AMALGAMATION PROPERTY IN QUASI-MODAL ALGEBRAS

SERGIO ARTURO CELANI

Abstract. In this paper we will give suitable notions of Amalgamation and Super-amalgamation properties for the class of quasi-modal algebras intro-duced by the author in his paper Quasi-Modal algebras.

1. Introduction and preliminaries

The class of quasi-modal algebras was introduced by the author in [2], as a generalization of the class of modal algebras. A quasi-modal algebra is a Boolean algebraAendowed with a map ∆ that sends each elementa∈Ato an ideal ∆aof

A, and satisfies analogous conditions to the modal operatorof modal algebras
[3]. This type of maps, called *quasi-modal operators, are not operations on the*
Boolean algebra, but have some similar properties to modal operators.

It is known that some varieties of modal algebras have the Amalgamation Prop-erty (AP) and Superamalgamation PropProp-erty (SAP). These properties are connected with the Interpolation property in modal logic (see [3]). The aim of this paper is to introduce a generalization of these notions for the class of quasi-modal algebras, topological quasi-modal algebras, and monadic quasi-modal algebras.

We recall some concepts needed for the representation for quasi-modal algebras. For more details see [2] and [1].

Let A = hA,∨,∧,¬,0,1i be a Boolean algebra. The set of all ultrafilters is denoted by Ul (A).The ideal (filter) generated inAby some subsetX⊆A,will be denoted byIA(X),(FA(X)). The complement of a subsetY ⊆Awill be denoted

byYc _{or} _{A}_{−}_{Y.}_{The lattice of ideals (filters) of}_{A}_{is denoted by} _{Id}_{(}_{A}_{) (}_{F i}_{(}_{A}_{)).}

Definition 1. LetAbe a Boolean algebra. A*quasi-modal operator* defined onA

is a function ∆ :A→Id(A) that verifies the following conditions for alla, b∈A:
*Q1* ∆ (a∧b) = ∆a∩∆b,

*Q2* ∆1 =A.

A*quasi-modal algebra, or*qm-algebra, is a pairhA,∆iwhereAis a Boolean algebra
and ∆ is a quasi-modal operator.

In every quasi-modal algebra we can define the dual operator ∇ : A→ F i(A) by∇a=¬∆¬a,where ¬∆x={¬y|y∈∆x}.It is easy to see that the operator

∇satisfies the conditions

2000 *Mathematics Subject Classification.* 06E25, 03G25.

(1) ∇(a∨b) =∇a∩ ∇b, and (2) ∇0 =A,

for alla, b∈A(see [2]). The class ofqm-algebras is denoted byQMA. LetAbe aqm-algebra. For eachP∈Ul (A) we define the set

∆−1_{(}_{P}_{) =}_{{}_{a}_{∈}_{A}_{|}_{∆}_{a}_{∩}_{P}_{6}_{=}_{∅}}_{.}
Lemma 2. [2] *Let*A∈ QMA*.*

(1) *For each* P∈Ul (A),∆−1_{(}_{P}_{)}_{∈}_{F i}_{(}_{A}_{)}_{,}

(2) a∈∆−1_{(}_{P}_{)}_{iff for all}_{Q}_{∈}_{Ul (}_{A}_{), if}_{∆}−1_{(}_{P}_{)}_{⊆}_{Q} _{then}_{a}_{∈}_{Q.}
LetA∈ QMA. We define on Ul (A) a binary relationRA by

(P, Q)∈RA ⇔ ∀a∈A: if ∆a∩P 6=∅, thena∈Q
⇔ ∆−1_{(}_{P}_{)}_{⊆}_{Q.}

Throughout this paper, we will frequently work with Boolean subalgebras of a given Boolean algebra. In order to avoid any confusion, if A ∈ QMA, then we will use the symbol ∆A to denote the corresponding operation of A. A function h:A→Bis an homomorphism of quasi-modal algebras, or aq-homomorphism, if

his an homomorphism of Boolean algebras, and

IB(h(∆Aa)) = ∆B(h(a)),

for anya∈A.A*quasi-isomorphism* is a Boolean isomorphism that is a
q-homomor-phism.

LetA∈ QMA. Let us consider the relational structure

F(A) =hUl (A), RAi.

