Serviços Personalizados
Journal
Artigo
Indicadores
-
Citado por SciELO
Links relacionados
-
Similares em SciELO
Compartilhar
Revista de la Unión Matemática Argentina
versão impressa ISSN 0041-6932
Rev. Unión Mat. Argent. vol.50 no.1 Bahía Blanca jun. 2009
The subvariety of Q-Heyting algebras generated by chains
Laura A. Rueda
Abstract. The variety of Heyting algebras with a quantifier [14] corresponds to the algebraic study of the modal intuitionistic propositional calculus without the necessity operator. This paper is concerned with the subvariety
of
generated by chains. We prove that this subvariety is characterized within
by the equations
and
. We investigate free objects in
.
1. Introduction and Preliminaries
Distributive lattices with a quantifier were considered as algebras for the first time by Cignoli in [7] who studied them under the name of -distributive lattices. A
-distributive lattice is an algebra
of type
such that
is a bounded distributive lattice and the unary operation
satisfies the following conditions, for any
,
:
,
,
and
. These conditions were introduced by Halmos [9] as an algebraic counterpart of the logical notion of an existential quantifier.
Various further investigations have been carried out since [7] (see R. Cignoli [8], H. Priestley [13], M. Adams and W. Dziobiak [4], M. Abad and J. P. Díaz Varela [2] and A. Petrovich [11]). As a natural generalization, the operation of quantification was considered for Heyting algebras in [3] and [15]. A Heyting algebra is an algebra of type
for which
is a bounded distributive lattice and for
,
,
is the relative pseudocomplement of
with respect to
, i.e.,
if and only if
. It is known that the class of Heyting algebras forms a variety. An important subvariety of Heyting algebras is the class of linear Heyting algebras [5]. A linear Heyting algebra is a Heyting algebra that satisfies the equation
. Throughout this paper
will denote the category of Heyting algebras and Heyting algebra homomorphisms and
will denote the subcategory of linear Heyting algebras.
A -Heyting algebra is an algebra
such that
is an object of
and
is a quantifier on
, that is,
is a unary operation defined as for
-distributive lattices. Monadic Boolean algebras are the simplest examples of
-Heyting algebras. The class of
-Heyting algebras forms a variety, which we denote
. The subvariety of
characterized within
by the equation
, that is, the subvariety of linear
-Heyting algebras will be denoted by
.
-Heyting algebras were first introduced in [14] and have been investigated in [14, 15, 3].
In this paper we investigate the subvariety of the variety of
-Heyting algebras generated by chains. We characterize
by identities in Section 2 and we investigate free objects in this variety in Section 3.
We will usually use the same notation for a variety and for the algebraic category associated with it. And, similarly we will use the same notation for a structure and for its universe.
Recall that Heyting algebras are algebraic models of the intuitionistic propositional logic and that the study of extensions of Intuitionistic Propositional Calculus (IPC) reduces to the study of subvarieties of the variety . The language of intuitionistic modal logic (MIPC) is the language of IPC enriched with two modal unary operators of necessity
and of posibility
. The algebraic models of MIPC are the monadic Heyting algebras.
Now, in MIPC the operators and
are independent from each other, that is
and
are not theorems in MIPC. Hence, the set of theorems of the propositional calculus without of the necessity operator
, called the
-free fragment of MIPC is different from that of MIPC. Similarly, the set of theorems of the propositional calculus without of the possibility operator
, called the
-free fragment of MIPC is different from that of MIPC.
It turns out that the behaviour of the -free fragment of MIPC is very much similar to that of MIPC. However, surprisingly enough, the
-free fragment of MIPC behaves pretty diferent from MIPC.
-Heyting algebras are the algebraic models of the
-free fragment of MIPC, that is,
-Heyting algebras are the
-free reducts of monadic Heyting algebras.
For a poset and
, let
and
. We write
,
instead of
,
respectively. We say that
is decreasing if
, increasing if
and convex if
. A mapping
is order preserving if
whenever
.
In order to describe the dual category of we recall that a Priestley space is a triple
such that
is a partially ordered set,
is a compact topological space, and the triple is totally order-disconnected (that is, for
,
, if
then there exists a clopen increasing
such that
and
). Priestley showed that the category of bounded distributive lattices and lattice homomorphisms is dually equivalent to the category of Priestley spaces and order preserving continuous functions (see the survey paper [12]).
A -Heyting space
(see [7, 14, 15]) is a Priestley space
together with an equivalence relation
defined on
such that
is clopen for every convex clopen
,
for each
, where
and
is the lattice of clopen increasing subsets of
, and
the blocks of
are closed in
. For
, let
denote the clopen increasing set that represents
, where
is the set of prime filters of
, ordered by set inclusion and with the topology having as a sub-basis the sets
and
for
. If
then, under the duality,
corresponds to the clopen increasing set
.
