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Revista de la Unión Matemática Argentina

versión On-line ISSN 1669-9637

Rev. Unión Mat. Argent. v.49 n.1 Bahía Blanca ene./jun. 2008

 

Saturated neighbourhood models of Monotonic Modal Logics

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.

1. Introduction

Monotonic neighbourhood semantics (cf. [1] and [6]) 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 □p ∧ □q → □ (p ∧ q) , do not hold. A monotonic neighbourhood model, or monotonic model, is a structure M = ⟨X, R,V ⟩ where R ⊆ X × P (X ), R (x) is closed under supersets for each x ∈ X , and V is a valuation defined on X .

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 m -saturated. We will define the image-compact, point-compact, point-closed, and modally saturated (or m -saturated) models. These notions are defined in topological terms. For each monotonic model M = ⟨X, R, V ⟩ we will define a topology TDV in the set

KR = {Y ⊆ X : ∃x ∈ X (Y ∈ R (x))},

using the Boolean algebra DV = {V(φ ) : φ ∈ F m } , and taking as sub-basis the collection of all sets of the form

L = {Y ∈ K : Y ∩ V (φ) ⁄= ∅} . V (φ) R

The topological space KR = ⟨KR, TDV ⟩ is called the hyperspace of ⟨X,DV ⟩ relative to KR . The notions of point-compact, point-closed, and m -saturated monotonic models are defined relative to this space. For instance, a model M is m -saturated if R (x) is a compact subset of KR for each x ∈ X , and for each Y ∈ R (x) , there exists a compact set of Z of ⟨X, DV ⟩ such that Z ⊆ Y and (x,Z ) ∈ R . With this notion of m -saturation we will be able to prove that the ultrafilter extension of a monotonic model is m -saturated. This question has already been addressed in [6], but with a different notion of m -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 m -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 M 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 m -saturation is invariant under surjective bounded morphism, we will deduce that the valuation extension is also m -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 [7] for Kripke models and [6] for monotonic models).

2. Preliminaries

A monotonic algebra is a pair A = ⟨A,♢ ⟩, where A is a Boolean algebra, and ♢ : A → A is a monotonic function, i.e. if a ≤ b then ♢ (a) ≤ ♢(b) , for all a,b ∈ A. The dual operator □ is defined as □a = ¬ ♢¬a . The filter (ideal) generated by a set H ⊆ A will be denoted by [H ) ((H ]) . The set of all prime filters or ultrafilters of A is denoted by U l(A) .

Given a set X , we denote by P (X ) the powerset of X , and for a subset Y of X , we write  c Y for the complement X \Y of Y in X . We will call a space a pair X = ⟨X, D ⟩, where X is a set, and D a subalgebra of the Boolean algebra of P (X ) . We note that D is a basis of a topology T D on X whose open sets are the unions of subsets of D . All the elements of D are clopen (closed and open) subsets of X , because D is a Boolean algebra, but an arbitrary clopen set does not to be need an element of D . Given a space ⟨X,D ⟩ and Y ⊆ X, we will use the notation cl(Y) to express the closure of Y. The set of all closed subsets (compact subsets) of X will be denoted by C(X ) (K (X ) ). We note that C (X ) and K (X ) are posets under the inclusion relation. Some topological properties of X can be characterized in terms of the map ɛD : X → U l(D ) given by ɛD (x) = {U ∈ D : x ∈ U} . The map ɛD is called the insertion map in [2]. A space X is called a Boolean space if it is compact and totally disconnected. If X is a Boolean space, then the family D of clopen subsets is a basis for X .

To each Boolean algebra A we can associate a Boolean space whose points are the elements of U l(A ) with the topology determined by the basis βA (A) = {βA (a) : a ∈ A } , where βA (a) = {x ∈ U l(A ) : a ∈ x } . It is known that βA (A ) is a Boolean subalgebra of P(U l(A )) . By the explanation above we have that, if X is a Boolean space, then  ~ X = U l(D ) , by means of the map ɛD , and if A is a Boolean algebra, then  ~ A = βA (A ) , by means of the map βA . Moreover, it is known that the map F → ˆF = {x ∈ U l(A ) : F ⊆ x } establishes a bijective correspondence between the lattice of all filters of A and the lattice C (U l(A )) of all closed subsets of U l(A ) .