From the result given in [2] it follows that the Boolean algebraP(Ul(A)) endowed with the operator

∆RA:P(Ul(A))→Id(P(Ul(A)))

defined by

∆RA(O) =I(∆RA(O)) ={U ∈ P(Ul(A))|U ⊆∆RA(O)},

is a quasi-modal algebra. Moreover, the mapβ :A→ P(Ul(A)) defined byβ(a) =

{P∈Ul(A)|a∈P}, for eacha∈A, is aq-homomorphism,i.e. IP(Ul(A))(β(∆a)) =

∆RA(β(a)), for eacha∈A.

Let A ∈ QMA. Let X ⊆ A. Define the ideal ∆X = I( S

x∈X

∆x) and the

filter ∇X = F S

x∈X∇x

. For a ∈ A we define recursively ∆0_{a} _{=} _{I}_{(}_{a}_{)}_{,} _{and}
∆n+1_{a}_{= ∆ (∆}n

a).

Theorem 3. [2] *Let* A ∈ QMA*. Then for all* a∈A, *the following equivalences*
*hold:*

1. ∆a⊆I(a)⇔RA *is reflexive.*

2. ∆a⊆∆2_{a}_{⇔}_{R}2

A⊆RA,*i.e.,* RA *is transitive.*

3. I(a)⊆∆∇a= T

x∈∇a

LetA∈ QMA.We shall say thatAis a topological*quasi-modal* *algebra* if for
every a ∈ A, ∆a ⊆ I(a) and ∆a ⊆ ∆2_{a.} _{We shall say that} _{A} _{is a monadic}
*quasi-modal* *algebra* if it is a quasi-topological algebra and I(a) ⊆ ∆∇a for
ev-ery a ∈ A. From the previous Theorem we get that a quasi-modal algebra A is
quasi-topological algebra iff the relationRAis reflexive and transitive. Similarly, a

quasi-modal algebraAis a monadic quasi-modal algebra iff the relationRA is an

equivalence.

LetAandBbe two*qm-algebras. We shall say that the structure*B=hB,∨,∧,

∆B,0,1i is a *quasi-modal subalgebra* of A, or *qm-subalgebra* for short, if B is a

Boolean subalgebra ofA,and for anya∈B,

IA(∆B(a)) = ∆A(a).

The following result is given in [1] for Quasi-modal lattices.

Lemma 4. *Let* A*be a qm-algebra. Let*B *be a Boolean subalgebra of*A*. Then the*
*following conditions are equivalent*

(1) *There is a quasi-modal operator,* ∆B : B → Id(B), *such that* B =
hB,∨,∧,∆B,0,1i*is a*qm*-subalgebra of*A.

(2) *For any* a∈B*,*∆B(a)*is defined to be* IA(∆A(a)∩B).

Remark 5. LetAbe aqm-algebra. LetBbe a Boolean subalgebra ofA. If there is a quasi-modal operator ∆B:B →Id(B),such thatB=hB,∨,∧,∆B,0,1iis a qm-subalgebra ofA, then ∆B is unique. The proof of this fact is as follows: First,

we note that ifH is an ideal ofB,andGis a filter ofB such thatH∩G=∅, then

IA(H)∩FA(G) =∅.We suppose now that there exist two quasi-modal operators

∆1 and ∆2 in B such that hB,∨,∧,∆1,0,1i and hB,∨,∧,∆2,0,1i are two qm -subalgebras ofA. Then

∆A(a) =IA(∆1(a)) =IA(∆2(a)).

If ∆1(a) ∆2(a), there exists b∈B such thatb ∈∆1(a) and b /∈∆2(a). Then ∆A(a)∩FA(b) =IA(∆1(a))∩FA(b)=6 ∅and ∆A(a)∩FA(b) =IA(∆2(a))∩FA(b) = ∅,which is a contradiction.

2. Amalgamation Property

It is known that the variety of modal algebras has the Amalgamation Property (AP) and the Superamalgamation Property (SAP) (see [3] for these properties and the connection with the Interpolation property in modal logic). In this section we shall give a generalization of these notions and prove that the classQMAhas these properties.