For -Heyting spaces
and
, a
-Heyting morphism is a continuous order-preserving mapping
such that
and
, for each
.
It can be proved in the usual way that the category of -Heyting algebras and homomorphisms is dually equivalent to the category of
-Heyting spaces and
-Heyting morphisms [14, 15]. For each
-Heyting algebra
the corresponding
-Heyting space is
, where
. Conversely, if
is a
-Heyting space, the corresponding
-Heyting algebra is
, where
is defined as in
.
In this section we will study the subvariety generated by chains within
. Observe that if
, then
, that is,
. Consequently,
.
Recall that in the variety of Heyting algebras, congruences are determined by filters. Precisely, if and
is a filter of
, then
is a congruence on
, and the correspondence
establishes an isomorphism from the lattice of filters of
on
, the lattice of congruences of
. If
is generated by an element
,
, we write
.
Observe that if is a Heyting chain and
is a filter of
,
if and only if
or
. Then,
. As a consequence of this, we have that if
is a chain,
is a subdirectly irreducible algebra in
if and only if
is a subdirectly irreducible algebra in
, that is,
has a unique dual atom.
A quantifier on an algebra
is said to be multiplicative if
, for every
.
Let be the subvariety of
characterized by the equation
.
Proof Let and
, i.e.,
. As
, then
, that is,
. So
. Thus
, so
. Therefore,
.
Observe that , that is,
. Let us see that
.
Lemma 2.2. Let be a subdirectly irreducible algebra in
. Then
is a chain.
Proof Let be a subdirectly irreducible algebra. Then
and hence
is subdirectly irreducible in
, that is,
has a unique dual atom. Since for every
,
, then
or
, that is,
or
. So
is a chain.
As a consequence of this corollary we have that is characterized within
by the identities
and
.
The following theorem characterizes the dual space of an algebra in .
Theorem 2.4. Let be a
-Heyting algebra, let
be the associated
-Heyting space and
the partition of
determined by
. Then,
if and only if each
has exactly one maximal element.
Proof Suppose that
and there exists
such that
has two maximal elements
,
,
. Let
be such that
and
. For each
, we have that
. Thus there exists
such that
and
. Consequently
As is closed, by a compacteness argument
and . So
and
. This implies that
and consequently
. On the other hand,
, which contradicts that
.
Conversely, we know that . Let us see that
. Since
is a quantifier,
. Let us prove the other inclusion. Let
and
such that
. Since
and
, if
, then
. Therefore
and so
Lemma 2.5. is the greatest subvariety of
such that every filter determines a congruence.
Proof Let such that
. We are going to construct a filter in
which does not determine a congruence. From
, there exist
such that
. Then there exists a prime ideal
such that
and
. Since
is an ideal we have that
and
. Consider the filter
. Then
, being that
. Let us see that
. Suppose on the contrary that
. Thus
, which implies that
(*) since the image of
is closed under implication. On the other hand,
, so
. Since
is a prime ideal and
we have that
. This, together with (*), implies that
, which is a contradiction.
In this section we characterize the free algebra in with
generators. Following a path analogous to that of M. Abad and L. Monteiro in [1], we will provide a method to construct the order set
of all join-irreducible elements of the free algebra, and as a consequence, we will obtain a formula to compute
.
It is clear that for any subset of a chain
, the subalgebra of
generated by
is
. Thus, every
-generated subalgebra of a chain of
has at most
elements, that is, the class of all chains in
is uniformly locally finite. So
is generated by a uniformly locally finite class, and consequently,
is a variety locally finite [6, Theorem 3.7].
If is a finite algebra, the
-Heyting space
has the discrete topology and
is anti-isomorphic to the ordered set
of join-irreducible elements of
. In this section we will use the set
instead of
and we will consider the relation
defined on
, that is we consider
. If
is the partition determined by
in
, we say that
if and only if
. This is an order relation.
Theorem 3.1. [10] A Heyting algebra is linear if and only if the family of prime filters which contain a prime filter is a chain.
Definition 3.2. Let
be a finite algebra. Let
and let
, such that
,
, where
. We say that
has coordinates
, if the chain
is of length
and if
,
.
Notice that the set of the previous definition is considered within
.
Let be a non negative integer. Let
be the chain with
elements. Let
,
, with
for
. Let
be the interval in
consisting of the elements
such that
. We denote
the algebra
, where
,
.
