aa r X i v : . [ m a t h . L O ] F e b CO-QUANTALE VALUED LOGICS
DAVID REYES AND PEDRO H. ZAMBRANOA
BSTRACT . In this paper, we propose a generalization of Contin-uous Logic ([BBHU08]) where the distances take values in suit-able co-quantales (in the way as it was proposed in [Fla97]). Byassuming suitable conditions (e.g., being co-divisible, co-Girardand a V -domain), we provide, as test questions, a proof of a ver-sion of the Tarski-Vaught test (Proposition 3.35) and Ło´s Theorem(Theorem 3.62) in our setting. Keywords. metric structures, lattice valued logics, co-quantales,co-Girard, co-divisibility, domains, Tarski-Vaught test, Ło´s theo-rem.
AMS classification 2010.
1. I
NTRODUCTION
S. Shelah and J. Stern proved in [SS78] that a first order attemptof study of classes of Banach Spaces has a “bad” behavior (this hasa very high Hanf number, having a model-theoretical behavior sim-ilar to a second order logic of binary relations). This led to developa suitable logic beyond first order logic, in order to do a suitablemodel-theoretic analysis of Banach Spaces.In [CK66], C. C. Chang and H. J. Keisler proposed a new logic withtruth values within a compact Hausdorff topological space, whichwas the first time where the term
Continuous Logic appeared. Theydeveloped basics on Mathematical Logic in this book, but by thenModel Theory had not been very developed (Morley’s first ordercategoricity theorem had just been proved, and there was no sta-bility theory by then). For some reason, people did not continueworking on this kind of logic until it was rediscovered in the 90’sby W. Henson and J. Iovino (see [HI02, Iov99]) and later by I. Ben-Yaacov et al (see [BBHU08]), but in the particular case by taking the
Date : February 12, 2021.The first author wants to thank the second author for the time devoted to advisehis undergraduate thesis, where this paper is one of its fruits. The second authorwants to thank Universidad Nacional de Colombia for the grant “Convocatoriapara el apoyo a proyectos de investigación y creación art´stica de la sede Bogotá dela Universidad Nacional de Colombia - 2019”. truth values in the unitary interval [
0, 1 ] , focusing on the study ofstructures based on complete metric spaces (e.g. Hilbert spaces to-gether with bounded operators -see [AB09]-, Banach spaces, Proba-bility spaces -see [BH04]-). This logic is known as Continuous Logic
Because of some technical reasons, people working on ContinuousLogic have to consider strong assumptions on the involved opera-tors (e.g. boundness) in order to axiomatize classes of metric struc-tures in this logic. This took us to the notion of
Metric Abstract Ele-mentary Class (see [HH09, Zam11]) for being able to deal with non-axiomatizable -in Continuous Logic- classes of complete metric struc-tures. However, this approach does not consider topological spacesin general.Independently, Lawvere provided in [Law73] a framework in Cat-egory Theory for being able to consider a logic with generalizedtruth values in order to study metric spaces from this point of view.However, there is no a deep model-theoretic study in this paper.There is an attempt of a study of first order Model Theory forTopological Spaces (see [FZ80]), but it was just suitable to studyparticular algebraic examples like Modules and Topological Groups,due to the algebraic nature of first order logic. Moreover, this ap-proach was left as a model-theoretic study of general TopologicalSpaces but it was the beginning of the model-theoretic study of Mod-ules (see [Pre88]).Quantales are a suitable kind of lattices introduced for being ableto deal with locales (a kind of lattices which generalizes the ideal ofopen sets of a topological space) and multiplicative lattices of idealsfrom Ring Theory and Functional Analysis (e.g., C ∗ -algebras andvon Neumann algebras). Considering the contravariant notions ina quantale (which we will called co-quantales ), Flagg gave in [Fla97]a way to deal with topological spaces as pseudo metric spaces wherethe distance takes values in a suitable quantale.In this paper, we will propose a generalization of Continuous Logicby defining distances with values in value co-quantales together withsuitable assumptions (e.g., being co-divisible, co-Girard and a V -domain).This paper is organized as follows: In the second section we willprovide basic facts in co-quantales. In the third section, we intro-duced our approach to co-quantales valued logics, analogously as itis done in Continuous Logic but by considering distances with val-ues in a suitable co-quantale. As test questions, we provide a proof O-QUANTALE VALUED LOGICS 3 of a version of the Tarski-Vaught test (Proposition 3.35) and a ver-sion of Ło´s Theorem (Theorem 3.62). A difference between our ap-proach and Continuous Logic lies on the fact that we can provideda version of Ło´s Theorem for D -products (before doing the quotientto force a D -product being an actual metric space -which is calleda D -ultraproduct in Continuous Logic-). We notice that the sameproof for D -products works for D -ultraproducts in our setting. Asconsequences of Ło´s Theorem, in a similar way as in first order andContinuous logics, we provide a proof of a version of CompactnessTheorem (Corollary 3.64) and of the existence of ω -saturated mod-els (Proposition 3.69).2. V ALUE CO - QUANTALES AND C ONTINUITY S PACES
In this section, we provide some basics on value co-quantales.2.1.
Valued lattices.Definition 2.1.
Given L a complete lattice and x, y ∈ L , we say that x is co-well below y (denoted by x ≺ y ), if and only, if for all A ⊆ X , if V A ≤ x then there exists a ∈ A such that a ≤ y . Remark 2.2. In [ ∞ ] , x ≺ y agrees with x < y . Lemma 2.3. ( [Fla97] ; Lemma 1.2) Let L be a complete lattice, then for all x, y, z ∈ L we have that (1) y ≺ x implies y ≤ x (2) z ≤ y and y ≺ x imply z ≺ x (3) y ≺ x and x ≤ z imply y ≺ z . Lemma 2.4. ( [Fla97] ; Lemma 1.3) If L is a complete lattice, then for all A ⊆ L and x ∈ L we have that V A ≺ x if and only if there exists a ∈ A such that a ≺ x . Definition 2.5.
A complete lattice L is said to be completely distribu-tive if a = V { b : a ≺ b } provided that a ∈ L . Lemma 2.6. ( [Fla97] ; Lema 1.6]) Given L a completely distributive latticeand x, y ∈ L provided that x ≺ y , there exists z ∈ L such that x ≺ z y z ≺ y . Definition 2.7. A value lattice is a completely distributive lattice L pro-vided that (1) ≺ (2) if δ, δ ′ ∈ L satisfy ≺ δ and ≺ δ ′ , then ≺ δ ∧ δ ′ . D. REYES AND P. ZAMBRANO
Examples 2.8. (1)
The 2-valued Boolean algebra := {
0, 1 } , where . (2) The ordinal number ω + := {
0, 1, ..., ω } together with the usualordering. (3) The unit interval I = [
0, 1 ] with the usual real ordering. (4) ([
0, 1 ] , ≥ ) = ([
0, 1 ] , ≤ ) op (which we will denote by E ). (5) ([ ∞ ] , ≤ usual ) . (6) ( [FK97] ; pg 115-117) Given a set R , let us denote P fin ( R ) = { X ∈ P ( X ) : X is finite } and for all X ∈ P fin ( R ) we denote ↓ ( X ) = { Y ∈ P ( X ) : Y ⊆ X } .Given Ω ( R ) = { p ∈ P ( P fin ( R )) : X ∈ p implies ↓ ( X ) ⊆ p } , then ( Ω ( R ) , ⊇ ) is a valued lattice. Value co-quantales.
In this paper, we do not work with theusual notion of quantale. We consider the contravariant notion (whichwe call co-quantale ) because this approach allows us to work with anotion of distance (pseudo-metric) with values in the co-quantale(see [Fla97]), in an analogous way as the metric structures given inContinuous Logic. We know that this is not the standard way tostudy quantales, but we chose this setting in order to do a similarstudy as it is done in Continuous Logic.
Definition 2.9. A co-quantale V is a complete lattice provided with acommutative monoid structure ( V , +) such that (1) The minimum element of V is the identity of ( V , +) ; i.e., a + = a for all a ∈ V and (2) for all a ∈ V and ( b i ) i ∈ I ∈ V I , a + V i ∈ I b i = V i ∈ I ( a + b i ) Proposition 2.10. ( [Fla97] ; Pg 6]) If V is a co-quantale, then for all a, b, c ∈ V we have that (1) a + = (2) a ≤ b implies c + a ≤ c + b Proposition 2.11. ( [Fla97] ; Thrm 2.2]) Given V a co-quantale and a, b ∈ V , define a · − b := V { r ∈ V : r + b ≥ a } . Therefore, for all c ∈ V we havethat (1) a · − b ≤ c if and only if a ≤ b + c (2) a ≤ ( a · − b ) + b (3) ( a + b ) · − b ≤ a (4) a · − b = if and only if a ≤ b (5) a · − ( b + c ) = ( a · − b ) · − c = ( a · − c ) · − b (6) a · − c ≤ ( a · − b ) + ( b · − c ) O-QUANTALE VALUED LOGICS 5
Since · − is the left adjoint of + and it preserves categorical limits,we have the following fact. Fact 2.12.
Let V be a co-quantale, then for any sequence ( b i ) i ∈ I and anyelement a ∈ V we have that ( W i ∈ I b i ) · − a = W i ∈ I ( b i · − a ) Lemma 2.13.
Given a co-quantale V , for any a, b ∈ V , a ≤ b impliesthat for all c we have that c · − a ≥ c · − b and a · − c ≤ b · − c .Proof. Let c ∈ V , then by Proposition 2.11 (2) we may say that c ≤ ( c · − a ) + a . So, c ≤ ( c · − a ) + b whenever a ≤ b (by Proposition 2.10(2)), and so by Proposition 2.11 (1) this is equivalent to c · − b ≤ c · − a .On the other hand, since a ≤ b then b ≤ ( b · − c ) + c ( Proposition 2.11 (2) ) a ≤ ( b · − c ) + c ( since a ≤ b ) a · − c ≤ b · − c ( Proposition 2.11 (1) ) (cid:3) Proposition 2.14.
Given V a co-quantale, a ∈ V and a sequence ( b i ) i ∈ I in V , we have that a · − V i ∈ I b i = W i ∈ I ( a · − b i ) .Proof. By Proposition 2.11 (1), a · − V i ∈ I b i ≤ W i ∈ I ( a · − b i ) implies that a ≤ W i ∈ I ( a · − b i ) + V i ∈ I b i = V i ∈ I (( W i ∈ I ( a · − b i )) + b i ) , which holdsby Proposition 2.11 (2) and Proposition 2.10 (2) allow us to say that a ≤ ( a · − b j ) + b j ≤ ( W i ∈ I ( a · − b i )) + b j for any j ∈ J .Since j ∈ I was taken arbitrarily, then a ≤ V i ∈ I (( W i ∈ I ( a · − b i )) + b i ) . (cid:3) Definition 2.15. A ( V , + , 0 ) co-quantale is said to be a value co-quantale if V is a value lattice. Definition 2.16.
Given V a value co-quantale, let set V + = { ǫ ∈ V : ≺ ǫ } , and we call it the positives filter of V . Lemma 2.17. ( [Fla97] ; Thrm 2.9]) If ( V , + , 0 ) is a value co-quantale,given ǫ ∈ V + there exists δ ∈ V + such that δ + δ ≺ ǫ By an obvious inductive argument, we can prove the followingfact.
Corollary 2.18 ([LRZ18]; Remark 2.26) . Given any ǫ ∈ V + and n ∈ N \ { } , there exists θ ∈ V + such that n times nθ := z }| { θ + · · · + θ ≺ ǫ . Fact 2.19. ( [Fla97] ; Thrm 2.10]) Given a value co-quantale V and p ∈ V , p = p + = p + V { ǫ ∈ V : ≺ ǫ } = V { p + ǫ : ≺ ǫ } . D. REYES AND P. ZAMBRANO
Lemma 2.20. ( [Fla97] ; Thrm 2.11]) If V is a value co-quantale and p, q ∈ V such that q ≺ p , then there exist r, ǫ ∈ V such that ≺ ǫ , q ≺ r and r + ǫ ≺ p Example 2.21.
The 2-valued Boolen algebra = {
0, 1 } is a value co-quantale, by taking + := ∨ Example 2.22. ([ ∞ ] , ≤ , +) is a value co-quantale. Notice that the oper-ation − · − a ( a ∈ [ ∞ ] ) is given by b · − a := max {
0, b − a } . We denotethis example by D . Example 2.23.