Definition 6. LetQbe a class of quasi-modal algebras. We shall say thatQ has
the*AP* if for any triple A0, A1, A2 ∈ Q and injective quasi-homomorphisms f1 :

A0→A1andf2:A0→A2there existsB ∈ Qand injective quasi-homomorphisms

We shall say thatQhas the*SAP* ifQhas the AP and in addition the mapsg1and

g2 above have the following property:

For all (a, b)∈A1×A2 such that g1(a)≤g2(b) there existsc ∈A0 such that

a≤f1(c) andf2(c)≤b.

Let Q be a class of quasi-modal algebras. Without losing generality, we can assume that in the above definitionA0 is aqm-subalgebra ofA1 andA2,i.e., that

f1 andf2 are the inclusion maps.

Theorem 7. *The class*QMA *has the AP.*

*Proof.* LetA0, A1, A2∈ QMAsuchA0is aqm-subalgebra ofA1and A2. Let us
consider the relational structures F(Ai) = hUl (Ai), RAi, β(Ai)i, 1 ≤i ≤2. We

shall define the set

X={(P, Q)∈Ul (A1)×Ul (A2)|P∩A0=Q∩A0}, and the binary relationRin X as follows:

(P1, Q1)R(P2, Q2) iff (P1, P2)∈RA1 and (Q1, Q2)∈RA2. Let us consider the quasi-modal algebra A(X) =

P(X),∪,c ,∆¯R,∅

where the operator

¯

∆R:P(X)→Id(P(X))

is defined by ¯

∆R(G) =I(∆R(G)) ={U ∈ P(X)|U ⊆∆R(G)}.

We note that in this caseI(∆R) = ∆R.

Let us define the mapsg1:A1→ A(X) andg2:A2→ A(X) by

g1(a) ={(P, Q)∈X :a∈P}andg1(b) ={(P, Q)∈X |b∈Q}, respectively. We prove thatg1andg2are injective q-homomorpshims.

First of all let us check the following property:

∀P ∈Ul (A1) ∃Q∈Ul (A2) such that (P, Q)∈X. (1) Let P ∈ Ul (A1). Since P ∩A0 ∈ Ul (A0) and A0 is a subalgebra of A2, we get by known results on Boolean algebras that there exists Q ∈ Ul (A2) such that

P∩A0=Q∩A0, i.e., (P, Q)∈X.

To see that g1 is injective, let a, b ∈ A1 such that a b. Then there exists

P ∈Ul (A1) such thata∈P andb /∈P.By (1), there existsQ∈Ul (A2) such that (P, Q)∈X. So, (P, Q)∈g1(a) and (P, Q)∈/ g1(b),i.e., g1(a) g1(b).Thus, g1 is injective. It is clear thatg1is a Boolean homomorphism.

We prove next that ∆R(g1(a))⊆g1(∆a), for anya∈A1.

Let (P, Q) ∈ X. Suppose that (P, Q) ∈/ g1(∆A1a), i.e., ∆A1a∩P = ∅. From Lemma 2 there exists P1 ∈ Ul (A1) such that (P, P1) ∈ RA1 and a /∈ P1. By property (1), there exists Q1 ∈Ul (A2) such that P1∩A0 =Q1∩A0. From the inclusion ∆−A11(P)⊆P1,it is easy to see that ∆

−1

We prove that there existsQ2∈Ul (A2) such thatP1∩A0=Q2∩A0and ∆−A12(Q)⊆

Q2.Let us consider the filter

F =FA2 ∆

−1

A2(Q)∪(P1∩A0)

.

The filterF is proper, because otherwise there existq∈∆−A12(Q) andd∈P1∩A0 such that q∧d= 0. So, q ≤ ¬d, and since ∆A2 is increasing, we get ∆A2(q) ⊆ ∆A2(¬d).AsA0 is a quasi-subalgebra ofA2 andd∈A0,we get

∆A0(¬d) =IA2(∆A2(¬d)∩A0). Thus,

IA2(∆A2(¬d)∩A0)∩Q6=∅.