Definition 1. Let X = ⟨X, D )⟩ be a space. The lower topology TD on a subset K of P (X ) has as sub-base the collection of all sets of the form LU = {Y ∈ K : Y ∩ U ⁄= ∅ }, for U ∈ D . The pair K = ⟨K, TD⟩ is called the hyperspace of X relative to K .

Let X be a set. A neighbourhood relation, or multirelation, defined on X is a relation R ⊆ X × P (X ) .

Definition 2. A monotonic neighbourhood frame, or monotonic frame, is a structure F = ⟨X, R ⟩ where R is a multirelation on X such that R (x ) = {Z ∈ P (X ) : (x,Z ) ∈ R } is an increasing subset of P (X ), for all x ∈ X .

Every monotonic frame F gives rise to a monotonic algebra of sets. Consider the monotonic map ♢R : P (X ) → P(X ) defined by:

♢R(U ) = {x ∈ X : ∃Y ⊆ X (Y ⊆ U and (x, Y) ∈ R )},

for each U ∈ P (X ) . It is clear that the pair ⟨P(X ),♢R ⟩ is a monotonic algebra. Using the notation introduced in Definition 1 the map ♢R can be defined also as ♢R (U ) = {x ∈ X : R (x) ∩ (LUc )c ⁄= ∅}, for each U ∈ P (X ) . The dual map □ R is defined by

□R (U ) = {x ∈ X : R (x) ⊆ LU } ,

for each U ∈ P (X ) .

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 ⟨A, ♢⟩ be a monotonic algebra. Let us define a multirelation R ♢ ⊆ U l(A ) × P (U l(A ) by:

(x,Y ) ∈ R ♢ ⇔ ∃F ∈ F i(A) ( ˆF ⊆ Y and F ⊆ ♢ -1(x).
(2.1)

where ˆF = {y ∈ Ul(A ) : F ⊆ x} . We note that for any filter F ∈ F i(A ), and for all x ∈ U l(A ) ,

(x, ˆF ) ∈ R ♢ iff F ⊆ ♢ -1(x).

Theorem 3. Let ⟨A, ♢⟩ be a monotonic algebra. Then

  1. ⟨Ul (A ),R ♢⟩ is a monotonic frame.
  2. ♢ (β (a)) = β (♢ (a )), R♢ A A for all a ∈ A.

Proof. (1) Clearly R ♢(x) is an increasing subset of P(U l(A)) , for each x ∈ U l(A) .

(2) We prove that for all a ∈ A , and for all x ∈ U l(A) ,

♢ (a ) ∈ x iff ∃F ∈ Fi(A ) : (x, ˆF ) ∈ R ♢ and a ∈ F.

If ♢ (a) ∈ x , then  -1 F(a) ⊆ ♢ (x) . So, (x, F(a)) ∈ R ♢ . If there exists F ∈ Fi (A) such that (x,Fˆ) ∈ R♢ and a ∈ F, then a ∈ F ⊆ ♢ -1(x). Thus, ♢ (a) ∈ x . As consequence of this we have that ♢R ♢(βA (a)) = βA (♢a ), for all a ∈ A . Now it is easy to prove ♢ (β (a )) = ♢ (β (a)) R♢ A R♢ A . □

3. m-saturated models

Let us consider a propositional language L♢ defined by using a denumerable set of propositional variables V ar , the connectives ∨ and ∧, the negation ¬ , the modal connective ♢ , and the propositional constant ⊤. We shall denote by □ the operator defined as □p = ¬ ♢¬p , for p ∈ Var . The set of all well formed formulas will be denoted by F m .

Definition 4. A monotonic model in the language L is a structure M = ⟨X, R, V ⟩, where ⟨X, R⟩ is a monotonic frame, and V : V ar → P(X ) is a valuation.

Every valuation can be extended to F m by means of the following clauses:

  1. V (⊤ ) = X,
  2. V (φ ∨ ψ ) = V(φ ) ∪ V (ψ),
  3.  c V (¬ φ) = X - V (φ) = V (φ) ,
  4. V (♢ φ) = ♢R (V (φ )).

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 φ is valid at point x in a model M , in symbols M ⊨x φ if x ∈ V (φ) . The formula φ is valid in a model M , in symbols M ⊨ φ , if V (φ ) = X.

