arXiv:2102.06067v1 [math.LO] 11 Feb 2021 CO-QUANTALE VALUED LOGICS DAVID REYES AND PEDRO H. ZAMBRANO A BSTRACT. In this paper, we propose a generalization of Continuous Logic ([BBHU08]) where the distances take values in suitable co-quantales (in the way as it was proposed in [Fla97]). By assuming suitable conditions (e.g., being co-divisible, co-Girard and a V-domain), we provide, as test questions, a proof of a version of the Tarski-Vaught test (Proposition 3.35) and Łoś Theorem (Theorem 3.62) in our setting. Keywords. metric structures, lattice valued logics, co-quantales, co-Girard, co-divisibility, domains, Tarski-Vaught test, Łoś theorem. AMS classification 2010. 03C95, 03B60, 03C90, 03C20, 06F07, 18B35 1. I NTRODUCTION S. Shelah and J. Stern proved in [SS78] that a first order attempt of study of classes of Banach Spaces has a “bad” behavior (this has a very high Hanf number, having a model-theoretical behavior similar to a second order logic of binary relations). This led to develop a suitable logic beyond first order logic, in order to do a suitable model-theoretic analysis of Banach Spaces. In [CK66], C. C. Chang and H. J. Keisler proposed a new logic with truth values within a compact Hausdorff topological space, which was the first time where the term Continuous Logic appeared. They developed basics on Mathematical Logic in this book, but by then Model Theory had not been very developed (Morley’s first order categoricity theorem had just been proved, and there was no stability theory by then). For some reason, people did not continue working on this kind of logic until it was rediscovered in the 90’s by W. Henson and J. Iovino (see [HI02, Iov99]) and later by I. BenYaacov 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 advise his undergraduate thesis, where this paper is one of its fruits. The second author wants to thank Universidad Nacional de Colombia for the grant “Convocatoria para el apoyo a proyectos de investigación y creación artśtica de la sede Bogotá de la Universidad Nacional de Colombia - 2019”. 1 2 D. REYES AND P. ZAMBRANO truth values in the unitary interval [0, 1], focusing on the study of structures based on complete metric spaces (e.g. Hilbert spaces together with bounded operators -see [AB09]-, Banach spaces, Probability spaces -see [BH04]-). This logic is known as Continuous Logic Because of some technical reasons, people working on Continuous Logic have to consider strong assumptions on the involved operators (e.g. boundness) in order to axiomatize classes of metric structures in this logic. This took us to the notion of Metric Abstract Elementary Class (see [HH09, Zam11]) for being able to deal with nonaxiomatizable -in Continuous Logic- classes of complete metric structures. However, this approach does not consider topological spaces in general. Independently, Lawvere provided in [Law73] a framework in Category Theory for being able to consider a logic with generalized truth 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 for Topological Spaces (see [FZ80]), but it was just suitable to study particular algebraic examples like Modules and Topological Groups, due to the algebraic nature of first order logic. Moreover, this approach was left as a model-theoretic study of general Topological Spaces but it was the beginning of the model-theoretic study of Modules (see [Pre88]). Quantales are a suitable kind of lattices introduced for being able to deal with locales (a kind of lattices which generalizes the ideal of open sets of a topological space) and multiplicative lattices of ideals from Ring Theory and Functional Analysis (e.g., C∗ -algebras and von Neumann algebras). Considering the contravariant notions in a quantale (which we will called co-quantales), Flagg gave in [Fla97] a way to deal with topological spaces as pseudo metric spaces where the distance takes values in a suitable quantale. In this paper, we will propose a generalization of Continuous Logic by defining distances with values in value co-quantales together with suitable assumptions (e.g., being co-divisible, co-Girard and a Vdomain). This paper is organized as follows: In the second section we will provide basic facts in co-quantales. In the third section, we introduced our approach to co-quantales valued logics, analogously as it is done in Continuous Logic but by considering distances with values in a suitable co-quantale. As test questions, we provide a proof CO-QUANTALE VALUED LOGICS 3 of a version of the Tarski-Vaught test (Proposition 3.35) and a version of Łoś Theorem (Theorem 3.62). A difference between our approach and Continuous Logic lies on the fact that we can provided a version of Łoś Theorem for D-products (before doing the quotient to force a D-product being an actual metric space -which is called a D-ultraproduct in Continuous Logic-). We notice that the same proof for D-products works for D-ultraproducts in our setting. As consequences of Łoś Theorem, in a similar way as in first order and Continuous logics, we provide a proof of a version of Compactness Theorem (Corollary 3.64) and of the existence of ω1 -saturated models (Proposition 3.69). 2. VALUE 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 V co-well below y (denoted by x ≺ y), if and only, if for all A ⊆ X, if A ≤ x then there exists a ∈ A such that a ≤ y. Remark 2.2. In [0, ∞], 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 V 1.3) If L is a complete lattice, then for all A ⊆ L and x ∈ L we have that A ≺ x if and only if there exists a ∈ A such that a ≺ x. DefinitionV 2.5. A complete lattice L is said to be completely distributive if a = {b : a ≺ b} provided that a ∈ L. Lemma 2.6. ([Fla97]; Lema 1.6]) Given L a completely distributive lattice and 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 provided that (1) 0 ≺ 1 (2) if δ, δ′ ∈ L satisfy 0 ≺ δ and 0 ≺ δ′ , then 0 ≺ δ ∧ δ′ . 4 D. REYES AND P. ZAMBRANO Examples 2.8. (1) The 2-valued Boolean algebra 2 := {0, 1}, where 0 < 1. (2) The ordinal number ω + 1 := {0, 1, ..., ω} together with the usual ordering. (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) ([0, ∞], ≤usual). (6) ([FK97]; pg 115-117) Given a set R, let us denote Pfin (R) = {X ∈ P(X) : X is finite} and for all X ∈ Pfin (R) we denote ↓ (X) = {Y ∈ P(X) : Y ⊆ X}. Given Ω(R) = {p ∈ P(Pfin (R)) : X ∈ p implies ↓ (X) ⊆ p}, then (Ω(R), ⊇) is a valued lattice. 