# Amalgamation property in quasi modal algebras

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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 , 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 . 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 ). 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  and .

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 AY.The lattice of ideals (filters) ofAis denoted by Id(A) (F i(A)).

Definition 1. LetAbe a Boolean algebra. Aquasi-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.

Aquasi-modal algebra, orqm-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.

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(1) ∇(a∨b) =∇a∩ ∇b, and (2) ∇0 =A,

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

∆−1(P) ={aA|aP6=∅}. Lemma 2.  LetA∈ QMA.

(1) For each P∈Ul (A),∆−1(P)F i(A),

(2) a∈∆−1(P)iff for allQUl (A), if−1(P)Q thenaQ. 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.Aquasi-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  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 ∆0a = I(a), and ∆n+1a= ∆ (∆n

a).

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

1. ∆a⊆I(a)⇔RA is reflexive.

2. ∆a⊆∆2aR2

A⊆RA,i.e., RA is transitive.

3. I(a)⊆∆∇a= T

x∈∇a

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LetA∈ QMA.We shall say thatAis a topologicalquasi-modal algebra if for every a ∈ A, ∆a ⊆ I(a) and ∆a ⊆ ∆2a. 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 twoqm-algebras. We shall say that the structureB=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  for Quasi-modal lattices.

Lemma 4. Let Abe a qm-algebra. LetB be a Boolean subalgebra ofA. Then the following conditions are equivalent

(1) There is a quasi-modal operator, ∆B : B → Id(B), such that B = hB,∨,∧,∆B,0,1iis aqm-subalgebra ofA.

(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  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 theAP if for any triple A0, A1, A2 ∈ Q and injective quasi-homomorphisms f1 :

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

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We shall say thatQhas theSAP 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 classQMA 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

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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 orc≤b. Then there exist P∈Ul (A1), Q∈Ul (A2)such that

a∈P, b /∈QandP∩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

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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 classQMA 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

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

 Celani, S. A., Quasi-Modal algebras, Mathematica Bohemica Vol. 126, No. 4 (2001), 721-736.  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

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