versión On-line ISSN 1669-9637
Rev. Unión Mat. Argent. v.49 n.1 Bahía Blanca ene./jun. 2008
Sergio Arturo Celani
Abstract. In this paper we shall introduce the notions of point-closed, point-compact, and m-saturated monotonic neighbourhood models. We will give some characterizations, and we will prove that the ultrafilter extension and the valuation extension of a model are m-saturated.
Key words and phrases. monotonic modal logic, neighbourhood frames and models, m-saturated models, ultrafilter extension, valuation extension.
2000 Mathematics Subject Classification. 06D05, 06D16, 03G10.
Monotonic neighbourhood semantics (cf.  and ) is a generalization of Kripke semantics. It is also the standard tool for reasoning about monotonic modal logics in which some (Kripke valid) principles such as , do not hold. A monotonic neighbourhood model, or monotonic model, is a structure where is closed under supersets for each , and is a valuation defined on .
The main objective of this paper will be the identification and study of some properties of saturation of monotonic models. It is also intented to prove that the ultrafilter and valuation extension of a monotonic model is -saturated. We will define the image-compact, point-compact, point-closed, and modally saturated (or -saturated) models. These notions are defined in topological terms. For each monotonic model we will define a topology in the set
using the Boolean algebra , and taking as sub-basis the collection of all sets of the form
The topological space is called the hyperspace of relative to . The notions of point-compact, point-closed, and -saturated monotonic models are defined relative to this space. For instance, a model is -saturated if is a compact subset of for each , and for each , there exists a compact set of of such that and . With this notion of -saturation we will be able to prove that the ultrafilter extension of a monotonic model is -saturated. This question has already been addressed in , but with a different notion of -saturation.
In Section 2 we will recall the principal results on the relational and algebraic semantics for monotonic modal logic. In Section 3 we will introduce the notions of compact, image-compact, point-compact, point-closed, and -saturated monotonic models. We will give a characterizations of point-compact models. We will show that some of the notions introduced are invariant by surjective bounded morphisms. Section 4 is the core of this paper. We shall prove that for a monotonic model there exists at least two saturated extensions: the valuation extension and the ultrafilter extension. The valuation extension of a model is a bounded image of the ultrafilter extension, and as the property of -saturation is invariant under surjective bounded morphism, we will deduce that the valuation extension is also -saturated. Saturated extensions of monotonic models may be seen as a completation of the underlying frame structure. For Kripke models, the saturated extensions are modally saturated structures, which implies that modally states are bisimilar (see  for Kripke models and  for monotonic models).
A monotonic algebra is a pair where is a Boolean algebra, and is a monotonic function, i.e. if then , for all The dual operator is defined as . The filter (ideal) generated by a set will be denoted by (. The set of all prime filters or ultrafilters of is denoted by .
Given a set , we denote by the powerset of , and for a subset of , we write for the complement of in . We will call a space a pair where is a set, and a subalgebra of the Boolean algebra of . We note that is a basis of a topology on whose open sets are the unions of subsets of . All the elements of are clopen (closed and open) subsets of , because is a Boolean algebra, but an arbitrary clopen set does not to be need an element of . Given a space and we will use the notation to express the closure of The set of all closed subsets (compact subsets) of will be denoted by (). We note that and are posets under the inclusion relation. Some topological properties of can be characterized in terms of the map given by . The map is called the insertion map in . A space is called a Boolean space if it is compact and totally disconnected. If is a Boolean space, then the family of clopen subsets is a basis for .
To each Boolean algebra we can associate a Boolean space whose points are the elements of with the topology determined by the basis , where . It is known that is a Boolean subalgebra of . By the explanation above we have that, if is a Boolean space, then , by means of the map , and if is a Boolean algebra, then , by means of the map . Moreover, it is known that the map establishes a bijective correspondence between the lattice of all filters of and the lattice of all closed subsets of .
Let be a set. A neighbourhood relation, or multirelation, defined on is a relation .
Every monotonic frame gives rise to a monotonic algebra of sets. Consider the monotonic map defined by:
for each . It is clear that the pair is a monotonic algebra. Using the notation introduced in Definition 1 the map can be defined also as for each . The dual map is defined by
for each .
Next we show that any monotonic algebra with monotone gives rise to a monotonic frame by invoking the basic Stone representation. In other words, we represent the elements of a monotonic algebra as subsets of some universal set, namely the set of all ultrafilters, and then define a multirelation over this universe.