Observe that if ,
is a chain. More precisely,
is of length
if and only if
is a chain with
elements [1, p. 7]. If
is the natural homomorphism and
are such that
, then there exist join-irreducible elements
in
such that
and
,
. Moreover, taking into account that
en
if and only if
en
, it follows that
has coordinates
,
if and only if
. Since
is the trivial relation, we have that
is a subdirect product of the chains
.
Let be the free
-algebra with a finite set of generators of cardinal
. For the sake of simplicity we will write
instead of
.
We know that every -generated subalgebra of a chain of
has at most
elements. Since
is a chain generated by at most
elements, we have the following
If then from Lemma 3.3,
has coordinates
, for some
,
and
such that
.
Consider the following sets:
and
For a subset of
to generate the algebra
, every non constant element must be contained in
, that is,
. Besides, every constant can be obtained from
, except the constants of
. So we have that
and consequently,
.
For every ,
, consider the sets
We will denote instead of
. Observe that
,
and
for every
, that is,
consists of one
-tuple whose
first coordinates are equal to 2. Moreover, if
, we have that
, but if
,
.
Let . It is clear that
and that for
.
Let be the set of all functions
from the set
of free generators of
into
such that
. Observe that every
is nonempty, as
if and only if
, that is
.
Recall that a filter in a finite Heyting algebra is prime if and only if
, where
is join-irreducible element.
If ,
can be extended to a unique homomorphism
from
onto
. If
is the kernel of
, it is well known that
is a prime filter in
, so
, with
. Thus, for each
, we have a function
defined by .
Lemma 3.4. The following holds .
Proof Let us see that is onto. For
, consider
the natural homomorphism from
onto
, and
the restriction of
to
. Then
and therefore
. Let
be the extension of
. Since
, then
and therefore
.
Let us prove that the function , is one-to-one. If the functions
,
satisfy
then there is an automorphism
of
such that
. But the only automorphism of
is the identity, then
and then
.
If is the number of functions from a set with
elements onto a set with
elements, then:
Let . Then, for each
,
In particular, for ,
and then
. For
,
,
and for
,
. So in both cases,
, then
. And for
,
,
,
, then
and
Consequently,
Consider the set
If , there is a unique
and a unique
such that
. If we put
we have a one-to-one mapping from
onto
.
The following lemma is immediate (recall that ).
Lemma 3.6. has coordinates
if and only if
for all
.
As a consequence, the set has
minimal elements.
Lemma 3.7. For , and
,
has coordinates
if and only if
.
Proof From the proof of Lemma 3.4, has coordinates
if and only if
, and from the comment preceding that lemma, this is equivalent to
.
Remark 3.8. We know that if , the extension homomorphism
and the natural homomorphism
from
into
satisfy
. Then if in
,
, we have
The proof of the following lemma will be omitted since it is an adaptation of that of [1, Lemma 3.13].
We say covers
if
and
implies
.
Lemma 3.9. If ,
,
covers
if and only if the following conditions hold:
-
(i)
,
-
(ii)
,
,
.
-
(iii)
or
,
,
and
.
In the following theorem we denote .
Theorem 3.10. Let ,
. Then
covers
if and only if
,
or
,
,
, and for
the following conditions hold:
-
(I)
if and only if
,
.
-
(II)
if and only if
or
.
Proof Suppose that covers
. The first part of the theorem is an immediate consequence of Lemma 3.9
Since in ,
, we have
if and only if
if and only if
if and only if
and
if and only if
In particular, we have the conditions and
.
Conversely, let ,
be such that
and satisfying and
. Then,
From Lemma 3.9 , we must prove that .
Consider in
and
the chains and
respectively and consider the following sets:
Then
We have that is a filter,
is an ideal and
,
, are nonempty sets, being that
,
,
.
is also nonempty. Indeed, if
, since
, there is
such that
, then from
and
,
and
, that is
. If
, there is
such that
, then from
and
,
and
, that is
.
It is clear that the sets ,
, are pairwise disjoint. Observe that
, and so it is a filter. Using these remarks it is a routine matter to show that the set
is a subalgebra of
.
Let us see that . If
,
.
If ,
and from
,
, that is,
. Then
.
If ,
and from
,
, that is
. So
.
If ,
, then
and from
,
, that is,
. Then
.
Therefore, and consequently
.
Then we can write, .
Since , we have
The previous theorem allows us to construct the ordered set of join-irreducible elements of the free algebra . By virtue of Lemma 3.6, there exists a one-to-one correspondence between the set of minimal elements of
and the set of functions
from
into
. Since
is a chain, for every
, then
is also a chain for
. So, the ordered-connected components of
are
, where
is minimal, that is, the order-connected components of
are the sets
, where
.
