{for all}\ \gamma\in \Gamma\\ &\Rightarrow&t\geq(t_{1}\wedge s)\vee (t_{1}\wedge x_{\gamma})\\ &\Rightarrow&t\geq t_{1}\wedge x_{\gamma}\\ &\Rightarrow& t\in \bar{b}\varphi\vee[x_{\gamma})\ \text{for all}\ \gamma\in \Gamma. \end{array} $$

Then \(\bar {b}\varphi \vee \cap _{\gamma \in \Gamma }[x_{\gamma })\subseteq \bar {b}\varphi \vee [x_{\gamma })\) implies \(\bar {b}\varphi \vee \cap _{\gamma \in \Gamma }[x_{\gamma })\subseteq \cap _{\gamma \in \Gamma }(\bar {b}\varphi \vee [x_{\gamma }))\). Conversely, let \(y\in \cap _{\gamma \in \Gamma }(\bar {b}\varphi \vee [x_{\gamma }))\). Then \(y\in \bar {b}\varphi \vee [x_{\gamma })\) for all γΓ. Hence ytz for \(t\in \bar {b}\varphi \) and z[xγ) for all γΓ. It follows that zxγ for all γΓ. This means that z is an upper bound of the set {xγ:γΓ}. Then sz as s= supD{xγ:γΓ}. Now

$$\begin{array}{@{}rcl@{}} y&\geq& t\wedge z\\ &=&t\wedge(s\vee z)\ \text{as}\ s\leq z\\ &=&(t\wedge s)\vee(t\wedge z)\ \text{by distributivity of}\ D\\ &\geq&t\wedge s\in \bar{b}\varphi\vee[s). \end{array} $$

Then \(y\in \bar {b}\varphi \vee [s)\). Therefore, \(\cap _{\gamma \in \Gamma }(\bar {b}\varphi \vee [x_{\gamma }))\subseteq \bar {b}\varphi \vee [s)\).

We prove the existence of infLN. First, we claim that

\(i=\left (a\wedge b,\overline {(a\wedge b)}\varphi \vee [z)\right)=\inf _{L} N~(\text {we put then} z=s)\).

First, we show that i is a lower bound of N. Let \((f,\bar {f}\varphi \vee [y))\in N.\) Since a= infMN, we get af. So, abaf. Then abf implies that \(\overline {a\wedge b}\geq \bar {f}\). Consequently, \(\overline {(a\wedge b)}\varphi =\bar {a}\varphi \vee \bar {b}\varphi \supseteq \bar {f}\varphi \). Moreover, \([y)\subseteq F\subseteq \bar {b}\varphi \vee [z)\) as yF. Then

$$\begin{array}{@{}rcl@{}} \overline{(a\wedge b)}\varphi\vee[z)&=&(\bar{a}\vee \bar{b})\varphi\vee[z)\\ &=&(\bar{a}\varphi\vee \bar{b}\varphi)\vee(\bar{b}\varphi\vee[z))\\ &\supseteq&\bar{f}\varphi\vee[y). \end{array} $$

Then \((a\wedge b,\overline {(a\wedge b)}\varphi \vee [z))\leq (f,\bar {f}\vee [y))\) for all \((f,\bar {f}\vee [y))\in N\). Therefore, i is a lower bound of N. It remains to show that i is the greatest lower bound of N. Let \((c,\bar {c}\varphi \vee [x))\) be a lower bound of N. Then, \((c,\bar {c}\varphi \vee [x))\leq (f,\bar {f}\varphi \vee [y)), \ \forall (f,\bar {f}\varphi \vee [y))\in N\). So, cf, fN. Then c is a lower bound of N. Thus ca as \(a=\inf \limits _{M} N^{\circ \circ }\). On the other hand, \(\bar {c}\varphi \vee [x)\supseteq \bar {f}\varphi \vee [y), \ \forall (f,\bar {f}\varphi \vee [y))\in N\). So, \(\bar {c}\varphi \vee [x)\supseteq [y), \ \forall y\in F\). Therefore, \(\bar {c}\varphi \vee [x)\supseteq F\). Hence, \(\bar {c}\varphi \vee [x)\supseteq \bar {b}\varphi \vee [z)\) by using equality (1). Then \(\bar {c}\varphi \vee [x)\supseteq F\) implies that cMF. So, cb as \(b=\max \limits _{M} M_{F}\in M\). Now, we have ca and cb. Then cab. Moreover, we have \(\bar {c}\varphi \supseteq \bar {a}\varphi \) because of ca. Also, \(\bar {c}\varphi \vee [x)\supseteq \bar {b}\varphi \vee [z)\). So, \(\bar {c}\varphi \vee [x)\supseteq \bar {a}\varphi \vee \bar {b}\varphi \vee [z)=\overline {(a\wedge b)}\varphi \vee [z)\). Therefore, \((c,\bar {c}\varphi \vee [x))\leq i\). Then i= infLN and L is complete. □

Corollary 2

If M and D are complete, then so is L.