2.2. Value co-quantales. In this paper, we do not work with the usual notion of quantale. We consider the contravariant notion (which we call co-quantale) because this approach allows us to work with a notion of distance (pseudo-metric) with values in the co-quantale (see [Fla97]), in an analogous way as the metric structures given in Continuous Logic. We know that this is not the standard way to study quantales, but we chose this setting in order to do a similar study as it is done in Continuous Logic. Definition 2.9. A co-quantale V is a complete lattice provided with a commutative monoid structure (V, +) such that (1) The minimum element 0 of V is the identity of (V, +); i.e., a+0 = a for all a ∈ V and V V (2) for all a ∈ V and (bi )i∈I ∈ V I , a + i∈I bi = i∈I (a + bi ) Proposition 2.10. ([Fla97]; Pg 6]) If V is a co-quantale, then for all a, b, c ∈ V we have that (1) a + 1 = 1 (2) a ≤ b implies c + a ≤ c + b Proposition 2.11. V ([Fla97]; Thrm 2.2]) Given V a co-quantale and a, b ∈ · b := {r ∈ V : r + b ≥ a}. Therefore, for all c ∈ V we have V, define a − that · b ≤ c if and only if a ≤ b + c (1) a − · b) + b (2) a ≤ (a − · b≤a (3) (a + b) − · (4) a − b = 0 if and only if a ≤ b · (b + c) = (a − · b) − · c = (a − · c) − · b (5) a − · · · (6) a − c ≤ (a − b) + (b − c) CO-QUANTALE VALUED LOGICS 5 · is the left adjoint of + and it preserves categorical limits, Since − we have the following fact. Fact 2.12. Let V be a co-quantale, then for any W W sequence (bi )i∈I and any · a = i∈I (bi − · a) element a ∈ V we have that ( i∈I bi ) − Lemma 2.13. Given a co-quantale V, for any a, b ∈ V, a ≤ b implies · a≥ c− · b and a − · c≤b− · c. that for all c we have that c − Proof. Let c ∈ V, then by Proposition 2.11 (2) we may say that c ≤ · a) + a. So, c ≤ (c − · a) + b whenever a ≤ b (by Proposition 2.10 (c − · b ≤ c− · a. (2)), and so by Proposition 2.11 (1) this is equivalent to c − On the other hand, since a ≤ b then · c) + c (Proposition 2.11 (2)) b ≤ (b − · c) + c ( since a ≤ b) a ≤ (b − · c ≤ b− · c (Proposition 2.11 (1)) a−  Proposition 2.14. Given a ∈ V and a sequence (bi )i∈I V V a co-quantale, W · · in V, we have that a − i∈I bi = i∈I (a − bi ). V W · i∈I bi ≤ i∈I (a − · bi ) implies that Proof.WBy Proposition a− V 2.11 (1), V W · · a ≤ i∈I (a − bi ) + i∈I bi = i∈I (( i∈I (a − bi )) + bi ), which holds by Proposition 2.11 (2) W and Proposition 2.10 (2) allow us to say that · · bi )) + bj for any j ∈ J. a ≤ (a − bj ) + bj ≤ ( i∈I (a − V W · bi )) + Since j ∈ I was taken arbitrarily, then a ≤ i∈I (( i∈I (a − bi ).  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 : 0 ≺ 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 following fact. Corollary 2.18 ([LRZ18]; Remark 2.26). Given any ǫ ∈ V+ and n ∈ n times z }| { + N \ {0}, there exists θ ∈ V such that nθ := θ + · · · + θ ≺ ǫ. Fact 2.19. ([Fla97]; V Thrm 2.10]) GivenVa value co-quantale V and p ∈ V, p = p + 0 = p + {ǫ ∈ V : 0 ≺ ǫ} = {p + ǫ : 0 ≺ ǫ}. 6 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 0 ≺ ǫ, q ≺ r and r+ǫ ≺p Example 2.21. The 2-valued Boolen algebra 2 = {0, 1} is a value coquantale, by taking + := ∨ Example 2.22. ([0, ∞], ≤, +) is a value co-quantale. Notice that the oper· a (a ∈ [0, ∞]) is given by b − · a := max{0, b − a}. We denote ation − − this example by D. Example 2.23. There are several ways to define a value co-quantale with underlying value lattice E. One of them is by taking + as the usual real product, another one consists by taking + := ∨. Also, the Łukasiewicz’s addition +L provides a value co-quantale structure with underlying lattice E -where a +L b = ∧{0, a + b − 1} by taking ∧ in ([0, 1], ≥)-. Denote the previous examples by E∗ , E∨ and EL respectively. Fact 2.24. ([FK97]; pg 115-117) Given a non empty set R, (Ω(R), ⊇, ∩) is a 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 associate to (X, τ). Definition 2.25. A co-quantale (V, ≤, +) is said to be co-divisible if for all 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). a, b ∈ V, a ≤ b implies b = a + (b − · a ⊣ a + −. Proof. This follow from − −  The following property allows us to approximate 0 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), if and only, if there is a sequence (un )n∈N such that V (1) n∈N un = 0. (2) for all n ∈ N, 0 ≺ un (3) for all n ∈ N, un+1 ≤ un 2.2.1. Co-Girard value co-quantales. In this section, we give the basic notions and results relative to co-Girard value co-quantales, which allows to consider a kind of pseudo complement relative to some · This notion will allow fixed element b (a dualizing element) and −. to prove our test questions (Proposition 3.35 and Theorem 3.62). CO-QUANTALE VALUED LOGICS 7 Definition 2.28. Given a co-quantale (V, +), an element d ∈ V is said to be a dualizing element, if and only if, for all a ∈ V we have that · (d − · a). a=d− Definition 2.29. A co-quantale V is said to be co-Girard, if and only, if it has a dualizing element. 2.3. Continuity spaces. In this section, we will provide some basics on continuity spaces, a framework given in [Fla97, FK97] in order to generalize (pseudo) metric spaces but considering distances with values in general co-quantales. Definition 2.30. Given X 6= ∅ 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 and only if, (1) (reflexivity) for all x ∈ X, d(x, x) = 0, 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 6= ∅ be a set. A 2-continuity space (X, d) codifies a binary relation R := {(x, y) ∈ X × X : d(x, y) = 0} on X which is reflexive and transitive. Example 2.32. Given D := ([0, ∞], ≤, +), a D-continuity space is just a pseudo metric space (where the distance betwwen two elements might be infinite). 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 · a. (V, d) is a V-continuity space. d(a, b) := b − Proposition 2.35. ([FK97]; Pg 119]) Given a value co-quantale V and (X, dX), (Y, dY ) V-continuity spaces, define dX×Y : (X × Y) × (X × Y) → V by dX×Y ((x1 , y1), (x2 , y2 )) := dX (x1 , x2 ) ∨ dY (y1 , y2 ), then (X × Y, dX×Y ) is a V-continuity space. Remark 2.36. There is another way to provide to a value co-quantale a V· b) ∨ (b − · a) for continuity space structure by defining dsV (a, b) := (a − s all a, b ∈ V. dV is called the symmetric distance for V. Notice that for 8 D. REYES AND P. ZAMBRANO all a ∈ V we have that dsV (a, 0) = dsV (0, a) = a: · 0) ∨ (0 − · a) dsV (a, 0) = (a − · 0) ∨ 0 (by Proposition 2.11 (4) and 0 = min V) = (a − · 0 = a− ^ · = {r : r + 0 ≥ a} (by definition of −) ^ = {r : r ≥ a} = a 2.3.1. The topology of a V-continuity space. In this subsection, we will give 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 Bǫ (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 Bǫ (x) ⊆ U. Fact 2.40. ([Fla97]; Thrm 4.2) The family of open subsets in a V-continuity space (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 open sets of (X, d) determines a topology on X, which we will call the topology induced by d and we will denote it by τd . Lemma 2.42. Given (X, d) a V-continuity space, for all x ∈ X and ǫ ∈ V+ we have that the disc Bǫ (x) is an open set of X. V Proof. Let y ∈ Bǫ (x), so d(x, y) ≺ ǫ. By Fact 2.19 d(x, y) = {d(x, y)+ δ : 0 ≺ δ} ≺ ǫ, then by Lemma 2.4 there exists δ ∈ V+ such that d(x, y) + δ ≺ ǫ. We may assure that Bδ (y) ⊆ Bǫ (x): If z ∈ X satisfies d(y, z) ≺ δ, then d(x, z) ≤ d(x, y) + d(y, z) ≤ d(x, y) + δ ≺ ǫ.  Fact 2.43. Given a V-continuity space (X, d), the family of open discs forms a base for τd . Definition 2.44. V Given a V-continuity space (X, d), A ⊆ X and x ∈ X, define d(x, A) := {d(x, a) : a ∈ A}. Proposition 2.45. Given a V-continuity space (X, d), then a subset A ⊆ X is τd -closed, if and only if, for all x ∈ X we have that d(x, A) = 0 implies x ∈ A. CO-QUANTALE VALUED LOGICS 9 Proof. Suppose that A ⊆ X is τd -closed and let x ∈ X be such that d(x, A) = 0. In case that x ∈ / A, since A is τd -closed, there exists + c ǫ ∈ V such that Bǫ (x) ⊆ A = {y ∈ X : y ∈ / A}. Since d(x, A) := V {d(x, a) : a ∈ A} = 0 ≺ ǫ, by Fact 2.4 there exists a ∈ A such that d(x, a) ≺ ε, so a ∈ Bǫ (x) ∩ A (contradiction). On the other hand, let A ⊆ X be such that for all x ∈ X, d(x, A) = 0 implies x ∈ A. Suppose that y ∈ X belongs to the adherence of A, so by Fact 2.43 given any ǫ ∈ V+ , we have that V A∩Bǫ (y) 6= ∅. Therefore, + for any ε ∈VV we have that d(y, A) := {d(y, a) : a ∈ A} ≤ ε, so d(y, A) ≤ {ε : 0 ≺ ε} = 0, hence by hypothesis we may say that y ∈ A. Therefore, A is closed.  