Let M be a monotonic model. As D = {V (φ ) : φ ∈ Fm } V is a Boolean algebra of sets, and V (♢φ ) = ♢R(V (φ)) ∈ DV , for every φ ∈ F m , ⟨DV ,♢R⟩ is a monotonic subalgebra of the algebra ⟨P(X ),♢R ⟩ . We shall denote by XV = ⟨X, DV ⟩ the space generated taking DV as a basis for a topology defined in X. Let

KR = {Y ⊆ X : ∃x ∈ X ((x, Y) ∈ R )}

the range of the multirelation R . Let K = ⟨K ,T ⟩ R R DV the hyperspace of XV = ⟨X, DV ⟩ relative to KR . Recall that K (XV ) denotes the set of all compact subsets of XV .

Definition 5. Let M = ⟨X, R,V ⟩  be a model. We shall say that:

  1. M  is compact if the space XV is compact,
  2. M is image-compact if for all x ∈ X and for all Y ∈ R (x) , there exists Z ∈ K (XV ) such that Z ⊆ Y and (x,Z ) ∈ R .
  3. M is point-compact if R (x) is compact subset in the topological space KR , for each x ∈ X ,
  4. M is point-closed, if R(x) is a closed subset of the space KR, for each x ∈ X .
  5. M  is modally saturated (or m-saturated) if it is image-compact and point-compact.

Remark 6. In [6] Hansen defines the notion of m -saturated model as follows:

A model M = ⟨X, R,V ⟩ is m -saturated if the following conditions hold:

  • (m1) For any Γ ⊆ F m , x ∈ X and Y ⊆ X such that (x, Y) ∈ R , if ⋂ {V (φ) : φ ∈ Γ 0} ∩ Y ⁄= ∅ , for all finite subset Γ 0 of Γ , then ⋂ {V (φ ) : φ ∈ Γ } ∩ Y ⁄= ∅ .
  • (m2) For any Γ ⊆ F m , and x ∈ X, if for every finite subset Γ 0 of Γ there exists an Y ∈ R (x) such that  ⋂ Y ⊆ {V (φ ) : φ ∈ Γ 0} , then there exists an Z ∈ R(x) such that  ⋂ Z ⊆ {V (φ ) : φ ∈ Γ } .

The condition (m2) is equivalent to say that the model M 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 M should be point-compact.

Proposition 7. Let M  be a model. Then M is point-compact iff it satisfies the condition (m2).

Proof. ⇒ ) Let Γ ⊆ F m and x ∈ X, such that for every finite subset Γ 0 of Γ there exists an Y ∈ R (x ) such that  ⋂ Y ⊆ {V (φ ) : φ ∈ Γ 0} . Suppose that  ⋂ Y ⊈ {V (φ) : φ ∈ Γ } , for any Y ∈ R (x) . Then for each Y ∈ R (x) there exists φy ∈ Γ such that Y ∩ V (¬ φy) ⁄= ∅ . So,

 ⋃ { } R (x) ⊆ LV (¬φy) : φy ∈ Γ .

As R (x) is a compact subset of ⟨KR, TDV ⟩, there exists a finite subset {φy1,...,φyn} ⊆ Γ such that

R (x) ⊆ LV (¬φy1) ∪ ...∪ LV(¬φyn).
(3.1)

By hypothesis, there exists Z ∈ R (x) such that Z ⊆ V (¬φ ) ∩ ...∩ V (¬ φ ), y1 yn which is a contradiction to (3.1). Thus, there exists an Z ∈ R (x) such that  ⋂ Z ⊆ {V (φ ) : φ ∈ Γ } .

⇐ ) Let x ∈ X . We prove that R(x ) is compact subset in the topological space ⟨KR, TDV ⟩ . Let Γ ⊆ F m such that  ⋃ { } R (x) ⊆ LV(φ) : φ ∈ Γ . Suppose that

 ⋃ { } R (x) ⊈ LV (φ) : φ ∈ Γ i ,

for any finite subset Γ i of Γ . So for each finite subset Γ i of Γ there exists Yi ∈ R (x) such that  ⋂ Yi ⊆ {V (¬ φ) : φ ∈ Γ i} . From condition (m1), there exists Z ∈ R(x ) such that  ⋂ Z ⊆ {V(¬ φ) : φ ∈ Γ } . Thus, Z ∕∈ ⋃ {L : φ ∈ Γ } V(φ) , which is a contradiction. □

Let M  be a model. Recall that the insertion map ɛ DV of ⟨X, D ⟩ V is defined by ɛDV (x) = {V (φ) : x ∈ V (φ)} , for each x ∈ X . We write ɛV by ɛDV .