Proof

. We need only to prove that the condition (iii) of Definition 8 holds. Let ED and t= infDE. Then, [t)=[ infDE)E. So, \((1,\bar {1}\varphi \vee [t))=(1,[t))\in L\). Therefore, 1ME. Hence, by the above Theorem, L is complete. □

Corollary 3

If M is finite and D is conditionally complete, then L is complete.

Proof

Since M is finite and ME is an ideal of M (see Lemma 1(1)), then M is complete and ME is a principal ideal of M. Therefore, ME contains the greatest element in M. So, the conditions (i)– (iii) of Definition 8 are satisfied and consequently, L is complete. □

Combining Theorems 3 and 4, we get the following theorem.

Theorem 5

Let L be a decomposable MS-algebra constructed from the decomposable MS-triple (M,D,φ). Then L is complete if and only if (M,D,φ) is complete.

Let L be a complete decomposable MS-algebra. In the proof of Theorem 4 arbitrary meets in L are described. In the following Lemma, we describe joins in L.

Lemma 4

Let L be a complete decomposable MS-algebra constructed from the decomposable MS-triple (M,D,φ). Let ϕNL and a= supMN. Then there exists an element zD such that \([z)=\bigcap \left \{\bar {c}\varphi \vee [t):(c,\bar {c}\varphi \vee [t))\in N\right \}\cap a\varphi \) and \(\sup N=(a,\bar {a}\varphi \vee [z))\).

Proof

Let ϕNL and \(\sup _{L} N=(b,\bar {b}\varphi \vee [z))\). We can assume that zaφ. We prove that b=a= supMN. Using Lemma 2(2), we get

\(\sup _{M}N^{\circ \circ }=(\sup _{L}N)^{\circ \circ }=(b,\bar {b}\varphi \vee [z))^{\circ \circ }=(b,\bar {b}\varphi)\).

But \(a=(a,\bar {a}\varphi)=\sup _{M}N^{\circ \circ }\). Then b=a. Hence, \(\bar {a}\varphi \vee [z)\) is the greatest filter of the form \(\bar {a}\varphi \vee [x), x\in D\) with

\(\bar {a}\varphi \vee [z))\subset \bar {c}\varphi \vee [t)\) for each \((c,\bar {c}\varphi \vee [t))\in N\).

The last condition is equivalent to

\([z)\subset \bigcap \left \{\bar {c}\varphi \vee [t):(c,\bar {c}\varphi \vee [t))\in N\right \}\cap a\varphi \).

Let \( \bigcap \left \{\bar {c}\varphi \vee [t):(c,\bar {c}\varphi \vee [t))\in N\right \}\cap a\varphi =R\). If [z)≠R, then there is yR,y̸z. It follows that yz<z and yzR. Then [z)[yz) implies \(\bar {a}\varphi \vee [z)\subset \bar {a}\varphi \vee [y\wedge z)\). Since yzR then \([y\wedge z)\subset \bar {c}\varphi \vee [t)\) for all \((c,\bar {c}\varphi \vee [t))\in N\). Since ac (as a= supMN) then \(\bar {a}\leq \bar {c}\). It follows that \(\bar {a}\varphi \leq \bar {c}\varphi \). Therefore, \(\bar {a}\varphi \vee [y\wedge z)\subset \bar {c}\varphi \vee [t)\) for all \((c,\bar {c}\varphi \vee [t))\in N\). Consequently,

\(\bar {a}\varphi \vee [z)\subset \bar {a}\varphi \vee [y\wedge z)\subset \bar {c}\varphi \vee [t)\) for all \((c,\bar {c}\varphi \vee [t))\in N\),

which contradicts the maximality of \(\bar {a}\varphi \vee [z)\). □

Complete homomorphisms via complete triple homomorphisms

In this section, we introduce complete triple homomorphisms of complete decomposable MS-algebras. Then, we characterize complete homomorphisms of complete decomposable MS-algebras in terms of complete triple homomorphisms. For this purpose, we recall from [4], the notion of triple homomorphism of decomposable MS-triples and related properties which will be used in rest of the paper.