Let be a monotonic algebra. Let us define a multirelation by:
where . We note that for any filter and for all ,
- is a monotonic frame.
- for all
Proof. (1) Clearly is an increasing subset of , for each .
(2) We prove that for all , and for all
If , then . So, . If there exists such that and then Thus, . As consequence of this we have that for all . Now it is easy to prove . □
Let us consider a propositional language defined by using a denumerable set of propositional variables , the connectives and the negation , the modal connective , and the propositional constant We shall denote by the operator defined as , for . The set of all well formed formulas will be denoted by .
Every valuation can be extended to by means of the following clauses:
The notions of truth at a point, validity in a model and validity in a frame for formulas are defined as is usual. A formula is valid at point in a model , in symbols if The formula is valid in a model , in symbols , if
Let be a monotonic model. As is a Boolean algebra of sets, and , for every , is a monotonic subalgebra of the algebra . We shall denote by the space generated taking as a basis for a topology defined in Let
the range of the multirelation . Let the hyperspace of relative to . Recall that denotes the set of all compact subsets of .
- is compact if the space is compact,
- is image-compact if for all and for all , there exists such that and .
- is point-compact if is compact subset in the topological space , for each ,
- is point-closed, if is a closed subset of the space for each .
- is modally saturated (or m-saturated) if it is image-compact and point-compact.
Remark 6. In  Hansen defines the notion of -saturated model as follows:
A model is -saturated if the following conditions hold:
- (m1) For any , and such that , if , for all finite subset of , then .
- (m2) For any , and if for every finite subset of there exists an such that , then there exists an such that .
The condition (m2) is equivalent to say that the model is point-compact, as we will see. But the condition (1) is not equivalent to the notion of image-compact. Hansen's definition has the disadvantage that the ultrafilter extension of a model (see definition 15) fails to meet the condition (m1). Now we will see that the condition (m2) is equivalent to require that should be point-compact.
Proof. Let and such that for every finite subset of there exists an such that . Suppose that , for any . Then for each there exists such that . So,
As is a compact subset of there exists a finite subset such that
By hypothesis, there exists such that which is a contradiction to (3.1). Thus, there exists an such that .
Let . We prove that is compact subset in the topological space . Let such that . Suppose that
for any finite subset of . So for each finite subset of there exists such that . From condition (m1), there exists such that . Thus, , which is a contradiction. □
Let be a model. Recall that the insertion map of is defined by , for each . We write by .
for all and for all .
Proof. Let and such that . Suppose that . Since is point-closed, there exists such that and Since , there exists such that , which is a contradiction, because . Therefore □
Let be a model. Let us consider the following property:
(P): For all and for all , if then there exists such that and ,
where is the topological closure in the space .
Proof. Let and let . Assume that
Suppose that for all . So for each there exists such that and , i.e. Thus,
Since is a compact subset of , there exists a some finite set of formulas such that
and By (3.2), i.e., which is a contradiction. Therefore, there exists such that □
Proof. Suppose that satisfies the property (P). Let . Suppose that for every finite subset of
We prove that . Suppose the contrary. Then . As is compact, , for some . From (3.3), there exists such that , i.e., , which is impossible. Thus, Let
It is clear that is a closed subset of , and by compacity, is compact. We prove that
If , then . By compacity, there exists a finite set such that
Then . From (3.3), there exists such that i.e., Thus, . So,
and by hypothesis there exists such that Then , for every Thus,
The other direction is followed by Proposition 9. □
Definition 11. () A bounded morphism between two monotonic models and is a function such that
- , for each propositional variable ,
- If , then , and
- If then there exists such that and .
It follows that truth of modal formulas is invariant under bounded morphisms ().
The following technical lemma is needed in the next results.
Proof. Let . We prove that Let us suppose the opposite. Then Since is compact, for some formulas . So, , which is impossible. It follows that there exists . Now, it is easy to see that .
Let Suppose that for any finite subset of , Let us consider the filter of generated by the set It is not difficult to prove that is proper. It follows that there exists an ultrafilter of such that Since the map is onto, there exists such that Then, for all So, which is a contradiction. Thus, there is a finite subset of , such that . □
The concepts of compact and point-compact models are preserved by surjective bounded morphisms.
- If is compact, then is compact.
- If is surjective and is compact, then is compact.
- If is point-compact, then is point compact.
- If is surjective and is point-compact then is point compact.