Lemma 8. Let M  be a model. If M is point-closed, then

 ⋂ FɛV(Y) = {ɛV(y) : y ∈ Y } ⊆ ♢-R 1(ɛV (x)) implies that (x,Y ) ∈ R,

for all x ∈ X and for all Y ⊆ X .

Proof. Let x ∈ X and Y ⊆ X such that  - 1 FɛV(Y) ⊆ ♢R (ɛV (x)) . Suppose that Y ∕∈ R (x) . Since R is point-closed, there exists V (φ) ∈ DV such that Y ⊆ V (φ) and x ∕∈ ♢R (V (φ )). Since F ɛV (Y ) ⊆ ♢ -R1(ɛV(x )) , there exists y ∈ Y such that y∈∕ V(φ ) , which is a contradiction, because Y ⊆ V (φ ) . Therefore (x,Y ) ∈ R.

Let M  be a model. Let us consider the following property:

(P): For all x ∈ X and for all Y ∈ P(X ) , if  -1 F ɛV (Y ) ⊆ ♢ R (ɛV(x)) then there exists Z ⊆ X such that (x,Z ) ∈ R and Z ⊆ cl(Y ) ,

where cl(Y) is the topological closure in the space X V .

Proposition 9. Let M  be a model. If M is point-compact, then M satisfies the property (P).

Proof. Let  x ∈ X and let Y ∈ P (X ) . Assume that

F ⊆ ♢- 1(ɛ (x)). ɛV(Y) R V
(3.2)

Suppose that for all Zi ∈ R (x) ,  ⋂ Zi ⊈ cl(Y ) = {V (φ ) : Y ⊆ V(φ )} . So for each Zi ∈ R (x) there exists V (φi ) ∈ DV such that Y ⊆ V (φi) and Zi ⊈ V (φi) , i.e. Zi ∩ V (¬ φi) ⁄= ∅. Thus,

 ⋃ { } R (x) ⊆ LV(¬φi) : Y ⊆ V (φi) .

Since R(x ) is a compact subset of ⟨KR, TDV ⟩ , there exists a some finite set of formulas {φ1,...,φn } such that

R (x) ⊆ LV (¬φ1) ∪ ⋅⋅⋅ ∪ LV (¬φn).

Then,

x ∕∈ ♢R (V (φ1) ∩ ...∩ V (φn)) = V (♢(φ1 ∧ ...∧ φn )),

and Y ⊆ V (♢(φ1 ∧ ...∧ φn )). By (3.2),  - 1 V (φ1 ∧ ...∧ φn) ∈ ♢R (ɛV (x)), i.e., x ∈ V(♢ (φ1 ∧ ...∧ φn)), which is a contradiction. Therefore, there exists Z ∈ R (x) such that Z ⊆ cl(Y ).

Proposition 10. Let M  be a compact model. Then M satisfies the property (P) iff M is point-compact.

Proof. Suppose that M satisfies the property (P). Let W ⊆ DV . Suppose that for every finite subset W0 of W

 ⋃ { } R (x) ⊈ LV (φ) : V (φ) ∈ W0 .
(3.3)

We prove that ⋂ {V (φ)c : V (φ) ∈ W } ⁄= ∅ . Suppose the contrary. Then  ⋃ X = {V (φ) : V (φ) ∈ W } . As M is compact, X = V (φ1) ∪ ...∪ V (φn) , for some {φ1, ...,φn} . From (3.3), there exists Y ∈ R (x) such that Y ∩ V (φ1) = ∅,...,Y ∩ V(φn ) = ∅ , i.e., Y ∩ (V (φ1) ∪ ...∪ V (φn)) = Y ∩ X = ∅ , which is impossible. Thus, ⋂ {V (φ)c : V (φ ) ∈ W } ⁄= ∅. Let

 ⋂ Z = {V (φ )c : V (φ) ∈ W } .