Corollary 2.46. Given a V-continuity space (X, d), the topological closure of A ⊆ X is given by cl(A) := A = {y ∈ X : d(y, A) = 0}. Definition 2.47. Given a V-continuity space (X, dX), ǫ ∈ V+ and x ∈ X, define the closed disc of radius ε centered in x by Cǫ (x) := {y ∈ X : dX (x, y) ≤ ǫ}. Fact 2.48. ([FK97]; Lemma 3.2 (2)) Let (X, dX ) be a V-continuity space and x ∈ X. The family {Cǫ (x) : ǫ ∈ V+ } determines a fundamental system of neighborhoods around x. Definition 2.49. Given a V-continuity space (X, d), define the dual distance 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 is related 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 τd -closed. Proof. Let x ∈ X and ǫ ∈ V+ , so by Proposition 2.45 it is enough to V ⋆ check that for any y ∈ X, d(y, Cǫ (x)) := {d(y, a) : a ∈ C⋆ǫ (x)} = 0 implies that y ∈ C⋆ǫ (x). Let y ∈ X be such that d(y, C⋆ǫ (x)) = 0 and δ ∈ V+ . Since d(y, C⋆ǫ (x)) = 0 ≺ δ, by Fact 2.4 there exists z ∈ C⋆ǫ (x) such that d(y, z) ≺ δ, so d⋆ (x, y) := d(y, x) ≤ Vd(y, z) + d(z, x) = d(y,V z) + d⋆ (x, z) ≤ δ + ǫ, therefore d⋆ (x, y) ≤ {ǫ + δ : 0 ≺ δ} = ǫ + {δ : 0 ≺ δ} = ǫ + 0 = ǫ, therefore y ∈ C⋆ǫ (x).  Definition 2.51. Given a V-continuity space (X, d), define the symmetric space relative to (X, d) by (X, ds ), where ds (x, y) := d(x, y)∨d⋆ (x, y). In general, we will denote the topological notions related to ds by adding the superscript s. Proposition 2.52. ([FK97]; Lemma 3) Given a V-continuity space (X, d), for the topology τs induced by ds we have that U ⊆ X belongs to τs , if and only if, there exist V, W ⊆ X such that V ∈ τd , W ∈ τ⋆ and U = V ∩ W. 10 D. REYES AND P. ZAMBRANO Lemma 2.53. ([FK97]; Lemma 3,4) If (X, d) is a V-continuity space, (X, τs ) satisfies the following separation properties: (1) (pseudo-Hausdorff) For all x, y ∈ X, if x ∈ / {y} according to (X, τd ) then there exist U, V ⊆ X such that x ∈ U, y ∈ V, U ∈ τd , V ∈ τ⋆ and U ∩ V = ∅. (2) (regularity) For all x ∈ X and A ⊆ X, if A ∈ τd and x ∈ A then there exist U, C ⊆ X such that U is τd -open, C es τ⋆ -closed and x ∈ U ⊆ C ⊆ A. 2.3.2. V-domains. In order to give a version of Łoś Theorem in our setting, following [CK66, BBHU08], we need to consider compact and Hausdorff topological spaces. The setting which involves these assumptions in continuity spaces corresponds to V-domains. Definition 2.54. A V-continuity space (X, d) is said to be T0 , if and only if, for any x, y ∈ X, d(x, y) = 0 and d(y, x) = 0 implies x = y. Remark 2.55 ([FK97], pg 120). A continuity space (X, d) is T0 , if and only if, (X, τd ) is T0 as a topological space and (X, τsd) is Hausdorff. Definition 2.56. A V-continuity space (X, d) is said to be a V − domain, if and only if, it is T0 and (X, τsd ) is compact. Remark 2.57. Let (X, d) be a V-domain, therefore by definition (X, τsd ) is compact. Since (X, d) is T0 , by Remark 2.55 (X, τsd ) is Hausdorff. The importance of the previous properties lies on the fact that these allow us to provide a proof of a version of Łos’s Theorem in the logic that we will introduce in this paper (Theorem 3.62). We will provide some examples which satisfy these properties. Proposition 2.58. ([FK97]; Thrm 4.14) The following examples are domains: (1) 2 = ({0, 1}, 0 ≤ 1, ∨). (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 X1 . (5) The free local associated to a set X: (Ω(X), ⊇, ∩)2. 3. VALUE CO - QUANTALE LOGICS In this section, we will introduce a logic with truth values within value co-quantales, generalizing Continuous Logic (see [BBHU08], 1 2 It is denoted by Λ(X)in [FK97] Tt is denoted by Γ (X) in [FK97] CO-QUANTALE VALUED LOGICS 11 where the truth values are taken in the unitary interval [0, 1], which is a particular case of our setting). Throughout the rest of this paper, we assume some technical conditions (Definitions 2.25, 2.29 and 2.56) that we need for providing a proof of a version of Tarski-Vaught test -Proposition 3.35- and a version of Łoś Theorem -Theorem 3.62- for the logics introduced in this paper. At some point, we require to work with the symmetric distance dsV of V. Assumption 3.1. Throughout this section, we assume that V is a value co-quantale which is co-divisible, co-Girard and a V-domain. 3.1. Modulus of uniform continuity. Modulus of uniform continuity are introduced in [BBHU08] as a technical way of controlling from the language the uniform continuity of the mappings considered in Continuous Logic. In this subsection, we develop an analogous study of modulus of uniform continuity but in the setting of mappings valued in value co-quantales. Remark 3.2. Given (M, dM ), (N, dN) V-continuity spaces and (x1 , y1 ), (x2, y2 ) ∈ M×N, we define dM×N ((x1 , y1 ), (x2 , y2)) := dM (x1 , y1 )∨ dN (x2 , y2 ). By Proposition 2.35, (M × N, dM×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, dM ) and (N, dN), we say that ∆ : V+ → V+ is a modulus of uniform continuity for f, if and only if, for any x, y ∈ M and any ǫ ∈ V+ , dM (x, y) ≤ ∆(ǫ) implies dN (f(x), f(y)) ≤ ǫ. As a basic consequence we have the following fact. Proposition 3.4. Given (M, dM ), (N, dN), (K, dK) V-continuity spaces and f : M → N, g : N → K uniformly continuous mappings, ∆ and Θ modulus of uniform continuity for f and g respectively, then ∆ ◦ Θ is a modulus of uniform continuity for g ◦ f. Definition 3.5. Given a sequence of mappings (fn )n∈N with domain (M, dM) and codomain (N, dN) (both of them V-continuity spaces), we say that (fn)n∈N uniformly converges to a mapping f : M → N, if and only if, for all ǫ ∈ V+ there exists n ∈ N such that for any m ≥ n and for any x ∈ M we may say that dN (fm (x), f(x)) ≤ ǫ. It is straightforward to see that uniform convergence behaves well with respect to composition of mappings. 12 D. REYES AND P. ZAMBRANO Proposition 3.6. Let (M, dM ), (N, dN), (K, dK ) be V-continuity spaces, f : M → N, (fn)n∈N be a sequence of mappings from M to N, g : N → K, and (gn )n∈N be a sequence of mappings from N to K such that (fn)n∈N uniformly converges to f and (gn )n∈N uniformly converges to g. If g is uniformly continuous, then (gn ◦ fn )n∈N uniformly converges to g ◦ f. V W 3.1.1. Uniform continuity of and . The following fact is very important because, as in Continuous Logic, it allows us to control V (by usingWdirectly the language) the uniform continuity of both (inf) and (sup), understood as quantifiers (in an analogous way as in Continuous Logic). Proposition 3.7. Let (M, dM ), (N, dN) be V-continuity spaces, f : M × N → V be a uniformly continuous mapping provided with a modulus of + uniform continuity ∆ : V+ → W V , then ∆ is also V a modulus of uniform continuity for the mappings : M → V and f : M → V defined by W V f x 7→ y∈N f(x, y) and x 7→ y∈N f(x, y) respectively. Proof. Let ǫ ∈ V be such that 0 ≺ ǫ, y ∈ N and a, b ∈ M be such that dM (b, a) ≤ ∆(ǫ). Then, dM×N ((b, y), (a, y)) := dM (b, a) ∨ dN (y, y) = dM (b, a) ≤ ∆(ǫ) Since ∆ is a modulus of uniform continuity for f, then · f(b, y) ≤ dV (f(b, y), f(a, y)) f(a, y) − ≤ ǫ By Proposition 2.11 (1) we may say f(a, y) ≤ f(b, y) + ǫ ≤ _ f(b, z) + ǫ z∈N Since y ∈ N was taken arbitrarily, then _ _ f(a, z) ≤ f(b, z) + ǫ z∈N z∈N and by Proposition 2.11 (1) _ _ _ _ · · f(a) − f(b) = f(a, z) − f(b, z) ≤ ǫ. f f z∈N z∈N In a similar way we prove the related statement for V f.  