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### Rev. Unión Mat. Argent. vol.50 no.1 Bahía Blanca June 2009

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 introduced by the author in his paper Quasi-Modal algebras.

2000 Mathematics Subject Classification. 06E25, 03G25.

Key words and phrases. Boolean algebras; Quasi-modal algebras; Amalgamation and Super-amalgamation properties.

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 algebra endowed with a map that sends each element to an ideal of , and satisfies analogous conditions to the modal operator of 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 Property (AP) and Superamalgamation Property (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 be a Boolean algebra. The set of all ultrafilters is denoted by The ideal (filter) generated in by some subset will be denoted by . The complement of a subset will be denoted by or The lattice of ideals (filters) of is denoted by ().

Definition 1. Let be a Boolean algebra. A quasi-modal operator defined on is a function that verifies the following conditions for all

• Q1

• Q2

A quasi-modal algebra, or -algebra, is a pair where is a Boolean algebra and is a quasi-modal operator.

In every quasi-modal algebra we can define the dual operator by where It is easy to see that the operator satisfies the conditions

• (1)   , and

• (2)   ,

for all (see [2]). The class of -algebras is denoted by .

Let be a -algebra. For each we define the set

Lemma 2. [2] Let .

• (1) For each ,

• (2) iff for all , if then

Let . We define on a binary relation by

Throughout this paper, we will frequently work with Boolean subalgebras of a given Boolean algebra. In order to avoid any confusion, if , then we will use the symbol to denote the corresponding operation of A function is an homomorphism of quasi-modal algebras, or a -homomorphism, if is an homomorphism of Boolean algebras, and

for any A quasi-isomorphism is a Boolean isomorphism that is a q-homomor-phism.

Let . Let us consider the relational structure

From the result given in [2] it follows that the Boolean algebra endowed with the operator

defined by

is a quasi-modal algebra. Moreover, the map defined by , for each , is a -homomorphism i.e. , for each .

Let Let . Define the ideal and the filter . For we define recursively and

Theorem 3. [2] Let . Then for all the following equivalences hold:

• 1. is reflexive.

• 2. i.e., is transitive.

• 3. is symmetrical.

Let We shall say that is a topological quasi-modal algebra if for every and We shall say that is a monadic quasi-modal algebra if it is a quasi-topological algebra and for every From the previous Theorem we get that a quasi-modal algebra is quasi-topological algebra iff the relation is reflexive and transitive. Similarly, a quasi-modal algebra is a monadic quasi-modal algebra iff the relation is an equivalence.

Let and be two qm-algebras. We shall say that the structure is a quasi-modal subalgebra of or qm-subalgebra for short, if is a Boolean subalgebra of and for any

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

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

• (1) There is a quasi-modal operator, such that is a -subalgebra of

• (2) For any , is defined to be .

Remark 5. Let be a -algebra. Let be a Boolean subalgebra of . If there is a quasi-modal operator such that is a -subalgebra of then is unique. The proof of this fact is as follows: First, we note that if is an ideal of and is a filter of such that , then We suppose now that there exist two quasi-modal operators and in such that and are two -subalgebras of Then

If , there exists such that and . Then and 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 class has these properties.

Definition 6. Let be a class of quasi-modal algebras. We shall say that has the AP if for any triple and injective quasi-homomorphisms and there exists and injective quasi-homomorphisms and such that and , for every .

We shall say that has the SAP if has the AP and in addition the maps and above have the following property:

For all such that there exists such that and .

Let be a class of quasi-modal algebras. Without losing generality, we can assume that in the above definition is a -subalgebra of and i.e., that and are the inclusion maps.

Theorem 7. The class has the AP.

Proof.  Let such is a -subalgebra of and  . Let us consider the relational structures We shall define the set

and the binary relation in as follows:

Let us consider the quasi-modal algebra where the operator

is defined by

We note that in this case

Let us define the maps and by

respectively. We prove that and are injective q-homomorpshims.

First of all let us check the following property:

 (1)

Let Since and is a subalgebra of we get by known results on Boolean algebras that there exists such that i.e.,

To see that is injective, let such that Then there exists such that and By (1), there exists such that . So, and i.e., Thus, is injective. It is clear that is a Boolean homomorphism.

We prove next that , for any .

Let Suppose that i.e., From Lemma 2 there exists such that and . By property (1), there exists such that  From the inclusion it is easy to see that Thus,

We prove that there exists such that and Let us consider the filter

The filter is proper, because otherwise there exist and such that . So, and since is increasing, we get As is a quasi-subalgebra of and we get

Thus,

So, there exists and such that Since Then we have which is a contradiction. Therefore, there exists such that

So, So, we have the inclusion . The inclusion is easy and left to the reader. Thus, , and consequently we get

i.e., is an injective q-homomorphism.

Similarly we can prove that is an injective q-homomorphism. Moreover, it is easy to check that for all , and . Thus, has the AP.

Lemma 8. Let and such that is a subalgebra of and Let , and let us suppose that there exists no such that or . Then there exist such that

Proof. Let us consider the filter in and the filter in We note that

because in otherwise there exists such that , which is a contradiction. Then by the Ultrafilter theorem, there exists that such that

Let us consider in the filter and the ideal . We prove that

Suppose that there exist elements , , and such that . This implies that

Thus, we get , which is a contradiction. So, there exists such that

Theorem 9. The class has the SAP.

Proof. The proof of the SAP is actually analogous to the previous one. Let such that is a qm-subalgebra of and of  . Let us consider the set and the quasi-modal algebra of the proof above.

Let Suppose that there exists no such that or . By Lemma 8 there exists and there exists such that

i.e., and Thus, So, has the SAP.

The results above can be applied to prove that other classes of quasi-modal algebras have the AP and SAP. For example, if is the class of the topological quasi-modal algebras, then in the proof of Theorem 7 the binary relation defined on the set is reflexive and transitive. Consequently, is a topological quasi-modal algebra and thus the class has the AP and the SAP. Similar considerations can be applied to the class 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.        [ Links ]

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

[3]    Kracht. M., Tools and Techniques in Modal Logic, Studies in Logic and the Foundations of Mathematics, Vol. 142, Elsevier.         [ Links ]

Sergio Arturo Celani
CONICET and Departamento de Matemáticas