It is clear that Z is a closed subset of X , and by compacity, Z is compact. We prove that

⋂ -1 {ɛV (z) : z ∈ Z} ⊆ ♢R (ɛV (x)) .

If  ⋂ V (ψ) ∈ {ɛV (z) : z ∈ Z} , then  ⋂ c Z = {V (φ ) : V (φ) ∈ W } ⊆ V(ψ ) . By compacity, there exists a finite set {φ1, ...,φn} such that

 c c V (φ1) ∩ ... ∩ V(φn ) ⊆ V (ψ ).

Then  c c ♢R (V(φ1 ) ∩ ...∩ V (φn) ) ⊆ ♢R (V(ψ )) . From (3.3), there exists T ∈ R (x ) such that T ∩ (V (φ1) ∪ ... ∪ V(φn )) = ∅, i.e.,  c c T ⊆ V (φ1) ∩ ...∩ V (φn ) ⊆ V (ψ). Thus, x ∈ ♢R(V (ψ)) . So,

⋂ -1 {ɛV (z) : z ∈ Z} ⊆ ♢R (ɛV (x)) ,

and by hypothesis there exists Y ∈ R (x) such that Y ⊆ cl(Z) = Z. Then Y ∩ V (φ ) = ∅ , for every V (φ ) ∈ W. Thus,

 ⋃ R (x) ⊈ {L : V(φ ) ∈ W } . V(φ)

The other direction is followed by Proposition 9. □

The maps between monotonic frames and monotonic models which preserve the modal structure will be referred to as bounded morphisms. These have previously been studied in [6] (see also [5]).

Definition 11. ([6]) A bounded morphism between two monotonic models M1 and M2 is a function f : X1 → X2 such that

  1. f -1(V2(p) = V1(p) , for each propositional variable p ,
  2. If (x,Y ) ∈ R1 , then (f(x),f(Y )) ∈ R2 , and
  3. If (f(x),Z ) ∈ R2, then there exists Y ⊆ X such that (x, Y) ∈ R1 and f (Y ) ⊆ Z .

It follows that truth of modal formulas is invariant under bounded morphisms ([6]).

Proposition 12. Let M1 and M2 be monotonic models. If f : X1 → X2 is a bounded morphism from M1 to M2 then for each formula φ , f-1(V (φ )) = V (φ) 2 1 .

The following technical lemma is needed in the next results.

Lemma 13. Let M be a model. Then XV is compact iff ɛV is surjective.

Proof. ⇒ ) Let P ∈ U l(DV ) . We prove that ⋂ {V (φ ) : V (φ ) ∈ P } ⁄= ∅. Let us suppose the opposite. Then  ⋃ X = {V (¬φ ) : V(φ ) ∈ P } . Since ⟨X, DV ⟩ is compact, X = V (¬ φ1) ∪ ...∪ V (¬φn ), for some formulas φ1,...,φn .  So, V (φ1 ∧ ...∧ φn ) = ∅ ∈ P , which is impossible. It follows that there exists  ⋂ x ∈ {V (φ) : V (φ ) ∈ P} . Now, it is easy to see that ɛV (x) = P .

⇐ ) Let  ⋃ X = {V (φ ) : φ ∈ Γ ⊆ F m. }. Suppose that for any finite subset Γ 0 of Γ ,  ⋃ X ⁄= {V (φ) : φ ∈ Γ 0}. Let us consider the filter F of DV generated by the set {V (¬φ ) : φ ∈ Γ } . It is not difficult to prove that F is proper. It follows that there exists an ultrafilter P of DV such that F ⊆ P. Since the map ɛV is onto, there exists x ∈ X such that ɛ (x ) = P. V Then, x ∈ V (¬ φ)  for all φ ∈ Γ . So,  ⋂ x ∈ V (¬φ ), φ∈Γ which is a contradiction. Thus, there is a finite subset Γ 0 of Γ , such that  ⋃ X = {V (φ ) : φ ∈ Γ ⊆ F m. } 0 . □

The concepts of compact and point-compact models are preserved by surjective bounded morphisms.

Proposition 14. Let f : X1 → X2 be a bounded morphism between the monotonic models M 1 and M 2 .