As an immediate consequence, we have the following useful facts. CO-QUANTALE VALUED LOGICS 13 Corollary 3.8. Given an arbitrarily set I 6= ∅ and I-sequences (ai )i∈I , (bi)i∈I in a V-continuity (M, dM ), if ǫ ∈ V+ satisfies d W spaceW V VV (ai , bi ) ≤ ǫ for all i ∈ I, then dV ( i∈I ai , i∈I bi ) ≤ ǫ and dV ( i∈I ai , i∈I bi ) ≤ ǫ. Corollary 3.9. Given a V-continuity space (M, dM ), I 6= ∅, and a Isequence of mappings (fi : M → V)i∈I , if ∆ : V+ → V+ is a modulus of uniform continuity forWfi ( i ∈ I), then ∆Vis also a modulus of uniform W continuity for both V i fi : M → V and i fi : M → V defined by x 7→ i∈I fi (x) and x 7→ i∈I fi (x), respectively. Proposition 3.10. Let (M, dM ), (N, dN) be V−continuity spaces, f : M × N → V, (fn )n∈N a sequence of mappings W from M ×  N to V such that (fn )n∈N uniformly converges to f, then uniformly y∈N fn (x, y) n∈N V  W converges to y∈N f(x, y) and uniformly converges to y∈N fn (x, y) n∈N V y∈N f(x, y). Proof. Since by hypothesis (fn)n∈N uniformly converges to f, given ǫ ∈ V+ there exists n ∈ N such that if m ≥ n, then dV (fm (x, y), f(x, y)) ≤ ǫ for any (x, y) ∈ M × N. For a fixed x ∈ M and m ≥ n, define the sequences (fm (x, y))y∈N and W (f(x, y))y∈N,Wwhich satisfy the hypothesis of Corollary 3.8, so dV ( y∈N fm (x, y), y∈N f(x, y)) ≤ ǫ whenever m ≥ n. Since this holds for all ǫ ∈ V+ , we got the uniform convergence desired. V In an analogous way, we prove the respective statement for .  3.2. Some basic notions. In first order logic, an n-ary relation in a set A is defined as a subset of An . In this way, a tuple (a1 , · · · , an ) might belong to A or not. We may codify this by using characteristic functions, dually, by the discrete distance from a tuple in An to R. In Continuous Logic, an n-ary relation in A is understood according to this second approach by taking a uniformly continuous mapping R : An → [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 topology of V. Definition 3.11. W Given a V-continuity space (M, dM ) and A ⊆ M, define diam(A) := {dM (a, b) : a, b ∈ A}. (which we will call the diameter of A. 3.2.1. Continuous structures. Given a V-continuity space (M, dM ) with diameter p ∈ V, we define a continuous structure with underline Vcontinuity space (M, dM ) as a tuple M = ((M, dM), (Ri )i∈I , (fj)j∈J , (ck)k∈K ), where: 14 D. REYES AND P. ZAMBRANO (1) For each i ∈ I, Ri : Mni → V is a uniformly continuous mapping (which we call a predicate), with modulus of uniform continuity ∆Ri : V+ → V+ . In this case, ni < ω is said to be the arity of Ri . (2) For each j ∈ J, fj : Mmj → M is a uniformly continuous mapping with modulus of uniform continuity ∆Fj : V+ → V+ . In this case, mj < ω is said to be the arity of Fj . (3) For each k ∈ K, Ck is an element M. 3.2.2. Languages for continuous structures. For a fixed continuous structure M := ((M, dM ), (Ri)i∈I , (fj)j∈J , (ck)k∈K ), we will define the language associated to M in the natural way, as follows. Predicate symbols: Ri 7→ (Pi , ni , ∆Ri ) (i ∈ I) Function symbols: fj 7→ (Fj, nj , ∆fj ) (j ∈ J) Constant symbols: ck 7→ ek (k ∈ K). This set of non logical symbols is denoted W V by NLM . Let us denote by LG := {d} ∪ X ∪ C ∪ { , } (which we call logical symbols), where: • X = {xi : i ∈ N} is a countable set of variables. • C is the set of all uniformly continuous mappings with domain Vn and codomain V (1 ≤ n < ω). As in Continuous Logic, we understand a uniformly continuous mapping u : Vn → V as a connective. • d is a symbol, which we will interpret as the V-valued distance given in (M, dM ). This symbol will play the role of the equality in first order logic, in a similar way as we do in Continuous Logic. Definition 3.12. Given a continuous structure M := ((M, dM), (Ri)i∈I , (fj)j∈J , (ck )k∈K ), we define the language based on M as LM := NLM ∪ 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, we define the notion of L-term recursively, as follows: • Any variable and any constant symbol is an L-term. • Given L-terms t1 , ..., tn and a function symbol f ∈ L of arity n, ft1, ..., tn is an L-term. Definition 3.14. An L-term is said to be closed, if and only if, it is built without use of variables. Now, we provide the notion of L-formulae in this new setting. We mimic the analogous notion given in Continuous Logic. CO-QUANTALE VALUED LOGICS 15 Definition 3.15. Given L a language based on a continuous structure, we define the notion of L-formula recursively, as follows: • Given L-terms t1 , t2, dt1 t2 is an L-formula. • Given L-terms t1 , ...tn and a predicate symbol P ∈ L of arity n, Pt1 , ...tn is an L-formula. • Given L-formulas ψ1 , ..., ψm and a connective (i.e., a uniformly continuous mapping) a : Vm → V, then aψ1 , ..., ψm is an Lformula. V W • Given an L-formula ψ and a variable x, both xψ and xψ are L-formulas. Remark 3.16. Let V be a co-Girard value co-quantale and b ∈ V be a · x. Denote the usual, dual and symdualizing element. Denote x ′ := b − ∗ s metric distances in V by d, d and d respectively. Notice that the mapping ·  : V → V defined by (b − · )(x) := b − · x is uniformly continub− ous (relative to the symmetric topology) provided with modulus of uniform continuity idV+ . In fact, given x, y ∈ V we have that · y d(y, x) = x − · (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)) = (b − ′ · ′ = 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 ds (x, y) = d(y, x) ∨ d∗ (y, x) = d∗ (y ′ , x ′ ) ∨ · ) respects the distance ds and therefore d(y ′ , x ′ ) = ds (y ′ , x ′ ), then (b − it is uniformly continuous relative to the symmetric topology provided with idV+ as a modulus of uniform continuity. W V Notation W 3.17. The subsequences x and x of an L-fórmula can be V written as x and x , respectively. d(t1, t2) denotes the sequence dt1 t2 . Definition 3.18. An L-formula W φ is said V to be quantifier-free, if and only if, there are no appearances of x and x inside φ. Definition 3.19. An appearance of a variable x inside W anVL-fórmula φ is said to be free whenever it is not under the scope of x o x inside φ. Definition 3.20. An L-formula φ is said to be an L-sentence, if and only if, all appearances of variables are not free. 16 D. REYES AND P. ZAMBRANO Notation 3.21. φ(x1 , ..., xn) means that the variables that appear free in φ are among x1 , ..., xn. 3.2.3. L-structures. Let L be a language based on a V-continuous structure M := ((M, dM), (Ri )i∈I , (fj)j∈J , (ck )k∈K ). Given a V-continuity space (N, dN ) with diameter (Definition 3.11) at most diam(M), we will interpret the symbols in NLM in (N, dN) as follows: • For any predicate symbol P of arity n and modulus of uniform continuity ∆P , associate a uniformly continuous mapping PN : Nn → V with modulus of uniform continuity ∆P . • For any function symbol F of arity m and with modulus of uniform continuity ∆F , associate a uniformly continuous mapping FN : Nm → N with modulus of uniform continuity ∆F . • For any constant symbol e, associate an element eN ∈ N. Also, the logical symbol d is interpreted in (N, dN) as the distance d := dN . N Definition 3.22. Given a language L based on a continuous structure M := ((M, dM), (Ri )i∈I , (fj)j∈J , (ck )k∈K ) and a V-continuity space (N, dN), the continuous structure obtained by interpreting the symbols of L on N = ((N, dN) as above, N := ((N, dN), (PiN )i∈I , (FN j )j∈J , (ek )k∈K ), is said to be an L-structure. 