Definition 9

[4] Let (M,D,φ) and (M1,D1,φ1) be decomposable MS-triples. A triple homomorphism of the triple (M,D,φ) into (M1,D1,φ1) is a pair (f,g), where f is a homomorphism of M into M1,g is a homomorphism of D into D1 preserving 1 such that for every aM,

$$ a\varphi g\subseteq af\varphi_{1} $$
(2)

Lemma 5

[4] Let (f,g) be a triple homomorphism of a decomposable MS-triple (M,D,φ) into a decomposable MS-triple (M1,D1,φ1). Let a,bM and x,y,tD. Then (i) aφ∩[y)=[t) implies afφ1∩[yg)=[tg), (ii) \(\left (\bar {a}f\varphi _{1}\vee \lbrack xg\right))\cap (\bar {b}f\varphi _{1}\vee \lbrack yg))=\overline {(a\vee b)}f\varphi _{1}\vee \lbrack tg).\)

Theorem 6

[4] Let L and L1 be decomposable MS-algebras, (M,D,φ) and (M1,D1,φ1) be the associated decomposable MS-triples, respectively. Let h be a homomorphism of L into L1 and hM,hD the restrictions of h to M and D, respectively. Then (hM,hD) is a triple homomorphism of the decomposable MS-triples. Conversely, every triple homomorphism (f,g) of the decomposable MS-triples uniquely determines a homomorphism h of L into L1 with hM=f,hD=g by the following rule:

$$ xh=x^{\circ \circ }f\wedge dg,\text{for all}\ x\in L, $$
(3)

where x=xd for some dD(L).

If L and L1 are represented as in the construction Theorem then (3) reads

$$ (a,\bar{a}\varphi\vee[x))h=(af,\overline{(af)}\varphi\vee[xg))\ \text{for all}\ (a,\bar{a}\varphi\vee[x))\in L. $$
(4)

In the following, we will write L=(M,D,φ) to indicate that (M,D,φ) is the decomposable MS-triple associated with L, that is, L=M,D(L)=D, and φ(L)=φ. Let L=(M,D,φ) and L1=(M1,D1,φ1) be decomposable MS-algebras, we will write h=(f,g) to indicate that (f,g):(M,D,φ)→(M1,D1,φ1) is the triple homomorphism of decomposable MS-triples corresponding to the homomorphism h of L into L1.

Lemma 6

Let h=(f,g) be a homomorphism of a decomposable MS-algebra L onto a decomposable MS-algebra L1. Then for each aL, we have

aφg=afφ1.

Proof

We have, aφgafφ1 by (2). It remains to show that afφ1aφg. Let yafφ1. Then

y[(af))∩D(L1)=[(ah))∩D(L1) implies y[(ah)) and yD(L1).

Then y≥(ah)=ah. Since h is onto, then g:D(L)→D(L1) is also onto. Hence, there exists xD(L) such that xh=y. Evidently, ax[a)∩D(L) and

(ax)h=ahxh=xh as xh=yah.

Therefore, y[ah)∩D(L1)=([a)hDg)=([a)∩D)g=aφg. □

Now, we introduce the concept of complete triple homomorphism.

Definition 10

A triple homomorphism (f,g) of a decomposable MS-triple (M,D,φ) into a decomposable MS-triple (M1,D1,φ1) is called complete if the following conditions are satisfied (i) f is a complete homomorphism of M and M1, (ii) g is a complete homomorphism of D and D1, (iii) (maxME)f= maxM1Eg for each ϕED.

Remark 1

First, we observe that the map g:DD1 is a complete means that \((\sup _{D}E)g=\sup _{D_{1}}Eg\) for any ED and if infDE and \(\inf _{D_{1}}Mg\) exist then \((\inf _{D}E)g=\inf _{D_{1}}Eg\).

Theorem 7

Let L=(M,D,φ) and L1=(M1,D1,φ1) be complete decomposable MS-algebras and let h=(f,g) be a homomorphism of L onto L1. Then h is complete if and only if (f,g) is complete.

Proof

The decomposable MS-triples (M,D,φ) and (M1,D1,φ1) are associated with L and L1, respectively. Let h=(f,g) be a complete homomorphism of L onto L1. Then f is a de Morgan homomorphism of M onto M1 and g is a lattice homomorphism of D onto D1 preserving 1. We have to verify that f and g are complete. Let ϕNM. Then

$$\begin{array}{@{}rcl@{}} \left(\inf_{M}N\right)f=\left(\inf_{L}N\right)f=\left(\inf_{L}N\right)h=\inf_{L_{1}}Nh=\inf_{L_{1}}Nf=\inf_{M_{1}}Nf\ \text{by Lemma 2(3)},\\ \left(\sup_{M}N\right)f=\left(\sup_{L}N\right)^{\circ\circ}f=\left(\left(\sup_{L}N\right)h\right)^{\circ\circ}=\left(\sup_{L_{1}}Nh\right)^{\circ\circ}=\sup_{M_{1}}Nf\ \text{by Lemma 2(2)}. \end{array} $$