  1. If M1 is compact, then M2 is compact.
  2. If f is surjective and M 2 is compact, then M 1 is compact.
  3. If M2 is point-compact, then M1 is point compact.
  4. If f is surjective and M1 is point-compact, then M2 is point compact.

Proof. (1) By Lemma 13 we need to prove that the map ɛV2 : X2 → U l(DV2) is surjective. Let Q ∈ U l(DV2 ) . Consider the set Q ′ = {V1(φ) : V2(φ ) ∈ Q } . It is easy to prove that Q′ ∈ U l(DV1 ) . Since ɛV 1 is surjective, there exists x ∈ X1 such that ɛV (x ) = Q ′ 1 . Let y = f (x) . Then, it is easy to see that ɛV2(y ) = Q .

(2) By Lemma 13 we need to prove that the map ɛV1 : X1 → U l(DV1) is surjective. Let P ∈ Ul(DV1 ) . Consider P ′ = {V2 (φ ) : V1(φ) ∈ P } . It is easy to see that P ′ ∈ U l(DV2) . As ɛV2 is surjective, there exists y ∈ X2 such that ɛV (y) = P ′ 2 . Since f is surjective, there exists x ∈ X1 such that f(x) = y . We prove that ɛV1(x) = P . For all φ ∈ F m ,  -1 x ∈ V1(φ ) = f (V2(φ)) iff f(x) = y ∈ V2(φ ) iff  ′ V2(φ ) ∈ ɛV2(y) = P iff V1 (φ) ∈ P . Thus, ɛV1(x ) = P .

(3) Let x ∈ X1 and let Γ ⊆ DV1. Suppose that

 ⋃ { } R1 (x ) ⊆ LV1(φ) : V1 (φ) ∈ Γ .

Taking into account that  - 1 f (V2 (φ )) = V1(φ) , for all φ ∈ Fm , it is easy to see that

 ⋃ { } R2 (f(x)) = R2(y ) ⊆ LV2(φ) : V1(φ ) ∈ Γ .

As M2 is point-compact, there exists a finite set {V2 (φ1),...,V2(φn)} such that R2 (y ) ⊆ LV2(φ1) ∪ ...∪ LV2(φ1). Let Z ∈ R1 (x) . Then f (Z ) ∈ R2(y) . So there exists V2(φi) ∈ {V2(φ1),...,V2(φn )} such that Z ∩ f-1(V2(φi)) = Z ∩ V1 (φi ) ⁄= ∅ . Thus R (x) ⊆ L ∪ ...∪ L 1 V1(φ1) V1(φn) .

(4) Let y ∈ X2 and let Γ ⊆ DV2. Suppose that

 ⋃ { } R2 (y ) ⊆ LV2(φ) : V2 (φ ) ∈ Γ .

As f is surjective there exists x ∈ X1 such that f(x) = y . We prove that

 ⋃ { } R1 (x ) ⊆ Lf-1(V2(φ)) : V2 (φ) ∈ Γ .
(3.4)

Let Z ∈ R1(x) . Since f is a bounded morphism, f (Z) ∈ R2 (f(x)) = R2(y) . So, there exists V2(φ ) ∈ Γ such that f(Z ) ∩ V2 (φ) ⁄= ∅ , i.e. Z ∩ f -1(V2(φ)) ⁄= ∅ . Thus (3.4) is valid. As M1 is point-compact, there exists {V2(φ1),...,V2(φn )} ⊆ Γ such that R1 (x) ⊆ Lf-1(V2(φ1)) ∪ ...∪ Lf- 1(V2(φn)) . We prove that

R2 (y) ⊆ LV2(φ1) ∪ ...∪ LV2(φn).

Let Y ∈ R2(y) = R2 (f(x)) . Since f is a bounded morphism, there exists Z ⊆ X1 such that Z ∈ R1(x) and f(Z ) ⊆ Y . As R1 (x) ⊆ LV2(φ1) ∪ ...∪ LV2(φ1) , there exists V2(φi) ∈ {V2(φ1 ),...,V2 (φn )} such that Z ∩ f -1(V2(φi)) ⁄= ∅ .i.e. f (Z) ∩ V2(φi) ⁄= ∅ . It follows Y ∩ V (φ ) ⁄= ∅ 2 i . Thus, R (y ) ⊆ L ∪ ...∪ L 2 V2(φ1) V2(φn) . So, M 2 is point-compact. □