There are several ways to define a value co-quantale withunderlying value lattice E . One of them is by taking + as the usual realproduct, another one consists by taking + := ∨ . Also, the Łukasiewicz’saddition + L provides a value co-quantale structure with underlying lattice E -where a + L b = ∧{
0, a + b − } by taking ∧ in ([
0, 1 ] , ≥ ) -. Denote theprevious examples by E ∗ , E ∨ and E L respectively. Fact 2.24. ( [FK97] ; pg 115-117) Given a non empty set R , ( Ω ( R ) , ⊇ , ∩ ) isa value co-quantale. In particular, if ( X, τ ) is a topological space, ( Ω ( τ ) , ⊇ , ∩ ) is a value co-quantale. This last example is called the free local asso-ciate to ( X, τ ) . Definition 2.25.
A co-quantale ( V , ≤ , +) is said to be co-divisible if forall a, b ∈ V , a ≤ b implies that there exists c ∈ V such that b = a + c . Lemma 2.26.
A co-quantale ( V , ≤ , +) is co-divisible,if and only, if for all a, b ∈ V , a ≤ b implies b = a + ( b · − a ) .Proof. This follow from − · − a ⊣ a + − . (cid:3) The following property allows us to approximate by means of a N -indexed sequence. Definition 2.27. ( [LRZ18] ) Given a co-quantale V , we say that it has the Sequential Approximation From Above property (shortly,
SAFA ), ifand only, if there is a sequence ( u n ) n ∈ N such that (1) V n ∈ N u n = (2) for all n ∈ N , ≺ u n (3) for all n ∈ N , u n + ≤ u n Co-Girard value co-quantales.
In this section, we give the basicnotions and results relative to co-Girard value co-quantales, whichallows to consider a kind of pseudo complement relative to somefixed element b (a dualizing element) and · − . This notion will allowto prove our test questions (Proposition 3.35 and Theorem 3.62). O-QUANTALE VALUED LOGICS 7
Definition 2.28.
Given a co-quantale ( V , +) , an element d ∈ V is saidto be a dualizing element , if and only if, for all a ∈ V we have that a = d · − ( d · − a ) . Definition 2.29.
A co-quantale V is said to be co-Girard , if and only, ifit has a dualizing element. Continuity spaces.
In this section, we will provide some ba-sics on continuity spaces , a framework given in [Fla97, FK97] in orderto generalize (pseudo) metric spaces but considering distances withvalues in general co-quantales.
Definition 2.30.
Given X = ∅ a set, V a value co-quantale and a mapping d : X × X → V , the pair ( X, d ) is said to be a V - continuity space , if andonly if, (1) (reflexivity) for all x ∈ X , d ( x, x ) = , and (2) (transitivity) for any x, y, z ∈ X , d ( x, y ) ≤ d ( x, z ) + d ( z, y ) .If there is no confusion about which co-quantale V we are considering,we may say that ( X, d ) is just a continuity space . Example 2.31.
Let V := (
2, , ≤ , ∨ ) and X = ∅ be a set. A -continuityspace ( X, d ) codifies a binary relation R := { ( x, y ) ∈ X × X : d ( x, y ) = } on X which is reflexive and transitive. Example 2.32.
Given D := ([ ∞ ] , ≤ , +) , a D -continuity space is justa pseudo metric space (where the distance betwwen two elements might beinfinite). Example 2.33.
Given ( X, τ ) a topological space and a, b ∈ X , define d ( a, b ) := { A ⊆ finite τ : for all U ∈ A, a ∈ U implies b ∈ U } . ( X, d ) is a Ω ( τ ) -continuity space. Example 2.34.
Given a value co-quantale V , define d : V × V → V by d ( a, b ) := b · − a . ( V, d ) is a V -continuity space. Proposition 2.35. ( [FK97] ; Pg 119]) Given a value co-quantale V and ( X, d X ) , ( Y, d Y ) V -continuity spaces, define d X × Y : ( X × Y ) × ( X × Y ) → V by d X × Y (( x , y ) , ( x , y )) := d X ( x , x ) ∨ d Y ( y , y ) , then ( X × Y, d X × Y ) is a V -continuity space. Remark 2.36.
There is another way to provide to a value co-quantale a V -continuity space structure by defining d sV ( a, b ) := ( a · − b ) ∨ ( b · − a ) forall a, b ∈ V . d sV is called the symmetric distance for V . Notice that for D. REYES AND P. ZAMBRANO all a ∈ V we have that d sV ( a, 0 ) = d sV (
0, a ) = a : d sV ( a, 0 ) = ( a · − ) ∨ ( · − a )= ( a · − ) ∨ ( by Proposition 2.11 (4) and = min V )= a · − = ^ { r : r + ≥ a } ( by definition of · −)= ^ { r : r ≥ a } = a The topology of a V -continuity space. In this subsection, we willgive some basics on the underlying topology of a V -continuity space. Note 2.37.
Throughout this subsection, V will denote a value co-quantale. Definition 2.38.
Given a V -continuity space ( X, d ) , ǫ ∈ V + and x ∈ X define B ǫ ( x ) := { y ∈ X : d ( x, y ) ≺ ǫ } (which we call the disc with radius ǫ centered in x . Definition 2.39.
A subset U of a V-continuity space ( X, d ) is said to be open , if and only if, given x ∈ U there exists some ǫ ∈ V + such that B ǫ ( x ) ⊆ U . Fact 2.40. ( [Fla97] ; Thrm 4.2) The family of open subsets in a V -continuityspace ( X, d ) is closed under finite interesections and arbitrary unions. Also, ∅ and X are open sets. Definition 2.41.
Given ( X, d ) a V -continuity space, the family of opensets of ( X, d ) determines a topology on X , which we will call the topologyinduced by d and we will denote it by τ d . Lemma 2.42.
Given ( X, d ) a V -continuity space, for all x ∈ X and ǫ ∈ V + we have that the disc B ǫ ( x ) is an open set of X .Proof. Let y ∈ B ǫ ( x ) , so d ( x, y ) ≺ ǫ . By Fact 2.19 d ( x, y ) = V { d ( x, y )+ δ : ≺ δ } ≺ ǫ , then by Lemma 2.4 there exists δ ∈ V + such that d ( x, y ) + δ ≺ ǫ . We may assure that B δ ( y ) ⊆ B ǫ ( x ) : If z ∈ X satisfies d ( y, z ) ≺ δ , then d ( x, z ) ≤ d ( x, y ) + d ( y, z ) ≤ d ( x, y ) + δ ≺ ǫ . (cid:3) Fact 2.43.
Given a V -continuity space ( X, d ) , the family of open discsforms a base for τ d . Definition 2.44.
Given a V -continuity space ( X, d ) , A ⊆ X and x ∈ X ,define d ( x, A ) := V { d ( x, a ) : a ∈ A } . Proposition 2.45.
Given a V -continuity space ( X, d ) , then a subset A ⊆ X is τ d -closed, if and only if, for all x ∈ X we have that d ( x, A ) = implies x ∈ A . O-QUANTALE VALUED LOGICS 9
Proof.
Suppose that A ⊆ X is τ d -closed and let x ∈ X be such that d ( x, A ) = . In case that x / ∈ A , since A is τ d -closed, there exists ǫ ∈ V + such that B ǫ ( x ) ⊆ A c = { y ∈ X : y / ∈ A } . Since d ( x, A ) := V { d ( x, a ) : a ∈ A } = ≺ ǫ , by Fact 2.4 there exists a ∈ A such that d ( x, a ) ≺ ε , so a ∈ B ǫ ( x ) ∩ A (contradiction).On the other hand, let A ⊆ X be such that for all x ∈ X , d ( x, A ) = implies x ∈ A . Suppose that y ∈ X belongs to the adherence of A , soby Fact 2.43 given any ǫ ∈ V + , we have that A ∩ B ǫ ( y ) = ∅ . Therefore,for any ε ∈ V + we have that d ( y, A ) := V { d ( y, a ) : a ∈ A } ≤ ε , so d ( y, A ) ≤ V { ε : ≺ ε } = , hence by hypothesis we may say that y ∈ A . Therefore, A is closed. (cid:3) Corollary 2.46.
Given a V -continuity space ( X, d ) , the topological closureof A ⊆ X is given by cl ( A ) := A = { y ∈ X : d ( y, A ) = } . Definition 2.47.
Given a V -continuity space ( X, d X ) , ǫ ∈ V + and x ∈ X ,define the closed disc of radius ε centered in x by C ǫ ( x ) := { y ∈ X : d X ( x, y ) ≤ ǫ } . Fact 2.48. ( [FK97] ; Lemma 3.2 (2)) Let ( X, d X ) be a V -continuity spaceand x ∈ X . The family { C ǫ ( x ) : ǫ ∈ V + } determines a fundamental systemof neighborhoods around x . Definition 2.49.
Given a V -continuity space ( X, d ) , define the dual dis-tance d ⋆ : X × X → V relative to d by d ⋆ ( x, y ) := d ( y, x ) . In general,if we add ⋆ as a superscript to any topological notion, it means that it isrelated to the distance d ⋆ ; e.g., the topology induced by d ⋆ is denoted by τ ⋆ d . Proposition 2.50.
Given a V -continuity space ( X, d ) , x ∈ X and ǫ ∈ V + , C ⋆ ǫ ( x ) is τ d -closed.Proof. Let x ∈ X and ǫ ∈ V + , so by Proposition 2.45 it is enough tocheck that for any y ∈ X , d ( y, C ⋆ ǫ ( x )) := V { d ( y, a ) : a ∈ C ⋆ ǫ ( x ) } = implies that y ∈ C ⋆ ǫ ( x ) . Let y ∈ X be such that d ( y, C ⋆ ǫ ( x )) = and δ ∈ V + . Since d ( y, C ⋆ ǫ ( x )) = ≺ δ , by Fact 2.4 there exists z ∈ C ⋆ ǫ ( x ) such that d ( y, z ) ≺ δ , so d ⋆ ( x, y ) := d ( y, x ) ≤ d ( y, z ) + d ( z, x ) = d ( y, z ) + d ⋆ ( x, z ) ≤ δ + ǫ , therefore d ⋆ ( x, y ) ≤ V { ǫ + δ : ≺ δ } = ǫ + V { δ : ≺ δ } = ǫ + = ǫ , therefore y ∈ C ⋆ ǫ ( x ) . (cid:3) Definition 2.51.
Given a V -continuity space ( X, d ) , define the symmet-ric space relative to ( X, d ) by ( X, d s ) , where d s ( x, y ) := d ( x, y ) ∨ d ⋆ ( x, y ) .In general, we will denote the topological notions related to d s by addingthe superscript s . Proposition 2.52. ( [FK97] ; Lemma 3) Given a V -continuity space ( X, d ) ,for the topology τ s induced by d s we have that U ⊆ X belongs to τ s , if andonly if, there exist V, W ⊆ X such that V ∈ τ d , W ∈ τ ⋆ and U = V ∩ W . Lemma 2.53. ( [FK97] ; Lemma 3,4) If ( X, d ) is a V -continuity space, ( X, τ s ) satisfies the following separation properties: (1) (pseudo-Hausdorff) For all x, y ∈ X , if x / ∈ { y } according to ( X, τ d ) then there exist U, V ⊆ X such that x ∈ U, y ∈ V, U ∈ τ d , V ∈ τ ⋆ and U ∩ V = ∅ . (2) (regularity) For all x ∈ X and A ⊆ X , if A ∈ τ d and x ∈ A thenthere exist U, C ⊆ X such that U is τ d -open, C es τ ⋆ -closed and x ∈ U ⊆ C ⊆ A . V -domains. In order to give a version of Ło´s Theorem in oursetting, following [CK66, BBHU08], we need to consider compactand Hausdorff topological spaces. The setting which involves theseassumptions in continuity spaces corresponds to V -domains . Definition 2.54. A V -continuity space ( X, d ) is said to be T , if and onlyif, for any x, y ∈ X , d ( x, y ) = and d ( y, x ) = implies x = y . Remark 2.55 ([FK97], pg 120) . A continuity space ( X, d ) is T , if andonly if, ( X, τ d ) is T as a topological space and ( X, τ sd ) is Hausdorff. Definition 2.56. A V -continuity space ( X, d ) is said to be a V − domain ,if and only if, it is T and ( X, τ sd ) is compact. Remark 2.57.