Thus, f is complete. We prove that g is complete. Let ϕED. Then

\((\sup _{D}E)g=(\sup _{L}E)g=(\sup _{L}N)h=\sup _{L_{1}}Nh=\sup _{D_{1}}Eg\ \text {by Lemma 2(4)}.\)

If infDE and \(\inf _{D_{1}}Eg\) exist, then

\((inf_{D}E)g=(\inf _{L}E)g=(\inf _{L}N)h=\inf _{L_{1}}Nh=\inf _{D_{1}}Eg\ \text {by Lemma 2(4)}.\)

Now, we prove (iii). Let ϕED. Consider E corresponding the set \(\acute {E}\) on D(L), where

\(\acute {E}=\left \{(1,[x)):x\in E\right \}\subseteq D(L)\).

By (4), we have

\(\acute {E}h=\{(1,[xg)):x\in E\}\subseteq D(L_{1})\).

Since h is complete, then \((\inf _{L}E)h=\inf _{L_{1}}Eh\) for each ϕEL. Hence, (infLE)= maxME (see the proof of Theorem 3) and similarly \((\inf _{L_{1}}Eh)^{\circ \circ }=\max M_{1Eg}\). Conversely, assume that (i)–(iii) hold and let h=(f,g) be a homomorphism of L onto L1. We have to show that h is complete. First we prove that for \(\phi \not = H\subseteq L, (\inf _{L}H)h=\inf _{L_{1}}Hh\) holds. Consider \(E=\bigcup \left \{[t):(c,\bar {c}\varphi \vee [x))\in M\right \}\). Let maxME=b and infMH=a. Then according to the proof of Theorem 4, we get

\(i=\left (a\wedge b,\overline {(a\wedge b)}\varphi \vee [z)\right)=\inf _{L}H\), where \(z=\sup _{D}\left \{x_{\gamma }:\bar {b}\varphi \vee [x_{\gamma })\supset E\right \}\). Using (4), we have

\(Hh=\left \{(cf,\bar {cf}\varphi \vee [xg)):(c,\bar {c}\varphi \vee [x))\in H\right \}\),

and

\(ih=\left ((a\wedge b)f,\overline {(a\wedge b)f}\varphi \vee [zg)\right)=(\inf _{L}H)h\).

Now, \(\inf _{L_{1}}(Hf)^{\circ \circ }=(\inf _{M}H^{\circ \circ })f= af\) by (i) and maxM1Eg=(maxME)f=bf by (iii). Since L1 is complete and HhL1 then again according to the proof of Theorem 4, we get

\(\inf _{L_{1}}Hh=\left ((a\wedge b)f,\overline {(a\wedge b)f}\varphi \vee [z_{1})\right)=ih\), where z1= sup{xγg:γΓ}=(sup{xγ:γΓ})g=zg as g is an onto homomorphism. Therefore, \(\inf _{L} Mh=(\inf _{L_{1}}M)h\).

Now, we prove that \((\sup _{L}H)h=\sup _{L_{1}}Hh\). By Lemma 4, \(\sup _{L}(M)=(a,\bar {a}\varphi \vee [z))\), where a= supMH and \([z)=\bigcap \left \{\bar {c}\varphi \vee [t):(c,\bar {c}\varphi \vee [t))\in H\right \}\cap a\varphi \). Then \(\sup _{L_{1}}Hh=(a_{1},\bar {a_{1}}\varphi _{1}\vee [z_{1}))\), where \(a_{1}=\sup _{M_{1}}(Hh)^{\circ \circ }=\sup _{L_{1}}(Hh)^{\circ \circ }=\sup _{L_{1}}H^{\circ \circ }h=(\sup _{L}M^{\circ \circ })h=(\sup _{M}H^{\circ \circ })h=ah=af\)(by using Lemma 2(2) and (i) of Definition 9) and \([z_{1})=\bigcap \left \{\bar {cf}\varphi _{1}\vee [tg):(c,\bar {c}\varphi \vee [t))\in H\right \}\cap a_{1}\varphi _{1}\). We show that zg=z1. We have cfφ1=cφg by Lemma 6 and \(\bar {c}\varphi g\vee [tg)=(\bar {c}\varphi \vee [t))g\) by Lemma 5(1). Then

$$\begin{array}{@{}rcl@{}} [z_{1})&=&\bigcap\{\bar{c}\varphi\vee[t))g:(c,\bar{c}\varphi\vee[t))\in H\}\cap a\varphi g\\ &=&\left(\bigcap\left\{\bar{c}\varphi\vee[t):(c,\bar{c}\varphi\vee[t))\in H\right\}\cap a\varphi\right)g\\ &=&[zg) \end{array} $$

which implies z1=zg. Therefore, \((\sup _{L}H)h=\sup _{L_{1}}Hh\) and h is complete. □

Availability of data and materials

Not applicable.