 

4. Ultrafilter and model extension

Let M = ⟨X, R, V⟩ be a model. Let U l(P (X )) = Ul(X ) . Let ⟨ ⟩ Ul(X ),βP(X )(P (X )) the Boolean space of the algebra P (X ) . Let us consider the ultrafilter frame ⟨ ⟩ Ul(X ) ,RP (X ) of the monotonic algebra ⟨P(X ),♢R ⟩ . Recall that RP (X) ⊆ U l(X ) × P (U l(X )) is defined as:

(P, Y ) ∈ R iff ∃C ∈ C (Ul(X )) (C ⊆ Y and F ⊆ ♢ -1(P )), P(X) C R

where F = ⋂ {Q : Q ∈ Y }, C and Y ⊆ U l(X ) . We write R = R P(X) U . We note that if Y is a closed subset of U l(X ) , then

 ⋂ (P, Y ) ∈ RU iff FY = {Q : Q ∈ Y } ⊆ ♢ -1(P). R

Definition 15. The ultrafilter extension of a monotonic model M = ⟨X, R, V⟩ is the structure

U e (M ) = ⟨U l(X ),RU ,VU ⟩,

where VU : V ar → P(U l(X )) is a map defined by:

VU (p) = {P ∈ U l(X ) : V (p) ∈ P} ,

for every p ∈ P.

It is easy to see that VU (p) = βP (X )(V (p)) , for each p ∈ V ar (see [6]).

Given a model M we can define another extension taking the set U l(DV ) as the base set of a model. Let ⟨U l(DV ),βD (DV )⟩ V be the Boolean space of D V . Let us consider the ultrafilter frame ⟨U l(D ),R ⟩ V DV of the monotonic algebra ⟨DV ,♢R⟩ , where RDV ⊆ U l(DV ) × P(U l(DV )) is defined by

 -1 (P, Y ) ∈ RDV iff ∃C ∈ C(U l(DV )) (C ⊆ Y and FC ⊆ ♢ R (P)),

where  ⋂ FC = {Q : Q ∈ Y }, and Y ⊆ U l(DV ) .

Definition 16. The valuation extension, or valuation model, of a model M is the structure

V e(M ) = ⟨U l(DV ),RDV ,VDV ⟩ ,

where the function VDV : V ar → P (Ul(DV )) is defined by

V (p) = {P ∈ U l(D ) : V (p) ∈ P }, DV V

for every p ∈ P.

We note that VDV (p ) = βDV (V (p)), for each p ∈ V ar .

Theorem 17. Let M be a monotonic model. Then for any φ ∈ Fm,

  1. VU (φ ) = βP(X)(V (φ)), and VDV (φ ) = βDV (V(φ )) ,
  2. M ⊨ φ iff U e(M ) ⊨ φ, and M ⊨ φ iff V e(M ) ⊨ φ.

Proof. We prove VD (φ ) = βD (V (φ)) V V , for any φ ∈ Fm . The other proof is very similar. The proof is by induction on the complexity of φ. We consider the case ♢ φ. Let P ∈ U l(DV ) . Let V (♢φ ) = ♢R (V (φ)) ∈ P. Let us consider the filter F = F (V (φ )) in the Boolean algebra DV . Then it is easy to see that ( ) P, ˆF ∈ R . DV For the other direction, suppose that P ∈ VDV (♢φ ). Then there exists a filter F of DV such that

 ˆ ˆ (P, F) ∈ RDV and F ⊆ VDV (φ).

By inductive hypothesis we have VD (φ) = βD (V (φ)) V V . Thus, V (φ ) ∈ F ⊆ ♢ -1(P ) R , i.e., V (♢φ ) = ♢ (V (φ)) ∈ P R .

(2). We prove M ⊨ φ iff V e (M ) ⊨ φ. The other proof is similar. Assume that M ⊨ φ . Then, V (φ) = X . But X belong to every ultrafilter of DV . Then, VDV (φ) = U l(DV ) , i.e., V e(M ) ⊨ φ . Now, if M ⊭ φ , then there exists x ∈ X such that x ∕∈ V (φ) . Then, V (φ ) ∕∈ {V (α ) : x ∈ V (α)} ∈ U l(DV ) . Thus, V e(M ) ⊭ φ.