Let ( X, d ) be a V -domain, therefore by definition ( X, τ sd ) iscompact. Since ( X, d ) is T , by Remark 2.55 ( X, τ sd ) is Hausdorff. The importance of the previous properties lies on the fact thatthese allow us to provide a proof of a version of Łos’s Theorem inthe logic that we will introduce in this paper (Theorem 3.62). Wewill provide some examples which satisfy these properties.
Proposition 2.58. ( [FK97] ; Thrm 4.14) The following examples are do-mains: (1) = ( {
0, 1 } , 0 ≤ ∨ ) . (2) ([
0, 1 ] , ≤ , +) . (3) The quantale of errors ([
0, 1 ] , ≥ , ⊗ ) , where a ⊗ b := max { a + b −
1, 0 } . (4) The quantale of fuzzy subsets associated to a set X . (5) The free local associated to a set X : ( Ω ( X ) , ⊇ , ∩ ) .
3. V
ALUE CO - QUANTALE LOGICS
In this section, we will introduce a logic with truth values withinvalue co-quantales, generalizing Continuous Logic (see [BBHU08], It is denoted by Λ ( X ) in [FK97] Tt is denoted by Γ ( X ) in [FK97] O-QUANTALE VALUED LOGICS 11 where the truth values are taken in the unitary interval [
0, 1 ] , whichis a particular case of our setting).Throughout the rest of this paper, we assume some technical con-ditions (Definitions 2.25, 2.29 and 2.56) that we need for providinga proof of a version of Tarski-Vaught test -Proposition 3.35- and aversion of Ło´s Theorem -Theorem 3.62- for the logics introduced inthis paper. At some point, we require to work with the symmetricdistance d sV of V . Assumption 3.1.
Throughout this section, we assume that V is a valueco-quantale which is co-divisible, co-Girard and a V -domain. Modulus of uniform continuity.
Modulus of uniform continu-ity are introduced in [BBHU08] as a technical way of controlling fromthe language the uniform continuity of the mappings consideredin Continuous Logic. In this subsection, we develop an analogousstudy of modulus of uniform continuity but in the setting of map-pings valued in value co-quantales.
Remark 3.2.
Given ( M, d M ) , ( N, d N ) V -continuity spaces and ( x , y ) , ( x , y ) ∈ M × N , we define d M × N (( x , y ) , ( x , y )) := d M ( x , y ) ∨ d N ( x , y ) . By Proposition 2.35, ( M × N, d M × N ) is a V -continuity space. Definition 3.3. (c.f. [BBHU08] ; pg 8) Given a mapping f : M → N between two V -continuity spaces ( M, d M ) and ( N, d N ) , we say that ∆ : V + → V + is a modulus of uniform continuity for f , if and only if, forany x, y ∈ M and any ǫ ∈ V + , d M ( x, y ) ≤ ∆ ( ǫ ) implies d N ( f ( x ) , f ( y )) ≤ ǫ . As a basic consequence we have the following fact.
Proposition 3.4.
Given ( M, d M ) , ( N, d N ) , ( K, d K ) V -continuity spacesand f : M → N , g : N → K uniformly continuous mappings, ∆ and Θ modulus of uniform continuity for f and g respectively, then ∆ ◦ Θ is amodulus of uniform continuity for g ◦ f . Definition 3.5.
Given a sequence of mappings ( f n ) n ∈ N with domain ( M, d M ) and codomain ( N, d N ) (both of them V -continuity spaces), we say that ( f n ) n ∈ N uniformly converges to a mapping f : M → N , if and onlyif, for all ǫ ∈ V + there exists n ∈ N such that for any m ≥ n and for any x ∈ M we may say that d N ( f m ( x ) , f ( x )) ≤ ǫ . It is straightforward to see that uniform convergence behaves wellwith respect to composition of mappings.
Proposition 3.6.
Let ( M, d M ) , ( N, d N ) , ( K, d K ) be V -continuity spaces, f : M → N , ( f n ) n ∈ N be a sequence of mappings from M to N , g : N → K ,and ( g n ) n ∈ N be a sequence of mappings from N to K such that ( f n ) n ∈ N uniformly converges to f and ( g n ) n ∈ N uniformly converges to g . If g isuniformly continuous, then ( g n ◦ f n ) n ∈ N uniformly converges to g ◦ f . Uniform continuity of V and W . The following fact is very im-portant because, as in Continuous Logic, it allows us to control (byusing directly the language) the uniform continuity of both V (inf)and W (sup), understood as quantifiers (in an analogous way as inContinuous Logic). Proposition 3.7.
Let ( M, d M ) , ( N, d N ) be V -continuity spaces, f : M × N → V be a uniformly continuous mapping provided with a modulus ofuniform continuity ∆ : V + → V + , then ∆ is also a modulus of uniformcontinuity for the mappings W f : M → V and V f : M → V defined by x → W y ∈ N f ( x, y ) and x → V y ∈ N f ( x, y ) respectively.Proof. Let ǫ ∈ V be such that ≺ ǫ , y ∈ N and a, b ∈ M be such that d M ( b, a ) ≤ ∆ ( ǫ ) . Then, d M × N (( b, y ) , ( a, y )) := d M ( b, a ) ∨ d N ( y, y ) = d M ( b, a ) ≤ ∆ ( ǫ ) Since ∆ is a modulus of uniform continuity for f , then f ( a, y ) · − f ( b, y ) ≤ d V ( f ( b, y ) , f ( a, y )) ≤ ǫ By Proposition 2.11 (1) we may say f ( a, y ) ≤ f ( b, y ) + ǫ ≤ _ z ∈ N f ( b, z ) + ǫ Since y ∈ N was taken arbitrarily, then _ z ∈ N f ( a, z ) ≤ _ z ∈ N f ( b, z ) + ǫ and by Proposition 2.11 (1) _ f f ( a ) · − _ f f ( b ) = _ z ∈ N f ( a, z ) · − _ z ∈ N f ( b, z ) ≤ ǫ. In a similar way we prove the related statement for V f . (cid:3) As an immediate consequence, we have the following useful facts.
O-QUANTALE VALUED LOGICS 13
Corollary 3.8.
Given an arbitrarily set I = ∅ and I -sequences ( a i ) i ∈ I , ( b i ) i ∈ I in a V-continuity space ( M, d M ) , if ǫ ∈ V + satisfies d V ( a i , b i ) ≤ ǫ for all i ∈ I , then d V ( W i ∈ I a i , W i ∈ I b i ) ≤ ǫ and d V ( V i ∈ I a i , V i ∈ I b i ) ≤ ǫ . Corollary 3.9.
Given a V -continuity space ( M, d M ) , I = ∅ , and a I -sequence of mappings ( f i : M → V ) i ∈ I , if ∆ : V + → V + is a modulusof uniform continuity for f i ( i ∈ I ), then ∆ is also a modulus of uni-form continuity for both W i f i : M → V and V i f i : M → V defined by x → W i ∈ I f i ( x ) and x → V i ∈ I f i ( x ) , respectively. Proposition 3.10.
Let ( M, d M ) , ( N, d N ) be V − continuity spaces, f : M × N → V , ( f n ) n ∈ N a sequence of mappings from M × N to V suchthat ( f n ) n ∈ N uniformly converges to f , then (cid:16)W y ∈ N f n ( x, y ) (cid:17) n ∈ N uniformlyconverges to W y ∈ N f ( x, y ) and (cid:16)V y ∈ N f n ( x, y ) (cid:17) n ∈ N uniformly converges to V y ∈ N f ( x, y ) .Proof. Since by hypothesis ( f n ) n ∈ N uniformly converges to f , given ǫ ∈ V + there exists n ∈ N such that if m ≥ n , then d V ( f m ( x, y ) , f ( x, y )) ≤ ǫ for any ( x, y ) ∈ M × N . For a fixed x ∈ M and m ≥ n , define thesequences ( f m ( x, y )) y ∈ N and ( f ( x, y )) y ∈ N , which satisfy the hypothe-sis of Corollary 3.8, so d V ( W y ∈ N f m ( x, y ) , W y ∈ N f ( x, y )) ≤ ǫ whenever m ≥ n . Since this holds for all ǫ ∈ V + , we got the uniform conver-gence desired.In an analogous way, we prove the respective statement for V . (cid:3) Some basic notions.
In first order logic, an n -ary relation in aset A is defined as a subset of A n . In this way, a tuple ( a , · · · , a n ) might belong to A or not. We may codify this by using characteristicfunctions, dually, by the discrete distance from a tuple in A n to R .In Continuous Logic, an n -ary relation in A is understood accordingto this second approach by taking a uniformly continuous mapping R : A n → [
0, 1 ] . In this setting, we generalize this approach replacing [
0, 1 ] by a suitable value co-quantale V .All topological notions about V are relative to the symmetric topol-ogy of V . Definition 3.11.
Given a V -continuity space ( M, d M ) and A ⊆ M , define diam ( A ) := W { d M ( a, b ) : a, b ∈ A } . (which we will call the diameter of A . Continuous structures.
Given a V -continuity space ( M, d M ) withdiameter p ∈ V , we define a continuous structure with underline V -continuity space ( M, d M ) as a tuple M = (( M, d M ) , ( R i ) i ∈ I , ( f j ) j ∈ J , ( c k ) k ∈ K ) ,where: (1) For each i ∈ I , R i : M n i → V is a uniformly continuous map-ping (which we call a predicate ), with modulus of uniformcontinuity ∆ R i : V + → V + . In this case, n i < ω is said to bethe arity of R i .(2) For each j ∈ J , f j : M m j → M is a uniformly continuous map-ping with modulus of uniform continuity ∆ F j : V + → V + . Inthis case, m j < ω is said to be the arity of F j .(3) For each k ∈ K , C k is an element M .3.2.2. Languages for continuous structures.
For a fixed continuous struc-ture M := (( M, d M ) , ( R i ) i ∈ I , ( f j ) j ∈ J , ( c k ) k ∈ K ) , we will define the lan-guage associated to M in the natural way, as follows.Predicate symbols: R i → ( P i , n i , ∆ R i ) ( i ∈ I )Function symbols: f j → ( F j , n j , ∆ f j ) ( j ∈ J ) Constant symbols: c k → e k ( k ∈ K ) .This set of non logical symbols is denoted by NL M .Let us denote by LG := { d } ∪ X ∪ C ∪ { W , V } (which we call logicalsymbols ), where: • X = { x i : i ∈ N } is a countable set of variables . • C is the set of all uniformly continuous mappings with do-main V n and codomain V ( ≤ n < ω ). As in Continu-ous Logic, we understand a uniformly continuous mapping u : V n → V as a connective . • d is a symbol, which we will interpret as the V -valued dis-tance given in ( M, d M ) . This symbol will play the role of theequality in first order logic, in a similar way as we do in Con-tinuous Logic. Definition 3.12.
Given a continuous structure M :=(( M, d M ) , ( R i ) i ∈ I , ( f j ) j ∈ J , ( c k ) k ∈ K ) , we define the language based on M as L M := NL M ∪ LG . We will drop M if it is clear from the context. We define the notion of terms as follows.
Definition 3.13.
Given a language based on a continuous structure L , wedefine the notion of L -term recursively, as follows: • Any variable and any constant symbol is an L -term. • Given L -terms t , ..., t n and a function symbol f ∈ L of arity n , ft , ..., t n is an L -term. Definition 3.14. An L -term is said to be closed , if and only if, it is builtwithout use of variables. Now, we provide the notion of L -formulae in this new setting. Wemimic the analogous notion given in Continuous Logic. O-QUANTALE VALUED LOGICS 15
Definition 3.15.
Given L a language based on a continuous structure, wedefine the notion of L -formula recursively, as follows: • Given L -terms t , t , dt t is an L -formula. • Given L -terms t , ...t n and a predicate symbol P ∈ L of arity n , Pt , ...t n is an L -formula. • Given L -formulas ψ , ..., ψ m and a connective (i.e., a uniformlycontinuous mapping) a : V m → V , then aψ , ..., ψ m is an L -formula. • Given an L -formula ψ and a variable x , both V xψ and W xψ are L -formulas. Remark 3.16.
