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Revista de la Unión Matemática Argentina
Print version ISSN 0041-6932
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.
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.
I would like to thank the referee for his observations and suggestions which have contributed to improve this paper.
[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
Universidad Nacional del Centro
Pinto 399, 7000 Tandil, Argentina
scelani@exa.unicen.edu.ar
Recibido: 9 de marzo de 2007
Aceptado: 19 de septiembre de 2008