We have constructed the -Heyting space
. The free algebra
with a finite set of generators of cardinality
, is the algebra obtained from
considering the decreasing subsets of
with the quantifier given by
, for each
decreasing set of
.
Example 3.11. In the next figure we give the free algebra generated by an element
, and the ordered set
of its join-irreducible elements, with the equivalence relation which determines the quantifier.
Where ,
,
,
and
. We denote
.
In the rest of this section we investigate the poset in order to obtain a recursive formula for the number of elements of
.
Let be the family of order-connected components
, with
minimal, such that
,
. It is clear that
, and if
, then
. In general,
.
For a given , all the order-connected components in
have the same number of elements. So if
and
for
, then
We are going to determine .
Consider . From Theorem 3.10, we know that
covers
if and only if
or
and
-
(I)
if and only if
.
-
(II)
if and only if
.
In particular, there are funtions
in
covering
, and similarly there are
funtions
in
covering
. So, there are
functions
covering
,
of which satisfy
,
.
In these conditions we have the following result.
-
(1) If
, there exists
with
such that
and
are order-isomorphic.
-
(2) If
and
covers
, with
and
, for every
, then there exists
,
such that
,
and
are order-isomorphic.
Proof
-
(1) If
is the function defined by:
,
is clearly a minimal element of
,
and
.
Let us see that
and
are order-isomorphic. Observe that if
, then
, where
and
. We define
by means of
, where
,
.
Clearly
is an isomorphism.
-
(2) Observe that, if
,
, then
,
,
and
. If
is the function defined by
, then
and
are order-isomorphic. Indeed, if we define
by means of
, where
and
defined as in
,
and it can be proved that
is an isomorphism.
Finally, consider
defined by
, where
,
,
,
and
,
Clearly
is an isomorphism.
From the previous proposition,
Therefore
and then
Acknowledgment: I gratefully acknowledge helpful comments of the referees. In particular, one of them suggested the algebraic proof of Lemma 2.5.
[1] M. Abad and L. Monteiro, On free L-algebras, Notas de Lógica Matemática 35(1987), 1-20. [ Links ]
[2] M. Abad and J. P. Díaz Varela, Free Q-distributive lattices from meet semilattices, Discrete Math. 224 (2000), 1-14. [ Links ]
[3] M. Abad, J. P. Díaz Varela, L. Rueda and A. Suardíaz, Varieties of three-valued Heyting algebras with a quantifier, Studia Logica 65 (2000), 181-198. [ Links ]
[4] M. E. Adams and W. Dziobiak, Quasivarieties of distributive lattices with a quantifier, Discrete Math. 135(1994), 12-28. [ Links ]
[5] R. Balbes and Ph. Dwinger, Distributive Lattices, University of Missouri Press, Columbia, Missouri, 1974. [ Links ]
[6] G. Bezhanishvili, Locally finite varieties, Algebra Universalis 46(2001), No.4, 531-548. [ Links ]
[7] R. Cignoli, Quantifiers on distributive lattices, Discrete Math. 96(1991), 183-197. [ Links ]
[8] R. Cignoli, Free Q-distributive lattices, Studia Logica 56(1996), 23-29. [ Links ]
[9] P. R. Halmos, Algebraic Logic I. Monadic Boolean algebras, Compositio Math. 12 (1955), 217-249. [ Links ]
[10] A. Monteiro, Sur les Algèbres de Heyting Symétriques, Portugaliae Mathematica, 39 (1980), 1-237. [ Links ]
[11] A. Petrovich, Equations in the theory of Q-distributive lattices, Discrete Math. 17(1997), 211-219. [ Links ]
[12] H. A. Priestley, Ordered sets and duality for distributive lattices, Ann. Discrete Math. 23(1984), 39-60. [ Links ]
[13] H. A. Priestley, Natural dualities for varieties of distributive lattices with a quantifier, Banach Center Publications, Volume 28, Warszawa 1993, 291-310. [ Links ]
[14] L. Rueda, Algebras de Heyting lineales con un cuantificador Ph. D. Thesis, Universidad Nacional del Sur, Bahía Blanca, 2000. [ Links ]
[15] L. Rueda, Linear Heyting algebras with a quantifier, Annals of Pure and Applied Logic 108 (2001), 327-343. [ Links ]
Laura A. Rueda
Departamento de Matemática,
Universidad Nacional del Sur
Bahía Blanca, Argentina
larueda@criba.edu.ar
Recibido: 12 de noviembre de 2007
Aceptado: 2 de junio de 2009