Let V be a co-Girard value co-quantale and b ∈ V be adualizing element. Denote x ′ := b · − x . Denote the usual, dual and sym-metric distances in V by d , d ∗ and d s respectively. Notice that the mapping b · − (cid:3) : V → V defined by ( b · − (cid:3) )( x ) := b · − x is uniformly continu-ous (relative to the symmetric topology) provided with modulus of uniformcontinuity id V + . In fact, given x, y ∈ V we have that d ( y, x ) = x · − y = ( b · − ( b · − x )) · − y ) ( b is a dualizing element) = b · − (( b · − x ) + y ) (by Prop. 2.11 (5)) = b · − ( y + ( b · − x )) ( + is commutative) = ( b · − y ) · − ( b · − x ) (by Prop. 2.11 (5)) = y ′ · − x ′ = d ( x ′ , y ′ ) Therefore, d ( y, x ) = d ( x ′ , y ′ ) = d ∗ ( y ′ , x ′ ) .Exchanging the role of y and x above, we may say that d ( x, y ) = d ∗ ( y, x ) = d ∗ ( x ′ , y ′ ) = d ( y ′ , x ′ ) . Since d s ( x, y ) = d ( y, x ) ∨ d ∗ ( y, x ) = d ∗ ( y ′ , x ′ ) ∨ d ( y ′ , x ′ ) = d s ( y ′ , x ′ ) , then ( b · − (cid:3) ) respects the distance d s and thereforeit is uniformly continuous relative to the symmetric topology provided with id V + as a modulus of uniform continuity. Notation 3.17.
The subsequences W x and V x of an L -fórmula can bewritten as W x and V x , respectively. d ( t , t ) denotes the sequence dt t . Definition 3.18. An L -formula φ is said to be quantifier-free , if and onlyif, there are no appearances of W x and V x inside φ . Definition 3.19.
An appearance of a variable x inside an L -fórmula φ issaid to be free whenever it is not under the scope of W x o V x inside φ . Definition 3.20. An L -formula φ is said to be an L - sentence , if and onlyif, all appearances of variables are not free. Notation 3.21. φ ( x , ..., x n ) means that the variables that appear free in φ are among x , ..., x n . L -structures. Let L be a language based on a V -continuous struc-ture M := (( M, d M ) , ( R i ) i ∈ I , ( f j ) j ∈ J , ( c k ) k ∈ K ) . Given a V -continuityspace ( N, d N ) with diameter (Definition 3.11) at most diam ( M ) , wewill interpret the symbols in NL M in ( N, d N ) as follows: • For any predicate symbol P of arity n and modulus of uni-form continuity ∆ P , associate a uniformly continuous map-ping P N : N n → V with modulus of uniform continuity ∆ P . • For any function symbol F of arity m and with modulus ofuniform continuity ∆ F , associate a uniformly continuous map-ping F N : N m → N with modulus of uniform continuity ∆ F . • For any constant symbol e , associate an element e N ∈ N .Also, the logical symbol d is interpreted in ( N, d N ) as the distance d N := d N . Definition 3.22.
Given a language L based on a continuous structure M := (( M, d M ) , ( R i ) i ∈ I , ( f j ) j ∈ J , ( c k ) k ∈ K ) and a V -continuity space ( N, d N ) ,the continuous structure obtained by interpreting the symbols of L on N =(( N, d N ) as above, N := (( N, d N ) , ( P N i ) i ∈ I , ( F N j ) j ∈ J , ( e k ) k ∈ K ) , is said to bean L - structure . Semantics.
Given an L -structure N and A ⊆ N , we extend thelanguage L by adding new constant symbols c a ( a ∈ A ), interpreting c N a := a . Abusing of notation, we will write a instead of c a , butunderstood as a constant symbol. Let us denote this language by L ( A ) . Definition 3.23.
Given an L − structure N , for any L − term t we definerecursively its interpretation in N , denoted by t N , as follows: (1) If t is a constant symbol c , define t N := c N . (2) If t is a variable x , define t N : N → N as the identity function of N . (3) If t is of the form ft , ...t n provided that f is an n -ary function sym-bol and t ( x ) , ..., t n ( x ) are L -terms, define t N := f N ( t N , ..., t N n ) asthe mapping f N ( t N , ..., t N n ) : N m → N where ( a ) → f N ( t N ( a ) , ..., t N n ( a )) for all a ∈ N m . Definition 3.24.
Let N be an L -structure. We define recursively the in-terpretation of L ( N ) -sentences in N , as follows. (1) ( d ( t , t )) N := d N ( t N , t N ) , where t , t are L ( N ) -terms O-QUANTALE VALUED LOGICS 17 (2) ( P ( t , ..., t n )) N := P N ( t N , ..., t N n ) , where P is an n -ary predicatesymbol and t , · · · , t n are L ( N ) -terms. (3) ( u ( φ , .., φ n )) N := u ( φ N , ..., φ N n ) for any uniformly continuousmapping (connective) u : V n → V and all L ( N ) -sentences φ , ..., φ n . (4) ( W x φ ) N := W a ∈ N φ N ( a ) , whenever φ ( x ) is an L ( N ) -formula. (5) ( V x φ ) N := V a ∈ N φ N ( a ) , whenever φ ( x ) is an L ( N ) -formula. Analogously as in Continuous Logic, all terms and all formulaehave a modulus of uniform continuity, which do not depend of thestructures.
Proposition 3.25.
Given L a language based on a continuous structure, φ ( x , ..., x n ) an L − formula and t ( x , ..., x m ) an L -term, then there exist ∆ φ : V + → V + and ∆ t : V + → V + such that for any L -structure N , ∆ φ isa modulus of uniform continuity for φ N and ∆ t is a modolus of continuityfor t N .Proof. The basic cases are given by definition, since constant sym-bols’s interpretations can be viewed as constant functions, variablesare interpreted as the identity function and predicate symbols areinterpreted as a uniformly continuous mapping with the respectivemodulus of uniform continuity. The connective case correspondsto compose uniformly continuous mappings (and we get the de-sired result by Proposition 3.4), and the quantifier cases follow fromProposition 3.7. (cid:3)
Definition 3.26.
Given L a language based on a continuous structure and M , N L − structures , we say that M is an L - substructure of N , if andonly if, M ⊆ N and the interpretations of all non logical symbols and of d in M correspond to the respective restrictions of the interpretations in N ofthose symbols. L -conditions. Fix L a language based on a continuous struc-ture. Definition 3.27. (c.f. [BBHU08] ; Def 3.9) Given φ ( x , ..., x n ) , φ ( x , ..., x n ) L − formulas , we say that φ is logically equivalent to φ , if andonly if, for any L -structure M and any a , ..., a n ∈ M we have that φ M ( a , ..., a n ) = φ M ( a , ..., a n ) . Definition 3.28.
Let φ ( x , ..., x n ) , φ ( x , ..., x n ) be L -formulas and M bean L -structure, we define the logical distance between φ and φ relativeto M as follows: d ( φ , φ ) M := W { d V ( φ M ( a , ...a n ) , φ M ( a , ...a n )) : a , ...a n ∈ M } The logical distance between φ , φ is defined as follows: d ( φ , φ ) := W { d ( φ , φ ) M : M is an L − structure } We define the notion of satisfiability in an L -structure in an analo-gous ways as in Continuous Logic, by using the notion of L -conditions. Definition 3.29. (c.f. [BBHU08] ; pg 19) An
L-condition E is a formalexpression of the form φ = , where φ ( x , ..., x n ) is an L -fórmula. An L -condition E is said to be closed if it is of the form φ = , where φ is an L -sentence. Given an L -formula φ ( x , ..., x n ) , the related condition E : φ ( x , ..., x n ) = is denoted by E ( x , ..., x n ) . Definition 3.30.
Given φ ( x , ..., x n ) an L -fórmula, M an L -structure and a , .., a n ∈ M , the L -condition E ( x , ..., x n ) : φ ( x , ..., x n ) = is said tobe satisfied in M for a , ..., a n , if and only if, φ M ( a , ..., a n ) = . Wedenote this by M | = E ( a , ..., a n ) . Notation 3.31.
Given φ, ψ L -formulae, we denote by φ = ψ the L -condition ( φ · − ψ ) ∨ ( ψ · − φ ) = and we denote by φ ≤ ψ the L -condition φ · − ψ = . Definition 3.32. An L - theory is a set of closed L -conditions. Definition 3.33.
Given an L -theory T and an L -structure M , we say that M is a model of T , if and only if, for any L -condition E ∈ T we have that M | = E . Tarski-Vaught test.Definition 3.34. (c.f. [BBHU08] ; Def 4.3)Let M , N be L -structures. (1) We say that M is elementary equivalent to N (denoted by M ≡ N ), if and only if, any L -sentence ϕ satisfies ϕ M = ϕ N . (2) Let M be an L -substructure of N . We say that M is an L - elementarysubstructure of N (denoted by M N ), if and only if, any L -formula ϕ ( x , ..., x n ) satisfies ϕ M ( a , ..., a n ) = ϕ N ( a , ..., a n ) forall a , ..., a n ∈ M . In this case, we also say that N is an L -elementary extension of M . We will provide a version of the well-known Tarski-Vaught test,as a equivalence of being an L -elementary substructure, as it holdsin both first order and Continuous logics. We need to assume that V is co-Girard (Definition 2.29). Proposition 3.35. (Tarski-Vaught test, c.f. [BBHU08]
Prop 4.5) Assumethat V is a co-Girard value co-quantale and let b a dualizing element of V .Let M , N be L -structures such that M ⊆ N . The following are equivalent: (1) M N . O-QUANTALE VALUED LOGICS 19 (2)
For any L -formula ϕ ( x, x , ..., x n ) and a , ..., a n ∈ M , we havethat V { ϕ M ( c, a , ..., a n ) : c ∈ M } = V { ϕ N ( c, a , ..., a n ) : c ∈ N } Proof.
Suppose that M N . Let ϕ ( x, x , ..., x n ) be an L -formula and a , ...a n ∈ M , so ^ { ϕ M ( c, a , ..., a n ) : c ∈ M } = ( ^ x ϕ ( x, a , ..., a n )) M ( by Definition 3.24 ( ))= ( ^ x ϕ ( x, a , ..., a n )) N ( since M N )= ^ { ϕ N ( c, a , ..., a n ) : c ∈ N } ( by Definition 3.24 ( )) . On the other hand, suppose that for any L -formula ϕ ( x, x , ..., x n ) and any a , ..., a n ∈ M we have that ^ { ϕ M ( c, a , ..., a n ) : c ∈ M } = ^ { ϕ N ( c, a , ..., a n ) : c ∈ N } By Definition 3.34 (2), we need to do an inductive argument on L -formulas in order to prove M N . It is straightforward to see that M ⊆ N guarantees the basic cases, and from the hypothesis if followsthe inductive step by using connectives and the quantifier V , so wehave just to check the inductive step by using W . Let ϕ ( x, x , ..., x n ) be an L -formula and a , ..., a n ∈ M . Let b ∈ V be a dualizing ele-ment. Notice that by Remark 3.16 b · − (cid:3) is uniformly continuous inthe symmetric topology and so it is a connetive. Theferore, ( _ x ϕ ( x, x , ..., x n )) M ( a , ..., a n ) = _ { ϕ ( c, a , ..., a n ) : c ∈ M } ( by Definition 3.24 (4) )= _ { b · − ( b · − ϕ M ( c, a , ..., a n )) : c ∈ M } ( b is a dualizing element) = b · − ^ { b · − ϕ M ( c, a , ..., a n ) : c ∈ M } ( by Proposition 2.14 )= b · − ^ { b · − ϕ N ( c, a , ..., a n ) : c ∈ N } ( hypothesis induction on ϕ and by applying this statement to b · − ϕ ( x, x , ..., x n ) and ^ x ( b · − ϕ ( x, x , ..., x n ) - b · − (cid:3) is a connective by Remark 3.16-) = _ { b · − ( b · − ϕ N ( c, a , ..., a n )) : c ∈ N } ( by Proposition 2.14 )= _ { ϕ N ( c, a , ..., a n ) : c ∈ N } ( b is a dualizing element) = ( _ x ϕ ( x, x , ..., x n )) N ( a , ..., a n )( by Definition 3.24 ) (cid:3) D -products and Ło´s Theorem in co-quantale valued logics. Changand Keisler ([CK66] ) defined some logics with truth values on Haus-dorff compact topological spaces. In that context, they provided aversion of Łos’ Theorem, which implies a Compactness Theorem intheir logic and the existence of saturated models (as it holds in firstorder logic). This approach is rediscovered in [BBHU08], but by tak-ing the particular case of truth values in the unit interval [
0, 1 ] . Wepropose to generalize the version of Ło´s Theorem in our context ofvalue co-quantale valued logics, as a test question of the logics pro-posed in this paper. O-QUANTALE VALUED LOGICS 21 D -limits. Let us fix V a V -domain value co-quantale providedwith its symmetric topology. By Remark 2.57, ( V , τ s ) is compact andHausdorff, therefore we may apply Lemma 3.37 to the symmetrictopology of V . Let I be a non empty set and D an ultrafilter over I . Remark 3.36.