Proof. (1) By Lemma 13 we need to prove that the map is surjective. Let . Consider the set . It is easy to prove that . Since is surjective, there exists such that . Let . Then, it is easy to see that .
(2) By Lemma 13 we need to prove that the map is surjective. Let . Consider . It is easy to see that . As is surjective, there exists such that . Since is surjective, there exists such that . We prove that . For all , iff iff iff . Thus, .
(3) Let and let Suppose that
Taking into account that , for all , it is easy to see that
As is point-compact, there exists a finite set such that Let . Then . So there exists such that . Thus .
(4) Let and let Suppose that
As is surjective there exists such that . We prove that
Let . Since is a bounded morphism, . So, there exists such that , i.e. . Thus (3.4) is valid. As is point-compact, there exists such that . We prove that
Let . Since is a bounded morphism, there exists such that and . As , there exists such that .i.e. . It follows . Thus, . So, is point-compact. □
Let be a model. Let . Let the Boolean space of the algebra . Let us consider the ultrafilter frame of the monotonic algebra . Recall that is defined as:
where and . We write . We note that if is a closed subset of , then
where is a map defined by:
It is easy to see that , for each (see ).
Given a model we can define another extension taking the set as the base set of a model. Let be the Boolean space of . Let us consider the ultrafilter frame of the monotonic algebra , where is defined by
where and .
where the function is defined by
We note that for each .
- and ,
- iff and iff
Proof. We prove , for any . The other proof is very similar. The proof is by induction on the complexity of We consider the case Let . Let Let us consider the filter in the Boolean algebra Then it is easy to see that For the other direction, suppose that Then there exists a filter of such that
By inductive hypothesis we have . Thus, , i.e., .
(2). We prove iff The other proof is similar. Assume that . Then, . But belong to every ultrafilter of . Then, , i.e., . Now, if , then there exists such that Then, . Thus, □
Proof. Let be a monotonic model. We prove that is compact i.e., is a compact space. From Lemma 13 it is enough to prove that the map is surjective. Let . Consider the filter in generated by . It is clear that is proper. So there exists such that . By construction . So, , and consequently is surjective.
We prove that is point-compact in the hyperspace , where . As is a compact space we will apply Proposition 10. Let and let be a subset of Suppose that
We need to prove that there exists a subset of such that and , where is the topological closure of in the topological space .
Consider the filter generated by the set . We prove that
Let . Then there exists such that
So, . Then
i.e. . Then . Thus . Taking into account that the topological closure of in the space is , it is easy to prove that . Therefore is -saturated. □
Proof. Since is a subalgebra of the algebra , we can define a map by . By well-established results in the duality theory of Boolean algebras the map is the dual map of the inclusion homomorphism between and . From the results given by H. Hansen  we get that is a surjective bounded morphism between the ultrafilter extension and the valuation extension . Since is -saturated by Proposition 14 we get that is also -saturated. □
 B. F. Chellas, Modal Logic: an introduction, Cambridge Univ. Press,1980. [ Links ]
 R. Goldblatt, Maps and Monads for Modal Frames, Studia Logica Volume 83, Numbers 1-3 (2006), pp.309-331. [ Links ]
 R. Goldblatt, Mathematics of Modality, CSLI Lectures Notes 43, 1993. [ Links ]
 R. Goldblatt, Logics of Time and Computation, CSLI Lectures Notes 7, Second Edition, 1992. [ Links ]
 H. H. Hansen and C. Kupke. A Coalgebraic Perspective on Monotone Modal Logic, Electronic Notes in Theoretical Computer Science, 106, pages 121-143, Elsevier, 2004. [ Links ]
 H. H. Hansen. Monotonic modal logic (Master's thesis). Preprint 2003-24, ILLC, University of Amsterdam, 2003. [ Links ]
 M. J. Hollenberg, Hennessey-Milner Classes and Process Algebra. In M. de Rijke A. Ponse and Y. Venema, editors, Modal Logic and Process Algebra: a Bisimulation Perspective, volume 53 of CSLI Lecture Notes, pages 187-216. CSLI Publications, 1995. [ Links ]
Sergio Arturo Celani
CONICET and Departamento de Matemáticas,
Facultad de Ciencias Exactas,
Universidad Nacional del Centro, Pinto 399,
7000 Tandil, Argentina
Recibido: 10 de abril de 2008
Aceptado: 28 de abril de 2008