References

  1. 1

    Blyth, T. S., Varlet, J. C.: On a common abstraction of de Morgan algebras and Stone algebras. Proc. Roy. Soc. Edinburgh. 94A, 301–308 (1983).

    MathSciNet  Article  Google Scholar 

  2. 2

    Blyth, T. S., Varlet, J. C.: Subvarieties of the class of MS-algebras. Proc. Roy. Soc. Edinburgh. 95A, 157–169 (1983).

    MathSciNet  Article  Google Scholar 

  3. 3

    Badawy, A., Guffova, D., Haviar, M.: Triple construction of decomposable MS-algebras. Acta Univ. Palacki. Olomuc. Fac. Rer. Nat. Math. 51(2), 53–65 (2012).

    MathSciNet  MATH  Google Scholar 

  4. 4

    Badawy, A., El-Fawal, R.: Homomorphism and Subalgebras of decomposable MS-algebras. J. Egypt. Math. Soc. 25, 119–124 (2017).

    MathSciNet  Article  Google Scholar 

  5. 5

    Badawy, A.: d L-Filters of principal MS-algebras. J. Egypt. Math. Soc. 23, 463–469 (2015).

    MathSciNet  Article  Google Scholar 

  6. 6

    Badawy, A.: Congruences and de Morgan filters of decomposable MS-algebras. SE Asia. Bull. Math. 43, 13–25 (2019).

    Google Scholar 

  7. 7

    Badawy, A., Sambasiva Rao, M.: Closure ideals of MS-algebras. Chamchuri J. Math. 6, 31–46 (2014).

    MathSciNet  Google Scholar 

  8. 8

    Badawy, A.: δ-ideals in MS-algebras. J. Comput. Sci. Syst. Biol. 9(2), 28–32 (2016).

    MathSciNet  Google Scholar 

  9. 9

    Chen, C. C., Grätzer, G.: Stone lattices II. Structure Theorems. Can. J. Math. 21, 895–903 (1969).

    MathSciNet  Article  Google Scholar 

  10. 10

    El-Assar, S., Atallah, M.: Completeness of distributive p-algebras. Qater Univ. Sci. Bull. 8, 29–33 (1988).

    MathSciNet  MATH  Google Scholar 

  11. 11

    Mederly, P.: A characterization of complete modular p-algebras. Colloq. Math. Soci. János Bolyai. 17, 211–329 (1975). Contributions to Universal Algebra (Hungary).

    MathSciNet  Google Scholar 

  12. 12

    Blyth, T. S., Varlet, J. C.: Ockham Algebras. Oxford University Press, Oxford (1994).

    Google Scholar 

  13. 13

    Blyth, T. S.: Lattices and ordered algebraic structures. Springer-Verlag, London (2005).

    Google Scholar 

  14. 14

    Grätzer, G.: Lattice theory, first concepts and distributive lattices, Lecture Notes. Freeman, San Francisco (1971).

    Google Scholar 

  15. 15

    Badawy, A.: Extensions of the Glivenko-type congruences on a Stone lattice. Math. Meth. Appl. Sci. 41, 5719—5732 (2018).

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

We thank the referees for valuable comments and suggestions for improving the paper.

Funding

No fund.

Author information

Affiliations

Authors

Contributions

The manuscript is being submitted by me (Corresponding author) on behalf of all the authors. The manuscript is the original work of all authors. All authors made a significant contribution to this study. All authors have read and approved the final version of the manuscript.

Corresponding author

Correspondence to Abd El-Mohsen Badawy.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Badawy, A., Gaber, A. Complete decomposable MS-algebras. J Egypt Math Soc 27, 23 (2019). https://doi.org/10.1186/s42787-019-0027-8

Download citation

Keywords

  • MS-algebras
  • Complete lattice
  • Complete decomposable MS-algebras
  • Complete decomposable MS-triples
  • Triple homomorphisms
  • Complete homomorphisms

AMS Mathematics Subject Classification (2010)

  • Primary 06D30
  • Secondary 06D15.