We need to assume that V is provided with its symmetricdistance d sV ( p, q ) := ( p · − q ) ∨ ( q · − p ) , because we need to guarantee that p · − q ≤ d sV ( p, q ) , which might fail for the original distance d ( p, q ) := q · − p in V . The inequality q · − p ≤ d ( p, q ) always holds for both the originaland the symmetric distances of V .From now, for the sake of simplicity, let us denote the symmetric distanceof V by d V . The following is a very known fact about convergence of sequencesin Hausdorff Compact topological spaces.
Lemma 3.37. ( [CK66] ; Thrm 1.5.1.) If ( X, τ ) is a Hausdorff compact topo-logical space, given a sequence ( x i ) i ∈ I in X there exists a unique x ∈ X suchthat for any neighborhood V of x , then { i ∈ I | x i ∈ V } ∈ D . Fact 2.48 and Lemma 3.37 allow us to give the following notion ofconvergence in our setting.
Definition 3.38.
Given ( a i ) i ∈ I a sequence in V , the unique a ∈ V whichsatisfies that for any ǫ ∈ V + we have that { i ∈ I | d V ( a, a i ) ≤ ǫ } ∈ D issaid to be the D -ultralimit of the sequence ( a i ) i ∈ I , which we denote it by lim i,D a i . Definition 3.39.
Given ǫ ∈ V + , define A ( ǫ ) := { j ∈ I | d V ( lim i,D a i , a j ) ≤ ǫ } . Proposition 3.40.
Let ( a i ) i ∈ I be a sequence in V and b ∈ V . (1) If there exists A ∈ D such that for all j ∈ A we have that b ≤ a j ,then b ≤ lim i,D a i . (2) If d V is the symmetric distance of V and there exists A ∈ D suchthat for all j ∈ A we have that b ≥ a j , then b ≥ lim i,D a i .Proof. (1) It is enough to prove that for any ǫ ∈ V such that ≺ ǫ we have that b · − lim i,D a i ≤ ǫ , because V is completelydistributive (Definition 2.5) and then we would have that b · − lim i,D a i ≤ V { ǫ ∈ V : ≺ ǫ } = and by Proposition 2.11 (4) b ≤ lim i,D a i holds, as desired.Let ǫ ∈ V + , so by definition of lim i,D a i we know that { j ∈ I : d V ( lim i,D a i , a j ) ≤ ǫ } =: A ( ǫ ) ∈ D . By hypothesis A ∈ D , therefore A ( ǫ ) ∩ A ∈ D and so there exists j ∈ A ( ǫ ) suchthat b ≤ a j (because j ∈ A ). Notice that a j · − lim i,D a i ≤ d V ( lim i,D a i , a j ) ≤ ǫ (by Remark 3.36 and since j ∈ A ( ǫ ) ). ByLemma 2.13 and since b ≤ a j , we may say that b · − lim i,D a i ≤ a j · − lim i,D a i ≤ ǫ (2) It is enough to prove that whenever ≺ ǫ we have that lim i,D a i · − b ≤ ǫ . As above, A ( ǫ ) ∈ D . Since A ∈ D , thereexists j ∈ A ( ǫ ) such that a j ≤ b , since d V is the symmetric dis-tance of V and by Remark 3.36 we have that lim i,D a i · − a j ≤ d V ( lim i,D a i , a j ) := ( lim i,D a i · − a j ) ∨ ( a j · − lim i,D a i ) ≤ ǫ ; byLemma 2.13 and since a j ≤ b we have that lim i,D a i · − b ≤ lim i,D a i · − a j ≤ d V ( lim i,D a i , a j ) ≤ ǫ , as desired. (cid:3) The following fact is a kind of converse of the previous result, byassuming co-divisibility (Definition 2.25).
Proposition 3.41.
Suppose that V is co-divisible (Definition 2.25) andthat d V is the symmetric distance of V . Let ( a i ) i ∈ I be a sequence in V and b ∈ V such that lim i,D a i ≤ b and ≺ b · − lim i,D a i . Therefore, thereexists A ∈ D such that i ∈ A , a i ≤ b .Proof. By hypothesis ≺ b · − lim i,D a i , therefore by Lemma 2.6 thereexists some ǫ ∈ V such that ≺ ǫ ≺ b · − lim i,D a i . By Lemma 2.3(1) we may say ≺ ǫ ≤ b · − lim i,D a i . By taking A ( ǫ ) := { j ∈ I : d V ( lim i,D a i , a j ) ≤ ǫ } , we have that A ( ǫ ) ∈ D (by definition of lim i,D a i ). By Remark 3.36 and definition of A ( ǫ ) , for all j ∈ A ( ǫ ) we have that a j · − lim i,D a i ≤ d V ( lim i,D a i , a j ) ≤ ǫ ≤ b · − lim i,D a i .By Proposition 2.11 (1) and Lemma 2.26 (by hypothesis, V is co-divisible), we may say that a j ≤ ( b · − lim i,D a i ) + lim i,D a i = b , so A := A ( ǫ ) is the required set. (cid:3) Lemma 3.42.
Let K = ∅ a set and ( a k ) k ∈ K bea K -sequence in X . Therefore, V k ∈ K ( W l ∈ K ( a l · − a k )) = , if and only if, for all ǫ ∈ V + there exists k ∈ K such that W l ∈ K a l · − ǫ ≤ a k . Also, V k ∈ K ( W l ∈ K ( a k · − a l )) = , if and onlyif, for all ǫ ∈ V + there exists k ∈ K such that a k · − ǫ ≤ V l ∈ K a l .Proof. Suppose that V k ∈ K ( W l ∈ K ( a l · − a k )) = and let ǫ ∈ V + (i.e., ≺ ǫ ), by definition of ≺ and since ≥ V k ∈ K ( W l ∈ K ( a l · − a k )) we maysay that there exists k ∈ K such that W l ∈ K ( a l · − a k ) ≤ ǫ . By Fact 2.12, W l ∈ K ( a l · − a k ) = ( W l ∈ K a l ) · − a k , so ( W l ∈ K a l ) · − a k ≤ ǫ , and by Propo-sition 2.11 we have that ( W l ∈ K a k ) · − ǫ ≤ a k .Conversely, suppose that for all ǫ ∈ V + there exists k ∈ K such that W l ∈ K a l · − ǫ ≤ a k . By Proposition 2.11 (1) and Fact 2.12 we may say O-QUANTALE VALUED LOGICS 23 that W l ∈ K ( a l · − a k ) = ( W l ∈ K a l ) · − a k ≤ ǫ . Therefore (by Proposi-tion 2.4), V k ∈ K ( W l ∈ K ( a l · − a k )) ≤ V { ǫ ∈ V : ≺ ǫ } = A similar idea works for proving the second statement. Suppose V k ∈ K ( W l ∈ K ( a k · − a l )) = and let ǫ ∈ V + (i.e., ≺ ǫ ). By definition of ≺ there exists k ∈ K such that W l ∈ K ( a k · − a l ) ≤ ǫ , and by Proposition 2.14we may say that a k · − V l ∈ K a l ≤ ǫ . By Proposition 2.11 (1), this impliesthat a k · − ǫ ≤ V l ∈ K a l .Conversely, suppose that for all ǫ ∈ V + exist k ∈ K such that a k · − ǫ ≤ V l ∈ K a l , so by Proposition 2.11 (1) and Proposition 2.14 we have that W l ∈ K ( a k · − a l ) = a k · − V l ∈ K a l ≤ ǫ , therefore (by Proposition 2.4) V k ∈ K ( W l ∈ K ( a k · − a l )) ≤ V { ǫ ∈ V : ≺ ǫ } = . (cid:3) Remark 3.43.
The following fact is very important to deal with the quan-tifier cases in the proof of Ło´s Theorem in this setting. The idea of theseproofs is quite similar to the one presented in [BBHU08] , but adapted toour setting.
Proposition 3.44. (c.f. [BBHU08] ; Lemma 5.2) Let d V be the symmetricdistance of V . Let S = ∅ and ( F i ) i ∈ I be a sequence of mappings with do-main S and codomain V , then: V { lim i,D F i ( x ) : x ∈ S } ≥ lim i,D ( V { F i ( x ) : x ∈ S } ) and W { lim i,D F i ( x ) : x ∈ S } ≤ lim i,D ( W { F i ( x ) : x ∈ S } ) .Also, given ǫ ∈ V + there exist sequences ( b i ) i ∈ I and ( c i ) i ∈ I in S such that lim i,D F i ( b i ) + ǫ ≥ lim i,D ( W { F i ( x ) : x ∈ S } ) and lim i,D F i ( c i ) · − ǫ ≤ lim i,D ( V { F i ( x ) : x ∈ S } ) , whenever there exist B, C ∈ D such that V y ∈ S ( W x ∈ S ( F i ( x ) · − F i ( y ))) = for all i ∈ B and V y ∈ S ( W x ∈ S ( F i ( y ) · − F i ( x ))) = for all i ∈ C .Proof. Let r i := V { F i ( x ) : x ∈ S } , r := lim i,D r i and ǫ ∈ V + . Define A ( ǫ ) = { j ∈ I : d V ( r, r j ) ≤ ǫ } , so by Definition of lim i,D r i we may saythat A ( ǫ ) ∈ D . Notice that if j ∈ A ( ǫ ) and by Remark 3.36 we havethat r · − r j ≤ d V ( r, r j ) ≤ ǫ . So, by Proposition 2.11 (1) it follows that r ≤ r j + ǫ and then r · − ǫ ≤ r j .Let x ∈ S , then r · − ǫ ≤ r j := V { F j ( y ) : y ∈ S } ≤ F j ( x ) . Since A ( ǫ ) ∈ D , by Proposition 3.40 (1) we have that r · − ǫ ≤ lim i,D F i ( x ) ,and since it holds for any x ∈ S then r · − ǫ ≤ V { lim i,D F i ( x ) : x ∈ S } .By Proposition 2.11 (1) and by commutativity of + , it follows that r ≤ V { lim i,D F i ( x ) : x ∈ S } + ǫ . Since ≺ ǫ was taken arbitrarily, byFact 2.19 we may say that r ≤ V { lim i,D F i ( x ) | x ∈ S } .Let s i := W { F i ( x ) : x ∈ S } , s := lim i,D s i and ǫ ∈ V + . By taking A ′ ( ǫ ) = { j ∈ I : d V ( s, s j ) ≤ ǫ } , then A ′ ( ǫ ) ∈ D (definition of lim i,D s i ).Given j ∈ A ′ ( ǫ ) , by Remark 3.36 we have that s j · − s ≤ d V ( s, s j ) ≤ ǫ and by Proposition 2.11 (1) we may say that s j ≤ s + ǫ . Given x ∈ S ,it follows that F j ( x ) ≤ s j ≤ s + ǫ . Since d V is symetric, by Proposi-tion 3.40 (2) and since A ′ ( ǫ ) ∈ D , we have that lim i,D F i ( x ) ≤ r + ǫ ;since x ∈ S was taken arbitrarily, then W { lim i,D F i ( x ) | x ∈ S } ≤ r + ǫ .Since ≺ ǫ is arbitrary, by Fact 2.19 we have that W { lim i,D F i ( x ) | x ∈ S } ≤ r .Let us continue with the proof of the two last facts. Let ǫ ∈ V + , so byLemma 2.17 there exists some θ ∈ V + such that ≺ θ y θ + θ ≤ ǫ . Byhypothesis, suppose there exists B ∈ D such that V y ∈ S ( W x ∈ S ( F i ( x ) · − F i ( y ))) = for all i ∈ B . Let i ∈ B , since ≺ θ and by Lemma 3.42take b i ∈ S such that W x ∈ S F i ( x ) · − θ ≤ F i ( b i ) , and so by Proposi-tion 2.11 we may say that W x ∈ S F i ( x ) · − F i ( b i ) ≤ θ . If i / ∈ B , choose b i as any element in S . Let B ( θ ) := { i ∈ I : d V ( lim i,D F i ( b i ) , F i ( b i )) ≤ θ } ,so by definition of lim i,D F i ( b i ) we know that B ( θ ) ∈ D . Therefore, if j ∈ B ∩ B ( θ ) then W x ∈ S F j ( x ) ≤ lim i,D F i ( b i ) + ǫ . In fact, if j ∈ B ∩ B ( θ ) then _ x ∈ S F j ( x ) · − lim i ∈ I F i ( b i ) ≤ d V ( _ x ∈ S F j ( x ) , lim i ∈ I F i ( b i )) (by Remark 3.36) ≤ d V ( _ x ∈ S F j ( x ) , F j ( b j )) + d V ( F j ( b j ) , lim i ∈ I F i ( b i ))= " _ x ∈ S F j ( x ) · − F j ( b j ) ! ∨ F j ( b j ) · − _ x ∈ S F j ( x ) ! + d V ( F j ( b j ) , lim i ∈ I F i ( b i ))= _ x ∈ S F j ( x ) · − F j ( b j ) + d V ( F j ( b j ) , lim i ∈ I F i ( b i )) ≤ θ + θ ≤ ǫ. Therefore, W x ∈ S F j ( x ) · − lim i ∈ I F i ( b i ) ≤ ǫ , and by Proposition 2.11(1) we may say that W x ∈ S F j ( x ) ≤ lim i ∈ I F i ( b i ) + ǫ . Hence, since B ∩ B ( θ ) ∈ D by Proposition 3.40 (2) we have that lim i,D ( W { F i ( x ) : x ∈ S } ) ≤ lim i,D F i ( b i ) + ǫ .Let us construct the sequence ( c i ) i ∈ I as follows: Let ǫ ∈ V + . Byhypothesis and Lemma 3.42, there for all j ∈ C there exists c j ∈ S such that F j ( c j ) · − θ ≤ V x ∈ S F j ( x ) , where θ ∈ V + satisfies θ + θ ≤ ǫ . By Proposition 2.11 (1), it follows that F j ( c j ) · − V x ∈ S F j ( x ) ≤ O-QUANTALE VALUED LOGICS 25 θ ; since V x ∈ S F j ( x ) ≤ F j ( c j ) , it implies that d V ( F j ( c j ) , V x ∈ S F j ( x )) ≤ θ . If j / ∈ C , take c j as any element of S . Let C ( θ ) := { i ∈ I : d V ( lim i ∈ I F i ( c i ) , F i ( c i )) ≤ θ } . Therefore, if j ∈ C ∩ C ( θ ) we have that lim i,D F i ( c i ) · − ^ x ∈ S F j ( x ) ≤ d V ( lim i,D F i ( c i ) , ^ x ∈ S F j ( x )) ≤ d V ( lim i,D F i ( c i ) , F j ( c j )) + d V ( F j ( c j ) , ^ x ∈ S F j ( x )) ≤ θ + θ ≤ ǫ By Proposition 2.11 (1), we have that lim i,D F i ( c i ) · − ǫ ≤ V x ∈ S F j ( x ) .Since C ∩ C ( θ ) ∈ D and by Proposition 3.40 (1), lim i,D F i ( c i ) · − ǫ ≤ lim i,D ( V { F i ( x ) : x ∈ S } ) . (cid:3) D -product and D -ultraproduct of spaces and mappings. Proposition 3.45.