3.2.4. Semantics. Given an L-structure N and A ⊆ N, we extend the language L by adding new constant symbols ca (a ∈ A), interpreting cN a := a. Abusing of notation, we will write a instead of ca , but understood 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 define recursively its interpretation in N, denoted by tN , as follows: (1) If t is a constant symbol c, define tN := cN . (2) If t is a variable x, define tN : N → N as the identity function of N. (3) If t is of the form ft1 , ...tn provided that f is an n-ary function symN bol and t1 (x), ..., tn(x) are L-terms, define tN := fN (tN 1 , ..., tn ) as N N N N N m the mapping f (t1 , ..., tn ) : N → N where (a) 7→ f (t1 (a), ..., tN n (a)) m for all a ∈ N . Definition 3.24. Let N be an L-structure. We define recursively the interpretation of L(N)-sentences in N, as follows. N (1) (d(t1, t2))N := dN (tN 1 , t2 ), where t1 , t2 are L(N)-terms CO-QUANTALE VALUED LOGICS 17 N (2) (P(t1, ..., tn))N := PN (tN 1 , ..., tn ), where P is an n-ary predicate symbol and t1 , · · · , tn are L(N)-terms. N (3) (u(φ1 , .., φn))N := u(φN 1 , ..., φn ) for any uniformly continuous n mapping (connective) u : V → V and all L(N)-sentences φ1 , ..., φn. W W (4) (Vx φ)N := Va∈N φN (a), whenever φ(x) is an L(N)-formula. (5) ( x φ)N := a∈N φN (a), whenever φ(x) is an L(N)-formula. Analogously as in Continuous Logic, all terms and all formulae have a modulus of uniform continuity, which do not depend of the structures. Proposition 3.25. Given L a language based on a continuous structure, φ(x1 , ..., xn) an L − formula and t(x1 , ..., xm) an L-term, then there exist ∆φ : V+ → V+ and ∆t : V+ → V+ such that for any L-structure N, ∆φ is a modulus of uniform continuity for φN and ∆t is a modolus of continuity for tN . Proof. The basic cases are given by definition, since constant symbols’s interpretations can be viewed as constant functions, variables are interpreted as the identity function and predicate symbols are interpreted as a uniformly continuous mapping with the respective modulus of uniform continuity. The connective case corresponds to compose uniformly continuous mappings (and we get the desired result by Proposition 3.4), and the quantifier cases follow from Proposition 3.7.  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 and only 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 of those symbols. 3.2.5. L-conditions. Fix L a language based on a continuous structure. Definition 3.27. (c.f. [BBHU08]; Def 3.9) Given φ1 (x1 , ..., xn), φ2 (x1 , ..., xn) L − formulas, we say that φ1 is logically equivalent to φ2 , if and only if, for any L-structure M and any a1 , ..., an ∈ M we have that M φM 1 (a1 , ..., an) = φ2 (a1 , ..., an). Definition 3.28. Let φ1 (x1 , ..., xn), φ2 (x1 , ..., xn) be L-formulas and M be an L-structure, we define the logical distance between φ1 and φ2 relative to M as follows: W M d(φ1 , φ2 )M := {dV (φM 1 (a1 , ...an), φ2 (a1 , ...an)) : a1 , ...an ∈ M} The logical distance between φ1 , φ2 is defined as follows: 18 D. REYES AND P. ZAMBRANO d(φ1 , φ2 ) := W {d(φ1 , φ2 )M : M is an L − structure} We define the notion of satisfiability in an L-structure in an analogous 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 formal expression of the form φ = 0, where φ(x1 , ..., xn) is an L-fórmula. An L-condition E is said to be closed if it is of the form φ = 0, where φ is an L-sentence. Given an L-formula φ(x1 , ..., xn), the related condition E : φ(x1 , ..., xn) = 0 is denoted by E(x1 , ..., xn). Definition 3.30. Given φ(x1 , ..., xn) an L-fórmula, M an L-structure and a1 , .., an ∈ M, the L-condition E(x1 , ..., xn) : φ(x1 , ..., xn) = 0 is said to be satisfied in M for a1 , ..., an, if and only if, φM (a1 , ..., an) = 0. We denote this by M |= E(a1 , ..., an). Notation 3.31. Given φ, ψ L-formulae, we denote by φ = ψ the L· ψ) ∨ (ψ − · φ) = 0 and we denote by φ ≤ ψ the L-condition condition (φ − · φ − ψ = 0. 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. 3.3. 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 ϕM = ϕN . (2) Let M be an L-substructure of N. We say that M is an L-elementary substructure of N (denoted by M 4 N), if and only if, any Lformula ϕ(x1 , ..., xn) satisfies ϕM (a1 , ..., an) = ϕN (a1 , ..., an) for all a1 , ..., an ∈ M. In this case, we also say that N is an Lelementary 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 holds in 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) Assume that 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 4 N. CO-QUANTALE VALUED LOGICS 19 (2) For any L-formula ϕ(x, x1 , ..., xn) and a1 , ..., an ∈ M, we have that V {ϕM (c, a1, ..., an) : c ∈ M} = V {ϕN (c, a1, ..., an) : c ∈ N} Proof. Suppose that M 4 N. Let ϕ(x, x1 , ..., xn) be an L-formula and a1 , ...an ∈ M, so ^ ^ {ϕM (c, a1, ..., an) : c ∈ M} = ( ϕ(x, a1, ..., an))M x (by Definition 3.24(5)) ^ = ( ϕ(x, a1, ..., an))N x (since M 4 N) ^ = {ϕN (c, a1, ..., an) : c ∈ N} (by Definition 3.24(5)). On the other hand, suppose that for any L-formula ϕ(x, x1 , ..., xn) and any a1 , ..., an ∈ M we have that ^ {ϕM (c, a1, ..., an) : c ∈ M} = ^ {ϕN (c, a1, ..., an) : c ∈ N} By Definition 3.34 (2), we need to do an inductive argument on L-formulas in order to prove M 4 N. It is straightforward to see that M ⊆ N guarantees the basic cases, and from the hypothesis ifVfollows the inductive step by using connectives and the Wquantifier , so we have just to check the inductive step by using . Let ϕ(x, x1 , ..., xn) be an L-formula and a1 , ..., an ∈ M. Let b ∈ V be a dualizing ele·  is uniformly continuous in ment. Notice that by Remark 3.16 b − the symmetric topology and so it is a connetive. Theferore, 20 ( D. REYES AND P. ZAMBRANO _ x ϕ(x, x1 , ..., xn))M (a1 , ..., an) = _ {ϕ(c, a1, ..., an) : c ∈ M} (by Definition 3.24 (4)) _ · (b − · ϕM (c, a1, ..., an)) : c ∈ M} = {b − (b is a dualizing element) ^ · · ϕM (c, a1, ..., an) : c ∈ M} = b− {b − (by Proposition 2.14) ^ · · ϕN (c, a1, ..., an) : c ∈ N} = b− {b − (hypothesis induction on ϕ and by applying this statement to · ϕ(x, x1 , ..., xn) and b− ^ · ϕ(x, x1 , ..., xn) (b − x ·  is a connective by Remark 3.16-) -b − _ · (b − · ϕN (c, a1, ..., an)) : c ∈ N} = {b − (by Proposition 2.14) _ = {ϕN (c, a1, ..., an) : c ∈ N} (b is a dualizing element) _ = ( ϕ(x, x1 , ..., xn))N (a1 , ..., an) x (by Definition 3.24)  3.4. D-products and Łoś Theorem in co-quantale valued logics. Chang and Keisler ([CK66] ) defined some logics with truth values on Hausdorff compact topological spaces. In that context, they provided a version of Łos’ Theorem, which implies a Compactness Theorem in their logic and the existence of saturated models (as it holds in first order logic). This approach is rediscovered in [BBHU08], but by taking the particular case of truth values in the unit interval [0, 1]. We propose to generalize the version of Łoś Theorem in our context of value co-quantale valued logics, as a test question of the logics proposed in this paper. CO-QUANTALE VALUED LOGICS 21 3.4.1. D-limits. Let us fix V a V-domain value co-quantale provided with its symmetric topology. By Remark 2.57, (V, τs) is compact and Hausdorff, therefore we may apply Lemma 3.37 to the symmetric topology 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 symmetric · q) ∨ (q − · p), because we need to guarantee that distance dsV (p, q) := (p − s · · p−q ≤ dV (p, q), which might fail for the original distance d(p, q) := q−p · p ≤ d(p, q) always holds for both the original in V. The inequality q − and the symmetric distances of V. From now, for the sake of simplicity, let us denote the symmetric distance of V by dV . The following is a very known fact about convergence of sequences in Hausdorff Compact topological spaces. Lemma 3.37. ([CK66]; Thrm 1.5.1.) If (X, τ) is a Hausdorff compact topological space, given a sequence (xi )i∈I in X there exists a unique x ∈ X such that for any neighborhood V of x, then {i ∈ I|xi ∈ V} ∈ D. Fact 2.48 and Lemma 3.37 allow us to give the following notion of convergence in our setting. Definition 3.38. Given (ai )i∈I a sequence in V, the unique a ∈ V which satisfies that for any ǫ ∈ V+ we have that {i ∈ I|dV (a, ai) ≤ ǫ} ∈ D is said to be the D-ultralimit of the sequence (ai )i∈I , which we denote it by limi,D ai . Definition 3.39. Given ǫ ∈ V+ , define A(ǫ) := {j ∈ I|dV (limi,D ai , aj) ≤ ǫ}. Proposition 3.40. Let (ai )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 ≤ aj, then b ≤ limi,D ai . (2) If dV is the symmetric distance of V and there exists A ∈ D such that for all j ∈ A we have that b ≥ aj, then b ≥ limi,D ai . Proof. (1) It is enough to prove that for any ǫ ∈ V such that 0 ≺ · limi,D ai ≤ ǫ, because V is completely ǫ we have that b − · distributiveV(Definition 2.5) and then we would have that b − limi,D ai ≤ {ǫ ∈ V : 0 ≺ ǫ} = 0 and by Proposition 2.11 (4) b ≤ limi,D ai holds, as desired. Let ǫ ∈ V+ , so by definition of limi,D ai we know that {j ∈ I : dV (limi,D ai , aj) ≤ ǫ} =: A(ǫ) ∈ D. By hypothesis A ∈ D, 22 D. REYES AND P. ZAMBRANO therefore A(ǫ) ∩ A ∈ D and so there exists j ∈ A(ǫ) such · limi,D ai ≤ that b ≤ aj (because j ∈ A). Notice that aj − dV (limi,D ai , aj) ≤ ǫ (by Remark 3.36 and since j ∈ A(ǫ)). By · limi,D ai ≤ Lemma 2.13 and since b ≤ aj , we may say that b − · aj − limi,D ai ≤ ǫ (2) It is enough to prove that whenever 0 ≺ ǫ we have that · b ≤ ǫ. As above, A(ǫ) ∈ D. Since A ∈ D, there limi,D ai − exists j ∈ A(ǫ) such that aj ≤ b, since dV is the symmetric dis· aj ≤ tance of V and by Remark 3.36 we have that limi,D ai − · aj ) ∨ (aj − · limi,D ai ) ≤ ǫ; by dV (limi,D ai , aj) := (limi,D ai − · b ≤ Lemma 2.13 and since aj ≤ b we have that limi,D ai − · aj ≤ dV (limi,D ai , aj) ≤ ǫ, as desired. limi,D ai −  The following fact is a kind of converse of the previous result, by assuming co-divisibility (Definition 2.25). Proposition 3.41. Suppose that V is co-divisible (Definition 2.25) and that dV is the symmetric distance of V. Let (ai )i∈I be a sequence in V and · limi,D ai . Therefore, there b ∈ V such that limi,D ai ≤ b and 0 ≺ b − exists A ∈ D such that i ∈ A, ai ≤ b. · limi,D ai , therefore by Lemma 2.6 there Proof. By hypothesis 0 ≺ b − · limi,D ai . By Lemma 2.3 exists some ǫ ∈ V such that 0 ≺ ǫ ≺ b − · (1) we may say 0 ≺ ǫ ≤ b − limi,D ai . By taking A(ǫ) := {j ∈ I : dV (limi,D ai , aj) ≤ ǫ}, we have that A(ǫ) ∈ D (by definition of limi,D ai ). By Remark 3.36 and definition of A(ǫ), for all j ∈ A(ǫ) · limi,D ai ≤ dV (limi,D ai , aj ) ≤ ǫ ≤ b − · limi,D ai . we have that aj − By Proposition 2.11 (1) and Lemma 2.26 (by hypothesis, V is co· limi,D ai ) + limi,D ai = b, so divisible), we may say that aj ≤ (b − A := A(ǫ) is the required set.  Lemma in X. Therefore, V W 3.42. Let K 6= ∅ a set and (ak )k∈K bea K-sequence + · ( (a − a )) = 0, if and only if, for all ǫ ∈ V there exists k ∈ K k k∈K l∈K W l V W · · such that l∈K al − ǫ ≤ ak . Also, k∈K ( l∈K (ak − al )) V = 0, if and only · ǫ ≤ l∈K al . if, for all ǫ ∈ V+ there exists k ∈ K such that ak − V W · ak )) = 0 and let ǫ ∈ V+ (i.e., Proof. Suppose that k∈K ( l∈K (al − V W · ak )) we may 0 ≺ ǫ), by definition of ≺ and since 0W≥ k∈K ( l∈K (al − · say that there exists W W k ∈ K such thatW l∈K (al − ak ) ≤ ǫ. By Fact 2.12, · · · − ak , so ( l∈K al ) − ak ≤ ǫ, and by Propol∈K (al − ak ) = ( l∈K al )W · ǫ ≤ ak . sition 2.11 we have that ( l∈K ak ) − Conversely, suppose that for all ǫ ∈ V+ there exists k ∈ K such that W · l∈K al − ǫ ≤ ak . By Proposition 2.11 (1) and Fact 2.12 we may say CO-QUANTALE VALUED LOGICS 23 W W · ak ) = ( l∈K al ) − · that l∈KV (al − W V ak ≤ ǫ. Therefore (by Proposi· ak )) ≤ {ǫ ∈ V : 0 ≺ ǫ} = 0 tion 2.4), k∈K ( l∈K (al − idea works for proving the second statement. Suppose V A similar W · l )) = 0 and let ǫ ∈ V+ (i.e., 0 ≺ ǫ). By definition of ≺ ( (a k −a k∈K l∈K W · there exists k ∈ K such Vthat l∈K (ak −al ) ≤ ǫ, and by Proposition 2.14 · we may say that V ak − l∈K al ≤ ǫ. By Proposition 2.11 (1), this implies · that ak − ǫ ≤ l∈K al . · ≤ Conversely, suppose that for all ǫ ∈ V+ exist k ∈ K such that ak −ǫ V Wl∈K al , so by Proposition V 2.11 (1) and Proposition 2.14 we have that · · (a − a ) = a − l k Vl∈K Wk V l∈K al ≤ ǫ, therefore (by Proposition 2.4) · {ǫ ∈ V : 0 ≺ ǫ} = 0.  k∈K ( l∈K (ak − al )) ≤ Remark 3.43. The following fact is very important to deal with the quantifier cases in the proof of Łoś Theorem in this setting. The idea of these proofs is quite similar to the one presented in [BBHU08], but adapted to our setting. Proposition 3.44. (c.f.