So, there existsx∈Qandy∈∆A2(¬d)∩A0such thatx≤y.Sincey∈Q∩A0=

P∩A0, y∈∆A2(¬d)∩A0∩P.Then we have ¬d∈P1, which is a contradiction. Therefore, there existsQ2∈Ul (A2) such that

P1∩A0=Q2∩A0 and ∆−A21(Q)⊆Q2.

So, (P, Q) ∈/ ∆Rg1(a). So, we have the inclusion ∆Rg1(a) ⊆ g1(∆A1a). The inclusiong1(∆A1a)⊆∆Rg1(a) is easy and left to the reader. Thus,g1(∆A1a) = ∆Rg1(a), and consequently we get

I(g1(∆A1a)) =I(∆Rg1(a)), i.e.,g1 is an injective q-homomorphism.

Similarly we can prove that g2 is an injective q-homomorphism. Moreover, it is easy to check that for allc∈A0, g1(c) =g2(c), andg1(∆A0c) =g2(∆A0c). Thus,

QMAhas the AP.

Lemma 8. *Let* A0, A1 *and* A2 ∈ QMA *such that* A0 *is a subalgebra of* A1 *and*

A2. *Let* a ∈A1, b ∈A2 *and let us suppose that there exists no* c ∈A0 *such that*

a≤c *or*c≤b*. Then there exist* P∈Ul (A1), Q∈Ul (A2)*such that*

a∈P, b /∈Q*and*P∩A0=Q∩A0.

*Proof.* Let us consider the filter FA1(a) in A1 and the filterFA2(FA1(a)∩A0) in

A2. We note that

b /∈FA2(FA1(a)∩A0),

because in otherwise there existsc ∈A0 such that a ≤1 c ≤2 b, which is a con-tradiction. Then by the Ultrafilter theorem, there exists Q ∈ Ul (A2) that such that

FA1(a)∩A0⊂FA2(FA1(a)∩A0)⊆Q andb /∈Q.

Let us consider in A1 the filter F = FA1({a} ∪(Q∩A0)) and the ideal I =

IA1((A2−Q)∩A0). We prove that

F∩I=∅.

Suppose that there exist elementsx∈A1,y∈Q∩A0, andz∈(A2−Q)∩A0such thata∧y≤1x≤1z. This implies that

Thus, we get z ∈Q, which is a contradiction. So, there exists P ∈ Ul (A1) such that

a∈P, Q∩A0=P∩A0 andb /∈Q.

Theorem 9. *The class*QMA *has the SAP.*

*Proof.* The proof of the SAP is actually analogous to the previous one. Let A0,

A1, A2∈ QMAsuch thatA0 is a qm-subalgebra ofA1 and of A2. Let us consider the setX and the quasi-modal algebraA(X) of the proof above.

Let (a, b)∈A1×A2. Suppose that there exists noc ∈ A0 such thata ≤c or

c ≤b. By Lemma 8 there exists P ∈ Ul (A1) and there exists Q∈ Ul (A2) such that

a∈P, b /∈Q andP∩A0=Q∩A0,

i.e., (P, Q)∈g1(a) and (P, Q)∈/ g2(a). Thus, g1(a)*g2(b). So,QMA has the

SAP.

The results above can be applied to prove that other classes of quasi-modal alge-bras have the AP and SAP. For example, ifT QMAis the class of the topological quasi-modal algebras, then in the proof of Theorem 7 the binary relation R de-fined on the setX is reflexive and transitive. Consequently,A(X) is a topological quasi-modal algebra and thus the class T QMA has the AP and the SAP. Sim-ilar considerations can be applied to the class MQMA of monadic quasi-modal algebras.

Acknowledgement

I would like to thank the referee for his observations and suggestions which have contributed to improve this paper.

References

[1] Castro, J. and Celani, S., Quasi-modal lattices, Order 21 (2004), 107-129.

[2] Celani, S. A., Quasi-Modal algebras, Mathematica Bohemica Vol. 126, No. 4 (2001), 721-736. [3] Kracht.M., Tools and Techniques in Modal Logic, Studies in Logic and the Foundations of

Mathematics, Vol. 142, Elsevier.

*Sergio Arturo Celani*

CONICET and Departamento de Matem´aticas Universidad Nacional del Centro

Pinto 399, 7000 Tandil, Argentina

scelani@exa.unicen.edu.ar