Let d V be the symmetric distance of V . If ( M i ) i ∈ I is asequence of V continuity spaces such that for all i ∈ I all distances d M i aresymmetric, then in the cartesian product Q i ∈ I M i , the relation ∼ definedby ( x i ) i ∈ I ∼ ( y i ) i ∈ I , if and only if, lim i,D d M i ( x i , y i ) = , is an equivalencerelation.Proof. Reflexivity follows trivially and symmetry follows from sym-metry of all d M i . Let us focus on the transitivity. Let ( x i ) i ∈ I , ( y i ) i ∈ I , ( z i ) i ∈ I ∈ Q i ∈ I M i be such that lim i,D d M i ( x i , y i ) = and lim i,D d M i ( y i , z i ) = .Since V is a value co-quantale, it is enough to check that for any ǫ ∈ V + we have that lim i,D d M i ( x i , z i ) ≤ ǫ . Let ǫ ∈ V + , so byLemma 2.17 there exists θ ∈ V + such that θ + θ ≤ ǫ . Since ≺ θ and by definition of lim i,D d M i ( x i , y i ) and lim i,D d M i ( y i , z i ) , A ( θ ) := { i ∈ I : d V ( lim i,D d M i ( x i , y i ) , d M i ( x i , y i ) ≤ θ } and B ( θ ) := { i ∈ I : d V ( lim i,D d M i ( y i , z i ) , d M i ( y i , z i ) ≤ θ } belong to D . By hypothesis, lim i,D d M i ( x i , y i ) = y lim i,D d M i ( y i , z i ) = . Notice that by Proposi-tion 2.11 (4) and since = min V, it follows that d V ( lim i,D d M i ( x i , y i ) , d M i ( x i , y i )) = ∨{ lim i,D d M i ( x i , y i ) · − d M i ( x i , y i ) ,d M i ( x i , y i ) · − lim i,D d M i ( x i , y i ) } ( d M i is assumed to be symmetric) = ∨{ · − d M i ( x i , y i ) , d M i ( x i , y i ) · − } = ∨{
0, d M i ( x i , y i ) } = d M i ( x i , y i ) Therefore, A ( θ ) = { i ∈ I : d M i ( x i , y i ) ≤ θ } and B ( θ ) = { i ∈ I : d M i ( y i , z i ) ≤ θ } . So, if i ∈ A ( θ ) ∩ B ( θ ) then d M i ( x i , z i ) ≤ d M i ( x i , y i ) + d M i ( y i , z i ) ≤ θ + θ ≤ ǫ . Since A ( θ ) ∩ B ( θ ) ∈ D , by Lemma 3.40 (2)we may say that lim i,D d M i ( x i , z i ) ≤ ǫ . (cid:3) Remark 3.46.
In general, we do not require that all continuity spaces in thesequence ( M i ) i ∈ I are symmetric, where in that case ∼ might not be an equiv-alence relation. We just need this requirement if ( M i , d M i ) := ( V, d M ) forall i ∈ I . As we will see in the following proposition, we can provided acontinuity space structure to the cartesian product Q i ∈ I M i , without as-suming the symmetry on d M i . Proposition 3.47.
Suppose that d V is the symmetric distance. If ( M i ) i ∈ I is a sequence of V − continuityspaces , ( Q i ∈ I M i , d D ) is a V -continuityspace, where d D : Q i ∈ I M i × Q i ∈ I M i → V is defined by (( x i ) i ∈ I , ( y i ) i ∈ I ) → lim i,D d M i ( x i , y i ) .Proof. Given ( x i ) i ∈ I ∈ Q i ∈ I M i , then d Q i ∈ (( x i ) i ∈ I , ( x i ) i ∈ I ) = lim i,D d M i ( x i , x i ) = lim i,D = .Let ( x i ) i ∈ I , ( y i ) i ∈ I , ( z i ) i ∈ I ∈ Q i ∈ I M i , a i := d M i ( x i , y i ) , b i := d M i ( x i , z i ) , c i := d M i ( y i , z i ) , a := lim i,D d M i ( x i , y i ) , b := lim i,D d M i ( x i , z i ) and c := lim i,D d M i ( y i , z i ) . We want to see that a ≤ b + c , which byProposition 2.11 (1) it is enough to prove that a · − b ≤ c . Let ǫ ∈ V + , so by Corollary 2.18 there exists some θ ∈ V + such that θ + θ + θ ≺ ǫ . By Lemma 2.3 (1), θ + θ + θ ≤ ǫ . By definition of lim i,D d M i ( x i , y i ) =: a , lim i,D d M i ( x i , z i ) =: b and lim i,D d M i ( y i , z i ) =: c , A := { i ∈ I : d V ( a, a i ) ≤ θ } , B := { i ∈ I : d V ( b, b i ) ≤ θ } and C := { i ∈ I : d V ( c, c i ) ≤ θ } belong to D . So, A ∩ B ∩ C ∈ D . Let i ∈ A ∩ B ∩ C , so by Proposition 2.11 (1) and (2) we may say that ( a · − b ) ≤ ( a · − a i ) + ( a i · − b i ) + ( b i · − b ) . Since d M i satisfies tran-sitivity, by Proposition 2.11 (1) we have that a i · − b i ≤ c i . Since i ∈ C , we may say that c i · − c ≤ ∨{ c i · − c, c · − c i } = d V ( c, c i ) ≤ θ .By Proposition 2.11 (1), it follows that c i ≤ c + θ . By monotonic-ity, a · − b ≤ ( a · − a i ) + ( b i · − b ) + ( c + θ ) . Since i ∈ A ( θ ) ∩ B ( θ ) ,we also have that a · − a i ≤ ∨{ a · − a i , a i · − a } = d V ( a, a i ) ≤ θ and b i · − b ≤ ∨{ b i · − b, b · − b i } = d V ( b, b i ) ≤ θ . Hence, a · − b ≤ ( a · − a i ) +( b i · − b ) + ( c + θ ) ≤ θ + θ + ( c + θ ) = θ + θ + θ + c ≤ ǫ + c . Since ǫ ∈ V + was chosen arbitrarily, it follows that a · − b ≤ c . (cid:3) Definition 3.48.
We call
D-product of the sequence of V -continuity spaces ( M i , d M i ) i ∈ I to the V -continuity space ( M D , d M D ) := ( Q i ∈ I M i , d D ) de-fined in Proposition 3.47. Lemma 3.49.
Let d V be the symmetric distance of V . If ( M i , d M i ) is asequence of V -continuity spaces such that d M i is symmetric for all i ∈ I , O-QUANTALE VALUED LOGICS 27 then (cid:0) Q i ∈ I M i / ∼ , d D (cid:1) is a V -continuity space, provided that d D is de-fined by ([( x i ) i ∈ I ] , [( y i ) i ∈ I ]) → lim i,D d M i ( x i , y i ) , where ∼ is defined as inProposition 3.45.Proof. First, let us check that (( x i ) i ∈ I , ( y i ) i ∈ I ) → lim i,D d M i ( x i , y i ) iswell-defined. Let ( a i ) i ∈ I , ( b i ) i ∈ I , ( c i ) i ∈ I , ( d i ) i ∈ I ∈ Q i ∈ I M i be such that ( a i ) i ∈ I ∼ ( b i ) i ∈ I and ( c i ) i ∈ I ∼ ( d i ) i ∈ I . By Fact 2.48 and the definition ofD-ultralimits, in order to prove that lim i,D d M i ( a i , c i ) = lim i,D d M i ( b i , d i ) it is enough to prove that for every ǫ ∈ V + we have that { i ∈ I : d V ( lim i,D d M i ( a i , c i ) , d M i ( b i , d i )) ≤ ǫ } ∈ D .Let ǫ ∈ V + , so by Proposition 2.17 there exists θ ∈ V + such that θ + θ ≺ ǫ . By Lemma 2.3 (1), we may say that θ + θ ≤ ǫ . Insimilar way we can prove that there exists some δ ∈ V + such that δ + δ ≤ θ . Since ( a i ) i ∈ I ∼ ( b i ) i ∈ I , by definition of ∼ we have that lim i,D d M i ( a i , b i ) = , hence for all γ ∈ V + we have that { i ∈ I : d V ( lim i,D d M i ( a i , b i ) , d M i ( a i , b i )) ≤ γ } ∈ D . By Remark 2.36, we maysay that { i ∈ I : d M i ( a i , b i ) ≤ γ } ∈ D . Since δ ∈ V + , in particular A := { i ∈ I : d M i ( a i , b i ) ≤ δ } ∈ D . Analogously, since ( c i ) i ∈ I ( d i ) i ∈ I we may say that B := { i ∈ I : d M i ( c i , d i ) ≤ δ } ∈ D . So, if i ∈ A ∩ B then d M i ( a i , c i ) ≤ d M i ( a i , b i ) + d M i ( b i , d i ) + d M i ( d i , c i )( by transitivity of d M i ) ≤ δ + d M i ( b i , d i ) + δ ( since i ∈ A ∩ B ) ≤ d M i ( b i , d i ) + θ ( by Proposition 2.10 and since δ + δ ≤ θ ) In a similar way we may say that d M i ( b i , d i ) ≤ d M i ( a i , c i ) + θ , when-ever i ∈ A ∩ B . By Proposition 2.11 (1), d M i ( b i , d i ) · − d M i ( a i , c i ) ≤ θ and d M i ( a i , c i ) · − d M i ( b i , d i ) ≤ θ , therefore d V ( d M i ( a i , c i ) , d M i ( b i , d i )) ≤ θ . Notice that by definition of lim i,D d M i ( a i , c i ) , A ( θ ) := { i ∈ I : d V ( lim i,D d M i ( a i , c i ) , d M i ( a i , c i )) ≤ θ } ∈ D . So, if i ∈ A ∩ B ∩ A ( θ ) we have that d V ( lim i,D d M i ( a i , c i ) , d M i ( b i , d i )) ≤ d V ( lim i,D d M i ( a i , c i ) , d M i ( a i , c i ))+ d V ( d M i ( a i , c i ) , d M i ( b i , d i ))( by transitivity of d V ) ≤ θ + θ ( since i ∈ A ∩ B ∩ A ( θ )) ≤ ǫ ( by Proposition 2.10 and since θ + θ ≤ ǫ ) (cid:3) Definition 3.50.