[BBHU08]; Lemma 5.2) Let dV be the symmetric distance of V. Let S 6= ∅ and (Fi )i∈I be a sequence of mappings with domain V S and codomain V, then: V W{limi,D Fi (x) : x ∈ S} ≥ limi,D (W{Fi (x) : x ∈ S}) and {limi,D Fi (x) : x ∈ S} ≤ limi,D ( {Fi (x) : x ∈ S}). Also, given ǫ ∈ V+ there exist W sequences (bi )i∈I and (ci )i∈I in S such that limi,D Fi (bi ) + ǫ ≥ limi,D (V {Fi (x) : x ∈ S}) and · ǫ ≤ limi,D ( {Fi (x) : x ∈ S}), whenever there exist B, C ∈ limi,D Fi (ci )V− W V W · i (y))) = 0 for all i ∈ B and y∈S ( x∈S (Fi (y)− · D such that y∈S ( x∈S (Fi (x)−F Fi (x))) = 0 for all i ∈ C. V Proof. Let ri := {Fi (x) : x ∈ S}, r := limi,D ri and ǫ ∈ V+ . Define A(ǫ) = {j ∈ I : dV (r, rj) ≤ ǫ}, so by Definition of limi,D ri we may say that A(ǫ) ∈ D. Notice that if j ∈ A(ǫ) and by Remark 3.36 we have · rj ≤ dV (r, rj) ≤ ǫ. So, by Proposition 2.11 (1) it follows that that r − · ǫ ≤ rj . r ≤ rj + ǫ and then r − V · Let x ∈ S, then r − ǫ ≤ rj := {Fj (y) : y ∈ S} ≤ Fj (x). Since · A(ǫ) ∈ D, by Proposition 3.40 (1) we have that V r − ǫ ≤ limi,D Fi (x), · and since it holds for any x ∈ S then r − ǫ ≤ {limi,D Fi (x) : x ∈ S}. By Proposition 2.11 (1) and by commutativity of +, it follows that V r ≤ {limi,D Fi (x) : x ∈ S} + ǫ.VSince 0 ≺ ǫ was taken arbitrarily, by Fact 2.19 we may say that r ≤ {limi,D Fi (x)|x ∈ S}. W Let si := {Fi (x) : x ∈ S}, s := limi,D si and ǫ ∈ V+ . By taking A′ (ǫ) = {j ∈ I : dV (s, sj ) ≤ ǫ}, then A′ (ǫ) ∈ D (definition of limi,D si ). · s ≤ dV (s, sj ) ≤ ǫ Given j ∈ A′ (ǫ), by Remark 3.36 we have that sj − 24 D. REYES AND P. ZAMBRANO and by Proposition 2.11 (1) we may say that sj ≤ s + ǫ. Given x ∈ S, it follows that Fj (x) ≤ sj ≤ s + ǫ. Since dV is symetric, by Proposition 3.40 (2) and since A′ (ǫ) ∈ D, we have W that limi,D Fi (x) ≤ r + ǫ; since x ∈ S was taken arbitrarily, then {limi,D Fi (x)|x W ∈ S} ≤ r + ǫ. Since 0 ≺ ǫ is arbitrary, by Fact 2.19 we have that {limi,D Fi (x)|x ∈ S} ≤ r. Let us continue with the proof of the two last facts. Let ǫ ∈ V+ , so by Lemma 2.17 there exists some θ ∈ V+ such that 0 ≺Vθ y θW+ θ ≤ ǫ. By · hypothesis, suppose there exists B ∈ D such that y∈S ( x∈S (Fi (x) − Fi (y))) = 0 for all i ∈ B. W Let i ∈ B, since 0 ≺ θ and by Lemma 3.42 · take bi ∈ S such that x∈S W Fi (x) − θ ≤ Fi (bi ), and so by Proposi· Fi (bi ) ≤ θ. If i ∈ tion 2.11 we may say that x∈S Fi (x) − / B, choose bi as any element in S. Let B(θ) := {i ∈ I : dV (limi,D Fi (bi ), Fi (bi )) ≤ θ}, so by definition ofWlimi,D Fi (bi ) we know that B(θ) ∈ D. Therefore, if j ∈ B ∩ B(θ) then x∈S Fj (x) ≤ limi,D Fi (bi ) + ǫ. In fact, if j ∈ B ∩ B(θ) then _ _ · limi∈I Fi (bi ) ≤ dV ( Fj (x), limi∈I Fi (bi )) Fj (x) − x∈S x∈S (by Remark 3.36) _ ≤ dV ( Fj (x), Fj(bj )) + dV (Fj (bj ), limi∈IFi (bi )) x∈S = " _ x∈S · Fj (bj) Fj(x) − ! ∨ · Fj (bj ) − _ Fj (x) x∈S +dV (Fj (bj), limi∈I Fi (bi )) _ · Fj (bj) + dV (Fj(bj ), limi∈I Fi (bi )) = Fj(x) − x∈S ≤ θ+θ ≤ ǫ. W · Therefore, x∈S Fj (x) − W limi∈I Fi (bi ) ≤ ǫ, and by Proposition 2.11 (1) we may say that x∈S Fj (x) ≤ limi∈I Fi (bi ) + ǫ. Hence, W since B ∩ B(θ) ∈ D by Proposition 3.40 (2) we have that limi,D ( {Fi (x) : x ∈ S}) ≤ limi,D Fi (bi ) + ǫ. Let us construct the sequence (ci )i∈I as follows: Let ǫ ∈ V+ . By hypothesis and Lemma 3.42, V there for all j ∈ C there exists cj ∈ · θ ≤ x∈S Fj (x), where θ ∈ V+ satisfies θ + S such that Fj (cj ) − V · x∈S Fj (x) ≤ θ ≤ ǫ. By Proposition 2.11 (1), it follows that Fj(cj ) − !# CO-QUANTALE VALUED LOGICS 25 V V θ; since x∈S Fj (x) ≤ Fj (cj), it implies that dV (Fj (cj), x∈S Fj (x)) ≤ θ. If j ∈ / C, take cj as any element of S. Let C(θ) := {i ∈ I : dV (limi∈I Fi (ci ), Fi (ci )) ≤ θ}. Therefore, if j ∈ C ∩ C(θ) we have that ^ ^ · limi,D Fi (ci ) − Fj (x) ≤ dV (limi,D Fi (ci ), Fj (x)) x∈S x∈S ≤ dV (limi,D Fi (ci ), Fj(cj)) + dV (Fj(cj ), ^ Fj(x)) x∈S ≤ θ+θ ≤ ǫ V · ǫ ≤ x∈S Fj (x). By Proposition 2.11 (1), we have that limi,D Fi (ci ) − · ǫ ≤ Since CV∩ C(θ) ∈ D and by Proposition 3.40 (1), limi,D Fi (ci ) − limi,D ( {Fi (x) : x ∈ S}).  3.4.2. D-product and D-ultraproduct of spaces and mappings. Proposition 3.45. Let dV be the symmetric distance of V. If (Mi )i∈I is a sequence of Vcontinuity spaces such that for Qall i ∈ I all distances dMi are symmetric, then in the cartesian product i∈I Mi , the relation ∼ defined by (xi )i∈I ∼ (yi )i∈I , if and only if, limi,D dMi (xi , yi ) = 0, is an equivalence relation. Proof. Reflexivity follows trivially and symmetry follows from symmetry of all dMi . Let us focus on the transitivity. Let (xi )i∈I , (yi )i∈I , (zi )i∈I ∈ Q i∈I Mi be such that limi,D dMi (xi , yi ) = 0 and limi,D dMi (yi , zi ) = 0. Since V is a value co-quantale, it is enough to check that for any ǫ ∈ V+ we have that limi,D dMi (xi , zi ) ≤ ǫ. Let ǫ ∈ V+, so by Lemma 2.17 there exists θ ∈ V+ such that θ + θ ≤ ǫ. Since 0 ≺ θ and by definition of limi,D dMi (xi , yi ) and limi,D dMi (yi , zi ), A(θ) := {i ∈ I : dV (limi,D dMi (xi , yi ), dMi (xi , yi ) ≤ θ} and B(θ) := {i ∈ I : dV (limi,D dMi (yi , zi ), dMi (yi , zi ) ≤ θ} belong to D. By hypothesis, limi,D dMi (xi , yi ) = 0 y limi,D dMi (yi , zi ) = 0. Notice that by Proposition 2.11 (4) and since 0 = min V, it follows that dV (limi,D dMi (xi , yi ), dMi (xi , yi )) · dMi (xi , yi ), = ∨{limi,D dMi (xi , yi ) − · limi,D dMi (xi , yi )} dMi (xi , yi ) − (dMi is assumed to be symmetric) · dMi (xi , yi ), dMi (xi , yi ) − · 0} = ∨{0 − = ∨{0, dMi (xi , yi )} = dMi (xi , yi ) 26 D. REYES AND P. ZAMBRANO Therefore, A(θ) = {i ∈ I : dMi (xi , yi ) ≤ θ} and B(θ) = {i ∈ I : dMi (yi , zi ) ≤ θ}. So, if i ∈ A(θ) ∩ B(θ) then dMi (xi , zi ) ≤ dMi (xi , yi ) + dMi (yi , zi ) ≤ θ + θ ≤ ǫ. Since A(θ) ∩ B(θ) ∈ D, by Lemma 3.40 (2) we may say that limi,D dMi (xi , zi ) ≤ ǫ.  Remark 3.46. In general, we do not require that all continuity spaces in the sequence (Mi )i∈I are symmetric, where in that case ∼ might not be an equivalence relation. We just need this requirement if (Mi , dMi ) := (V, dM ) for all i ∈ I. As we will see in the following proposition, Q we can provided a continuity space structure to the cartesian product i∈I Mi , without assuming the symmetry on dMi . Proposition 3.47. Suppose that dV is the Q symmetric distance. If (Mi )i∈I is a sequence of V Q − continuityspaces, ( i∈I Mi , dD ) is a V-continuity Q space, where dD : i∈I Mi × i∈I Mi → V is defined by ((xi )i∈I , (yi )i∈I ) 7→ limi,D dMi (xi , yi ). Q Proof. Given (xi )i∈I ∈ i∈I Mi , then dQi∈ ((xi )i∈I , (xi )i∈I ) = limi,D dMi (xi , xi ) = limi,D 0 = 0. Q Let (xi )i∈I , (yi )i∈I , (zi )i∈I ∈ i∈I Mi , ai := dMi (xi , yi ), bi := dMi (xi , zi ), ci := dMi (yi , zi ), a := limi,D dMi (xi , yi ), b := limi,D dMi (xi , zi ) and c := limi,D dMi (yi , zi ). We want to see that a ≤ b + c, which by · b ≤ c. Let ǫ ∈ Proposition 2.11 (1) it is enough to prove that a − + 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 limi,D dMi (xi , yi ) =: a, limi,D dMi (xi , zi ) =: b and limi,D dMi (yi , zi ) =: c, A := {i ∈ I : dV (a, ai ) ≤ θ}, B := {i ∈ I : dV (b, bi) ≤ θ} and C := {i ∈ I : dV (c, ci ) ≤ 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 · b) ≤ (a − · ai ) + (ai − · bi ) + (bi − · b). Since dMi satisfies tran(a − · bi ≤ ci . Since sitivity, by Proposition 2.11 (1) we have that ai − · · · i ∈ C, we may say that ci − c ≤ ∨{ci − c, c − ci } = dV (c, ci) ≤ θ. By Proposition 2.11 (1), it follows that ci ≤ c + θ. By monotonic· b ≤ (a − · ai ) + (bi − · b) + (c + θ). Since i ∈ A(θ) ∩ B(θ), ity, a − · ai ≤ ∨{a − · ai , ai − · a} = dV (a, ai ) ≤ θ and we also have that a − · · · · b ≤ (a − · ai ) + bi − b ≤ ∨{bi − b, b − bi } = dV (b, bi) ≤ θ. Hence, a − · b) + (c + θ) ≤ θ + θ + (c + θ) = θ + θ + θ + c ≤ ǫ + c. Since (bi − · b ≤ c. ǫ ∈ V+ was chosen arbitrarily, it follows that a −  Definition 3.48. We call D-product of the sequence of V-continuity spaces Q (Mi , dMi )i∈I to the V-continuity space (MD , dMD ) := ( i∈I Mi , dD ) defined in Proposition 3.47. Lemma 3.49. Let dV be the symmetric distance of V. If (Mi , dMi ) is a sequence of V-continuity spaces such that dMi is symmetric for all i ∈ I, CO-QUANTALE VALUED LOGICS 27