Theorem 18. Let M be a model. Then the model U e (M ) is m -saturated.

Proof. Let M be a monotonic model. We prove that U e(M ) is compact i.e., ⟨U l(X ),DVU ⟩ is a compact space. From Lemma 13 it is enough to prove that the map ɛVU : Ul(X ) → U l(DVU ) is surjective. Let P ∈ U l(DVU ) . Consider the filter F in P (X ) generated by {V (φ ) : V (φ ) ∈ P } U . It is clear that F is proper. So there exists Q ∈ U l(X ) such that F ⊆ Q . By construction ɛVU(Q ) ⊆ P . So, ɛVU (Q ) ⊆ P , and consequently ɛVU is surjective.

We prove that RU is point-compact in the hyperspace ⟨ ⟩ KR ,TD U VU , where XV = ⟨U l(X ),DV ⟩ U U . As ⟨U l(X ),DV ⟩ U is a compact space we will apply Proposition 10. Let P ∈ U l(X ) and let Y be a subset of U l(X ). Suppose that

⋂ -1 {ɛU (Q ) : Q ∈ Y } ⊆ ♢ RU(ɛU (P )).

We need to prove that there exists a subset Z of U l(X ) such that Z ∈ RU (P ) and Z ⊆ cl(Y) , where cl(Y ) is the topological closure of Y in the topological space ⟨U l(X ),DVU ⟩ .

Consider the filter F generated by the set {V (φ ) : Y ⊆ VU (φ )} . We prove that

(P,Fˆ) ∈ RU .

Let W ∈ F . Then there exists φ1,...,φn ∈ F m such that

Y ⊆ VU (φ ) ∩ ...∩ VU (φn) and V (φ ) ∩ ...∩ V (φn) ⊆ W.

So,  ⋂ -1 VU (φ) ∩ ...∩ VU(φn ) ∈ {ɛU (Q) : Q ∈ Y} ⊆ ♢RU (ɛU (P )) . Then

♢ (V (φ ∧ ...φ ) = V (♢(φ ∧ ...∧ φ )) ∈ ɛ (P ), RU U n U 1 n U

i.e. V (♢(φ1 ∧ ...∧ φn )) = ♢R (V (φ1 ∧ ...∧ φn)) ∈ P . Then ♢ (W ) ∈ P R . Thus (P,Fˆ) ∈ R U . Taking into account that the topological closure of Y in the space ⟨U l(X ),DVU ⟩ is  ⋂ cl(Y ) = {VU (φ ) : Y ⊆ VU(φ)} , it is easy to prove that ˆF ⊆ cl(Y ) . Therefore U e (M ) is m -saturated. □

Theorem 19. Let M be a model. Then the model V e(M ) is m -saturated.

Proof. Since ⟨D ,♢ ⟩ V R is a subalgebra of the algebra ⟨P (X ),♢ ⟩ R , we can define a map f : U l(X ) → U l(DV ) by f (P ) = P ∩ DV . By well-established results in the duality theory of Boolean algebras the map f is the dual map of the inclusion homomorphism between DV and P (X ) . From the results given by H. Hansen [6] we get that f : U l(X ) → Ul (DV ) is a surjective bounded morphism between the ultrafilter extension U (M ) e and the valuation extension V (M ) e . Since U e(M ) is m -saturated, by Proposition 14 we get that V e (M ) is also m -saturated. □

References

[1]    B. F. Chellas, Modal Logic: an introduction, Cambridge Univ. Press,1980.        [ Links ]

[2]    R. Goldblatt, Maps and Monads for Modal Frames, Studia Logica Volume 83, Numbers 1-3 (2006), pp.309-331.        [ Links ]

[3]    R. Goldblatt,  Mathematics of Modality, CSLI Lectures Notes 43, 1993.        [ Links ]

[4]    R. Goldblatt, Logics of Time and Computation, CSLI Lectures Notes 7, Second Edition, 1992.        [ Links ]

[5]    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 ]

[6]    H. H. Hansen. Monotonic modal logic (Master's thesis). Preprint 2003-24, ILLC, University of Amsterdam, 2003.        [ Links ]

[7]    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
scelani@exa.unicen.edu.ar

Recibido: 10 de abril de 2008
Aceptado: 28 de abril de 2008