Let ( M i , d M i ) i ∈ I be a sequence of V -continuity spaces.The V -continuity space Q i ∈ I M i / ∼ provided with the symmetric distance d M D ([( x i ) i ∈ I ] , [( y i ) i ∈ I ]) := lim i,D d M i ( x i , y i ) is called the D - ultraproduct of the sequence ( M i , d M i ) i ∈ I . Definition 3.51. (c.f. [BBHU08] ; pag 25) Let ( M i ) i ∈ I , ( N i ) i ∈ I be se-quences of V -continuity spaces and K ∈ V be such that K ∈ V greaterthan the diameter of all the considered spaces. Given a fixed n ∈ N \ { } and ( f i : M ni → N i ) i ∈ I a sequence of uniformly continuous mappings pro-vided with the same modulus of uniform continuity. The mapping f D :( M D , d MD ) n → ( N D , d ND ) defined by (( x ) i ∈ I , ..., ( x ni ) i ∈ I ) → ( f i ( x , ..., x ni )) i ∈ I is said to be the D-product of the sequence ( f i : M ni → N i ) i ∈ I Proposition 3.52.
Let V be a co-divisible value co-quantale such that d V is the symmetric distance of V . If ( f i : M ni → N i ) i ∈ I is a sequence of uni-formly continuous mappings with the same modulus of uniform continuity ∆ : V + → V + , then ∆ is also a modulus of uniform continuity for the D -product f D : ( M D , d MD ) n → ( N D , d ND ) .Proof. For the sake of simplicity, let us take n = . Let x = ( x i ) i ∈ I , y =( y i ) i ∈ I ∈ Q i ∈ I M i such that d MD ( x , y ) ≤ ∆ ( ǫ ) , where ǫ ∈ V + (i.e., lim i,D d M i ( x i , y i ) ≤ ∆ ( ǫ ) ). By Lemma 3.41, there exists A ∈ D suchthat for all i ∈ A we have that d M i ( x i , y i ) ≤ ∆ ( ǫ ) . Since ∆ is a modu-lus of uniform continuity for f i for all i ∈ I , in particular we may saythat d N i ( f i ( x i ) , f i ( y i ) ≤ ǫ , whenever i ∈ A . Since d V is the symmet-ric distance of V and by Lemma 3.40 (2), d ND (( f i ) i ∈ I ( x ) , ( f i ) i ∈ I ( y )) ≤ ǫ . (cid:3) O-QUANTALE VALUED LOGICS 29
Ło´s Theorem in value co-quantale logics.
In order to prove a ver-sion of Ło´s Theorem in this setting, we do not require that the in-volved distances of the metric structures are necessarily symmet-ric. Because of that, it is enough to consider the D -product of a se-quence of V -continuity spaces ( M i , d M i ) i ∈ I instead of its respective D -ultraproduct, like we need to do in Continuous Logic for assur-ing that the obtained distance is actually symmetric. However, weneed to consider the respective quotient of D -powers of ( V , d V ) asa V -symmetric continuity space. From now, we assume that ( V , d V ) is a compact, Hausdorff, co-divisible value co-quantale, where d V isthe symmetric distance of V . Definition 3.53.
Given ( M, d M ) a V -continuity space where d M is sym-metric, define the ultrapower of ( M, d M ) as the D -ultraproduct (cid:0) Q i ∈ I M/ ∼ , d D (cid:1) of the constant sequence (( M, d M )) i ∈ I Definition 3.54.
Given a D -ultraproduct M D / ∼ := ( Q i ∈ I M i ) / ∼ , themapping θ : Q i ∈ I M i → M D / ∼ defined by ( x i ) i ∈ I → [( x i ) i ∈ I ] is called the canonical mapping . Notation 3.55.
We denote by ( V D / ∼ , d V/ ∼ ) the D -ultrapower of the V -continuity space ( V, d V ) (where d V is the symmetric distance). Definition 3.56.
We say that two V -continuity spaces ( X, d X ) and ( Y, d Y ) are V - equivalent , if and only if, there exists a bijection f : ( X, d x ) → ( Y, d Y ) such that d X ( x, y ) = d Y ( f ( x ) , f ( y )) for any x, y ∈ X . In this case,we say that f is an V -equivalence. Proposition 3.57. (c.f. [BBHU08] ; pg 26) Let ( V , d V ) a value co-quantaleprovided that d V is the symmetric distance, then the D -ultrapower ( V D / ∼ , d V/ ∼ ) is V − equivalent to ( V, d V ) .Proof. Defining T : V → V / ∼ by x → [( x ) i ∈ I ] , we have that T isa V -equivalence. In fact, T is injective: Let x, y ∈ V be such that T ( x ) = T ( y ) , so by definition of ∼ (Proposition 3.45) we have that d V/ ∼ ( T ( x ) , T ( y )) = d V/ ∼ ([( x ) i ∈ I ] , [( y ) i ∈ I ]) = lim i,D d V ( x, y ) = d V ( x, y ) =( x · − y ) ∨ ( y · − x ) = , therefore x · − y = and y · − x = . By Proposi-tion 2.11 (4) we may say that x ≤ y and y ≤ x , and so x = y .In order to prove that T is surjective, let [( x i ) i ∈ I ] ∈ V D . Let us seethat T ( lim i,D x i ) := [( lim i,D x i ) i ∈ I ] = [( x i ) i ∈ I ] (i.e., we will have that lim i,D d V ( lim i.D x i , x i ) = )): let ǫ ∈ V + , therefore d V (
0, d V ( lim i.D x i , x i )) = ∨{ · − d V ( lim i.D x i , x i ) , d V ( lim i.D x i , x i ) · − } = ∨{
0, d V ( lim i.D x i , x i ) } = d V ( lim i.D x i , x i ) , and by definition of lim i.D x i we may say that { i ∈ I : d V ( lim i.D x i , x i ) ≤ ǫ } ∈ D . Hence, { i ∈ I : d V (
0, d V ( lim i.D x i , x i )) ≤ ǫ } ∈ D and then lim i,D d V ( lim i.D x i , x i ) = . T preserves distances: In fact, d V D ( T ( x ) , T ( y )) = d V D ([( x ) i ∈ I ] , [( y ) i ∈ I ]) := lim i,D ( d V ( x, y )) i ∈ I = d V ( x, y ) .Therefore, T is an equivalence. (cid:3) Remark 3.58.
Notice that the mapping T ′ : V D / ∼ → V defined by [( x i ) i ∈ I ] → lim i,D x i is the inverse of T . Fact 3.59.
Given a sequence of V -continuity spaces (( M i , d M i )) i ∈ I pro-vided that all distances d M i are symmetric, the canonical mapping θ : Q i ∈ I M i → M D / ∼ is uniformly continuous with modulus of uniformcontinuity id V + . Definition 3.60.
Suppose that V is co-divisible and let L be a languagebased in a continuous structure. Given a sequence ( M i ) i ∈ I of L -structures,define the D-product of ( M i ) i ∈ I as the L -structure M D with underlying V -continuity space ( M D , d D ) , defined as follows: (1) For a predicate symbol R ∈ L , define R Q i ∈ I M i := T ′ ◦ θ ◦ R D ,where R D is the D -product of the mappings ( R M i ) i ∈ I , θ the canon-ical mapping given in Definition 3.54 and T ′ the mapping definedin Remark 3.58. (2) For a function symbol F ∈ L , defined F Q i ∈ I M i as the D -product ofthe mappings ( F M i ) i ∈ I . (3) For a constant symbol c ∈ L , define c Q i ∈ I M i := ( c M i ) i ∈ I . Remark 3.61.
Notice that Theorem 3.52 guarantees that the interpreta-tions of the symbols of L given above have the same modulus of uniformcontinuity given by the language. Theorem 3.62. (Ło´s Theorem; c.f. [BBHU08]
Thrm 5.4). Let ( V , d V ) bea co-divisible V -domain. If ( M i ) i ∈ I is a sequence of L -structures, then forany L -formula φ ( x , ..., x n ) (if φ has quantifiers, we require that its inter-pretations in any L -structure M i satisfy the hypothesis of Proposition 3.42)and any tuple (( a ) i ∈ I , ..., ( a ni ) i ∈ I ) ∈ ( Q i ∈ I M i ) n , we have that φ M D (( a ) i ∈ I , ..., ( a ni ) i ∈ I ) = lim i,D φ M i ( a , ..., a ni ) Proof.
We proceed by induction on L -formulae.(1) φ : d ( x , x )( d ( x , x )) M (( a ) i ∈ I , ( a ) i ∈ I ) := d M (( a ) i ∈ I , ( a ) i ∈ I )= lim i,D d M i ( a , a )( definition of a distance in a product )= lim i,D ( d ( x , x )) M i ( a , a )) O-QUANTALE VALUED LOGICS 31 (2) φ : R ( x , ..., x n ) , where R is a predicate symbol in L . ( R ( x , ..., x n )(( a ) i ∈ I , ..., ( a ni ) i ∈ I )) M := ( R (( a ) i ∈ I , ..., ( a ni ) i ∈ I )) M = T ′ ◦ θ (( R M i ( a , ..., a ni )) i ∈ I )( by Definition 3.60 (1) )= lim i,D R M i (( a , ..., a ni ))( by definition of θ and T ′ ) (3) φ : u ( σ , ..., σ m )( x , ..., x n ) , where u : V m → V is a uniformlycontinuous mapping and σ , · · · σ m are L -formulae such that σ M k (( a ) i ∈ I , ..., ( a ni ) i ∈ I ) = lim i,D σ M i k ( a , ..., a ni ) for all k ∈ {
1, ..., m } (induction hypothesis). For the sake of simplicity, denote (( a ) i ∈ I , ..., ( a ni ) i ∈ I ) =: a and ( a , ..., a ni ) = a i . ( u ( σ , ...σ m )) M ( a ) := u ( σ M ( a ) , ..., σ M m ( a ))= u ( lim i,D σ M i ( a i ) , ..., lim i,D σ M i m ( a i ))( induction hypothesis ) Define b i,k := σ M i k ( a i ) for any k ∈ { · · · , m } and i ∈ I . Noticethat { i ∈ I : d V ( u ( lim i,D b i,1 , ..., lim i,D b i,m ) , u ( b i,1 , ..., b i,m )) ≤ ǫ } contains the set { i ∈ I : d V n (( lim i,D b i,1 ..., lim i,D b i,m ) , ( b i,1 , ..., b i,m )) ≤ ∆ ( ǫ ) } = (cid:14) i ∈ I : n _ k = { d V ( lim i,D b i,k , b i,k ) } ≤ ∆ ( ǫ ) (cid:15) , whenever ∆ is a modulus of uniform continuity for u .Notice that this previous set belongs to D , because it contains T nk = { i ∈ I | d V ( lim i,D b i,k , b i,k ) ≤ ∆ ( ǫ ) } , which belongs to D bydefinition of a D -limit and since D is an ultrafilter.Therefore, u ( lim i,D σ M i ( a i ) , ..., lim i,D σ M i m ( a i )) = lim i,D u ( σ M i ( a i ) , · · · σ M i m ( a i )) lim i,D ( u ( σ , · · · , σ m )) M i ( a i ) . (4) φ : W x ϕ ( x, x , ..., x n ) Let ϕ ( x, x , ..., x n ) be an L -formula such that ϕ M (( b i ) i ∈ I , ( a ) i ∈ I , ..., ( a ni ) i ∈ I ) = lim i ∈ I ϕ M i ( b i , a , ..., a ni ) for all ( b i ) i ∈ I ∈ Q i ∈ I M i (induction hypothesis). For the sakeof simplicity, denote a := (( a ) i ∈ I , ..., ( a ni ) i ∈ I ) , a i := ( a , ..., a ni ) and b := ( b i ) i ∈ I . So, _ x ϕ ( x, x , ..., x n ) ! M ( a ) = _ x ϕ ( x, a ) ! M = _ { ϕ M ( b, a ) : b ∈ M} (by Definition 3.24 (4)) = _ { lim i,D ϕ M i ( b i , a i ) : b ∈ M} ( induction hypothesis ) ≤ lim i,D _ { ϕ M i ( b i , a i ) : b ∈ M} ( by Proposition 3.44 ) By Proposition 3.44, given ǫ ∈ V + there exists a sequence ( c j ) j ∈ I of tuples c j := ( c ji ) i ∈ I ∈ Q i ∈ I M i such that lim i,D ( _ { ϕ M i ( b i , a i ) : b ∈ M} ) ≤ lim i,D ϕ M i ( c ii , a i ) + ǫ ≤ _ { lim i,D ϕ M i ( b i , a ( i )) : b ∈ M} + ǫ where the last inequality follows from monotonicity of ≤ and since ( c ii ) i ∈ I ∈ Q i ∈ I M i . Since ǫ ∈ V + is arbitrary, we havethat lim i,D W { ϕ M i ( b i , a i ) : b ∈ M} ≤ lim i,D W { ϕ M i ( b i , a i ) : b ∈ M} . Therefore, by antisymmetry of ≤ we may say that lim i,D W { ϕ M i ( b ( i ) , a i ) : b ∈ M} = W { lim i,D ϕ M i ( b ( i ) , a ( i )) : b ∈ M} . Notice that W { ϕ M i ( b i , a i ) : b ∈ M} = W { ϕ ( c, a i ) : c ∈ M i } , then ( W x ϕ ( x, x , ..., x n )) M ( a ) = lim i,D W { ϕ ( c, a i ) : c ∈ M i } = lim i,D ( W x ϕ ( x, x , ..., x n )) M i ( a i ) , as desired.(5) φ : V x ϕ ( x, x , ..., x n ) Let ϕ ( x, x , ..., x n ) be an L -formula such that ϕ M (( b i ) i ∈ I , ( a ) i ∈ I , ..., ( a ni ) i ∈ I ) = lim i ∈ I ϕ M i ( b i , a , ..., a ni ) O-QUANTALE VALUED LOGICS 33 for all ( b i ) i ∈ I ∈ Q i ∈ I M i . For the sake of simplicity, denote a := (( a ) i ∈ I , ..., ( a ni ) i ∈ I ) , a i := ( a , ..., a ni ) and b := ( b i ) i ∈ I . So, ^ x ϕ ( x, x , ..., x n ) ! M ( a ) = ^ x ϕ ( x, a ) ! M = ^ { ϕ M ( b, a ) : b ∈ M} ( by Definition 3.24 (5) )= ^ { lim i,D ϕ M i ( b i , a i ) : b ∈ M} ( induction hypothesis ) ≥ lim i,D ^ { ϕ M i ( b i , a i ) : b ∈ M} ( by Proposition 3.44 ) By Proposition 3.44 and by hypothesis, given ǫ ∈ V + thereexists a sequence ( b j ) j ∈ I of tuples b j := ( b ji ) i ∈ I ∈ Q i ∈ I M i suchthat lim i,D ( ^ { ϕ M i ( b i , a i ) : b ∈ M} ) + ǫ ≥ lim i,D ϕ M i ( b ii , a i ) ≥ ^ { lim i,D ϕ M i ( b i , a i ) : b ∈ M} where the last inequality follows from the fact that ( b ii ) i ∈ I ∈ Q i ∈ I M i . Since ǫ ∈ V + was taken arbitrarily, we have that lim i,D V { ϕ M i ( b i , a i ) : b ∈ M} ≥ V { lim i,D ϕ M i ( b i , a i ) : b ∈ M} ≥ lim i,D V { ϕ M i ( b ( i ) , a i ) : b ∈ M} . By antisymmetry of ≤ ,we may say that lim i,D ^ { ϕ M i ( b i , a i ) : b ∈ M} = ^ { lim i,D ϕ M i ( b i , a i ) : b ∈ M} . Since, V { ϕ M i ( b i , a i ) : b ∈ M} = V { ϕ M i ( c, a i ) : c ∈ M i } ,then ( V x ϕ ( x, x , ..., x n )) M ( a ) = lim i,D V { ϕ ( c, a i ) : c ∈ M i } = lim i,D ( V x ϕ ( x, x , ..., x n )) M i ( a i ) , as desired. (cid:3) Note 3.63.
In case that we want to work in symmetric spaces, the sameargument as above works to prove a version of Ło´s Theorem for the D -ultraproduct (cid:0) Q i ∈ I M i (cid:1) ∼ . Some consequences of Ło´s Theorem.
In this setting, Ło´s Theo-rem implies a version of Compactness Theorem and the existenceof some kind of ω -saturated models, as it holds in both first orderand Continuous logics. First, we provide a proof of Compactness Theorem, up to Ło´s The-orem.
Corollary 3.64. (Compactness Theorem, c.f. [BBHU08]
Thrm 5.8) Let L be a language based on a continuous structure. Let T be an L − theorywhich conditions satisfy the hypothesis of Theorem 3.62 and C be a class of L -structures. Therefore, if T is finitary satisfiable in C , then there exists a D -product of structures in C that is a model of T.Proof. Let Λ be the collection of all finite subsets of T . By hypothesis,given λ ∈ Λ with λ := { E , ..., E n } , there exists some M λ ∈ C such that M λ | = E k for all k ∈ {
1, ..., n } .Fixed an L -condition E ∈ T , define S ( E ) := { λ ∈ Λ : E ∈ λ } .Notice that S ( E ) ∩ ... ∩ S ( E n ) = ∅ ( λ := { E , · · · , E n } ∈ S ( E ) ∩ ... ∩ S ( E n ) ), therefore { S ( E ) : E ∈ T } satisfies the Finite IntersectionProperty. Let D be an ultrafilter over Λ extending { S ( E ) : E ∈ T } .Let M := Q λ ∈ Λ M λ be the respective D -product of the sequence of L -structures ( M λ ) λ ∈ Λ . Given E ∈ T , where E : ψ = ( ψ an L -sentence).Notice that for all λ ∈ S ( E ) we have that M λ | = E ; i.e., ψ M λ = . Sinceby construction S ( E ) ∈ D , by Ło´s Theorem (Theorem 3.62) it followsthat lim λ,D ψ M λ = . Notice that ψ M = lim λ,D ψ M λ = , so M | = E .Therefore, M | = T . (cid:3) Keisler showed in [Kei64] the existence of saturated structures byusing ultraproducts. In [BBHU08], there is a proof of an analogousresult to Keisler’s construction by using metric ultraproducts. In thefollowing lines, we provide a proof of this result in the logic prop-posed in this paper, supposing that V is provided with the sym-metric distance (abusing of the notation, we will denote by d V ) andco-divisible value co-quantale satistying the SAFA Property (Defini-tion 2.27). Definition 3.65.
Let L be a language based on a continuous structure, Γ ( x , ..., x n ) be a set of L -conditions and M be an L -structure. We say that Γ ( x , ..., x n ) is satisfiable in M , if and only if, there exist a , ..., a n ∈ M such that M | = E ( a , ..., a n ) for all E ( x , · · · , x n ) ∈ Γ ( x , ..., x n ) . Definition 3.66.
Let L be a language based on a continuous structure , M be an L -structure and κ be an infinite cardinal. We say that M is κ -saturated , if and only if, given A ⊆ M such that | A | < κ and Γ ( x , ..., x n ) a set of L ( A ) -conditions with parameters in A , it holds that if Γ ( x , ..., x n ) is finitely satisfiable in M then Γ ( x , ..., x n ) is satisfiable in M . Definition 3.67.
An ultrafilter D over K = ∅ is said to be countably-incomplete , if and only if, there exists some { A n : n ∈ N } ⊆ D such that T n ∈ N A n = ∅ . O-QUANTALE VALUED LOGICS 35
Proposition 3.68.
Let D be an coutably-incomplete ultrafilter over K = ∅ ,then there exists a countable subcollection { J n : n ∈ N } of D such that J n + ⊆ J n for all n ∈ N and T n ∈ N J n = ∅ .Proof. By hypothesis, there exists a subcollection { A n : n ∈ N } ⊆ D of D tal que T n ∈ N A n = ∅ . Define J := A and J n + := J n ∩ A n + forany n ∈ N \ { } . (cid:3) Proposition 3.69. (c.f. [BBHU08] ; Prop. 7.6) Let V be a compact, Haus-dorff, co-divisible value co-quantale satisfying SAFA, provided with thesymmetric distance d V . Let L be a countable language based on a continu-ous structure and D be a countably-incomplete ultrafilter over a non emptyset Λ . Given any Λ -sequence of L -structures ( M λ ) λ ∈ Λ , its D -product M D is ω -saturated, assuming that all L -formulae satisfy the hypothesis in Ło´sTheorem (Theorem 3.62).Proof. For the sake of simplicity, let us analyze L -conditions with onevariable x . Let A ⊆ Q λ ∈ Λ M λ be countable and Γ ( x ) be a set of L ( A ) -conditions with parameters in A which is finitely satisfiable in M D .We will prove that Γ ( x ) is satisfiable in M D .Since L is countable, let Γ ( x ) := { ψ n ( x ) : n < ω } be an enumerationof Γ ( x ) . Since D is contably-incomplete, by Proposition 3.68 thereexists a sequence ( J n ) n ∈ N of elements in D such that J n + ⊆ J n for all n ∈ N and T n ∈ N J n = ∅ .By hypothesis, for all k ∈ N the set { ψ ( x ) , ..., ψ k ( x ) } is satisfiablein Q λ ∈ Λ M λ . By Ło´s Theorem (Theorem 3.62), there exists some a :=( a λ ) λ ∈ Λ ∈ Q λ ∈ Λ M λ such that for all n ∈ { · · · , k } we have that ψ M D n M λ ( a ) = lim λ,D ψ M λ n ( a λ ) = . Therefore, given ǫ ∈ V + we maysay that { λ ∈ Λ : d V (
0, ψ M λ n ( a λ )) ≤ ǫ } ∈ D . Since d V is the symmetricdistance, by Remark 2.36 we have that { λ ∈ Λ : ψ M λ n ( a λ ) ≤ ǫ } ∈ D . Since V satisfies SAFA Property (Definition 2.27), there exists asequence ( u k ) k ∈ N in V such that(1) V n ∈ N u n = (2) for all n ∈ N , ≺ u n (3) for all n ∈ N , u n + ≤ u n Therefore, { λ ∈ Λ : ψ M λ n ( a λ ) ≤ u l } ∈ D for all l ∈ N . This implies that A k := { λ ∈ Λ : M λ | = V x ∈ M λ W kn = ψ n ( x ) ≤ u k + } ∈ D whenever k ∈ N . Define the sequence ( X n ) n ∈ N of elements in D as follows: X := Λ and X k := J k ∩ A k if k ∈ N \ { } . Notice that V x ∈ M λ W kn = ψ n ( x ) ≤ W x ∈ M λ V k + = ψ n ( x ) ≤ u k + ≤ u k + , then A n + ⊆ A n for all n ∈ N andthen X n + ⊆ X n for all n ∈ N . Notice that T n ∈ N X n = ∅ , thereforeif λ ∈ Λ there exists k λ ∈ N such that k λ := max { n ∈ N : λ ∈ X n } . Define a := ( a λ ) λ ∈ Λ ∈ Q λ ∈ Λ M λ as follows: In case that k λ = ,take a λ as any element in M λ , otherwise take a λ ∈ M λ such that W { ψ M λ n ( a λ ) : n ≤ k λ } ≤ u k λ . So, if k ∈ N then for any n ∈ N suchthat k ≤ n and λ ∈ X n we have that n ≤ k λ , hence ψ M λ k ( a λ ) ≤ u k λ ≤ u n . Since X n ∈ D , by Ło´s Theorem (Theorem 3.62) we have that ψ Q λ ∈ Λ M λ k ( a ) = lim λ,D ψ M D k ( a λ ) = . Since ψ k ( x ) ∈ Γ ( x ) was takenarbitrarily, then a ∈ Q λ ∈ Λ M λ realizes Γ ( x ) , as desired. (cid:3) R EFERENCES [AB09] C. Argoty and A. Berenstein. Hilbert spaces with unitary operators.
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O-QUANTALE VALUED LOGICS 37
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