Q then i∈I Mi / ∼, dD is a V-continuity space, provided that dD is defined by ([(xi )i∈I ], [(yi )i∈I ]) 7→ limi,D dMi (xi , yi ), where ∼ is defined as in Proposition 3.45. Proof. First, let us check that ((xi )i∈I , (yi )i∈I ) 7→Qlimi,D dMi (xi , yi ) is well-defined. Let (ai )i∈I , (bi )i∈I , (ci)i∈I , (di )i∈I ∈ i∈I Mi be such that (ai )i∈I ∼ (bi )i∈I and (ci )i∈I ∼ (di )i∈I . By Fact 2.48 and the definition of D-ultralimits, in order to prove that limi,D dMi (ai , ci ) = limi,D dMi (bi , di ) it is enough to prove that for every ǫ ∈ V+ we have that {i ∈ I : dV (limi,D dMi (ai , ci ), dMi (bi , di )) ≤ 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. In similar way we can prove that there exists some δ ∈ V+ such that δ + δ ≤ θ. Since (ai )i∈I ∼ (bi )i∈I , by definition of ∼ we have that limi,D dMi (ai , bi ) = 0, hence for all γ ∈ V+ we have that {i ∈ I : dV (limi,D dMi (ai , bi ), dMi (ai , bi )) ≤ γ} ∈ D. By Remark 2.36, we may say that {i ∈ I : dMi (ai , bi ) ≤ γ} ∈ D. Since δ ∈ V+ , in particular A := {i ∈ I : dMi (ai , bi ) ≤ δ} ∈ D. Analogously, since (ci )i∈I (di )i∈I we may say that B := {i ∈ I : dMi (ci , di ) ≤ δ} ∈ D. So, if i ∈ A ∩ B then dMi (ai , ci ) ≤ dMi (ai , bi ) + dMi (bi , di ) + dMi (di , ci ) (by transitivity of dMi ) ≤ δ + dMi (bi , di ) + δ (since i ∈ A ∩ B) ≤ dMi (bi , di ) + θ (by Proposition 2.10 and since δ + δ ≤ θ) In a similar way we may say that dMi (bi , di ) ≤ dMi (ai , ci ) + θ, when· dMi (ai , ci ) ≤ θ ever i ∈ A ∩ B. By Proposition 2.11 (1), dMi (bi , di ) − · Mi (bi , di ) ≤ θ, therefore dV (dMi (ai , ci ), dMi (bi , di )) ≤ and dMi (ai , ci )−d θ. Notice that by definition of limi,D dMi (ai , ci ), A(θ) := {i ∈ I : dV (limi,D dMi (ai , ci ), dMi (ai , ci )) ≤ θ} ∈ D. So, if i ∈ A ∩ B ∩ A(θ) we 28 D. REYES AND P. ZAMBRANO have that dV (limi,D dMi (ai , ci ), dMi (bi , di )) ≤ dV (limi,D dMi (ai , ci), dMi (ai , ci )) +dV (dMi (ai , ci ), dMi (bi , di )) (by transitivity of dV ) ≤ θ+θ (since i ∈ A ∩ B ∩ A(θ)) ≤ ǫ (by Proposition 2.10 and since θ + θ ≤ ǫ)  Definition 3.50. Let (M Qi , dMi )i∈I be a sequence of V-continuity spaces. The V-continuity space i∈I Mi / ∼ provided with the symmetric distance dMD ([(xi )i∈I ], [(yi)i∈I ]) := limi,D dMi (xi , yi ) is called the D-ultraproduct of the sequence (Mi , dMi )i∈I . Definition 3.51. (c.f. [BBHU08]; pag 25) Let (Mi )i∈I , (Ni )i∈I be sequences of V-continuity spaces and K ∈ V be such that K ∈ V greater than the diameter of all the considered spaces. Given a fixed n ∈ N \ {0} and (fi : Mni → Ni )i∈I a sequence of uniformly continuous mappings provided with the same modulus of uniform continuity. The mapping fD : n N (MD , dM D ) → (ND , dD ) defined by ((x1i )i∈I , ..., (xni)i∈I ) 7→ (fi (x1i , ..., xni))i∈I is said to be the D-product of the sequence (fi : Mni → Ni )i∈I Proposition 3.52. Let V be a co-divisible value co-quantale such that dV is the symmetric distance of V. If (fi : Mni → Ni )i∈I is a sequence of uniformly continuous mappings with the same modulus of uniform continuity ∆ : V+ → V+ , then ∆ is also a modulus of uniform continuity for the n N D-product fD : (MD , dM D ) → (ND , dD ). Proof. For Q the sake of simplicity, let us take n = 1. Let x = (xi )i∈I , y = M + (yi )i∈I ∈ i∈I Mi such that dD (x, y) ≤ ∆(ǫ), where ǫ ∈ V (i.e., limi,D dMi (xi , yi ) ≤ ∆(ǫ)). By Lemma 3.41, there exists A ∈ D such that for all i ∈ A we have that dMi (xi , yi ) ≤ ∆(ǫ). Since ∆ is a modulus of uniform continuity for fi for all i ∈ I, in particular we may say that dNi (fi (xi ), fi (yi ) ≤ ǫ, whenever i ∈ A. Since dV is the symmetric distance of V and by Lemma 3.40 (2), dN D ((fi )i∈I (x), (fi )i∈I (y)) ≤ ǫ.  CO-QUANTALE VALUED LOGICS 29 3.4.3. Łoś Theorem in value co-quantale logics. In order to prove a version of Łoś Theorem in this setting, we do not require that the involved distances of the metric structures are necessarily symmetric. Because of that, it is enough to consider the D-product of a sequence of V-continuity spaces (Mi , dMi )i∈I instead of its respective D-ultraproduct, like we need to do in Continuous Logic for assuring that the obtained distance is actually symmetric. However, we need to consider the respective quotient of D-powers of (V, dV ) as a V-symmetric continuity space. From now, we assume that (V, dV ) is a compact, Hausdorff, co-divisible value co-quantale, where dV is the symmetric distance of V. Definition 3.53. Given (M, dM ) a V-continuity space where dM is symmetric, define
the ultrapower of (M, dM ) as the D-ultraproduct Q i∈I M/ ∼, dD of the constant sequence ((M, dM ))i∈I Q Definition 3.54. Given a D-ultraproduct M / ∼:= ( D i∈I Mi )/ ∼, the Q mapping θ : i∈I Mi → MD / ∼ defined by (xi )i∈I 7→ [(xi )i∈I ] is called the canonical mapping. Notation 3.55. We denote by (VD / ∼, dV/∼ ) the D-ultrapower of the Vcontinuity space (V, dV ) (where dV is the symmetric distance). Definition 3.56. We say that two V-continuity spaces (X, dX ) and (Y, dY ) are V-equivalent, if and only if, there exists a bijection f : (X, dx ) → (Y, dY ) such that dX (x, y) = dY (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, dV ) a value co-quantale provided that dV is the symmetric distance, then the D-ultrapower (VD / ∼ , dV/∼ ) is V−equivalent to (V, dV ). Proof. Defining T : V → V/ ∼ by x 7→ [(x)i∈I ], we have that T is a 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 dV/∼ (T (x), T (y)) = dV/∼ ([(x)i∈I], [(y)i∈I]) = limi,D dV (x, y) = dV (x, y) = · y) ∨ (y − · x) = 0, therefore x − · y = 0 and y − · x = 0. By Proposi(x − 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 [(xi )i∈I ] ∈ VD . Let us see that T (limi,D xi ) := [(limi,D xi )i∈I ] = [(xi )i∈I ] (i.e., we will have that limi,D dV (limi.D xi , xi ) = 0)): let ǫ ∈ V+ , therefore dV (0, dV (limi.D xi , xi )) = · dV (limi.D xi , xi ), dV (limi.D xi , xi ) − · 0} = ∨{0, dV (limi.D xi , xi )} = ∨{0 − dV (limi.D xi , xi ), and by definition of limi.D xi we may say that {i ∈ I : dV (limi.D xi , xi ) ≤ ǫ} ∈ D. Hence, {i ∈ I : dV (0, dV (limi.D xi , xi )) ≤ ǫ} ∈ D and then limi,D dV (limi.D xi , xi ) = 0. 30 D. REYES AND P. ZAMBRANO T preserves distances: In fact, dVD (T (x), T (y)) = dVD ([(x)i∈I], [(y)i∈I]) := limi,D (dV (x, y))i∈I = dV (x, y). Therefore, T is an equivalence.  Remark 3.58. Notice that the mapping T ′ : VD / ∼ → V defined by [(xi )i∈I ] 7→ limi,D xi is the inverse of T . Fact 3.59. Given a sequence of V-continuity spaces ((Mi , dMi ))i∈I provided that all distances dMi are symmetric, the canonical mapping θ : Q i∈I Mi → MD / ∼ is uniformly continuous with modulus of uniform continuity idV+ . Definition 3.60. Suppose that V is co-divisible and let L be a language based in a continuous structure. Given a sequence (Mi )i∈I of L-structures, define the D-product of (Mi )i∈I as the L-structure MD with underlying V-continuity space (MD , dD ), defined as follows: Q (1) For a predicate symbol R ∈ L, define R i∈I Mi := T ′ ◦ θ ◦ RD , where RD is the D-product of the mappings (RMi )i∈I , θ the canonical mapping given in Definition 3.54 and T ′ the mapping defined in Remark 3.58. Q (2) For a function symbol F ∈ L, defined F i∈I Mi as the D-product of the mappings (FMi )i∈I . Q (3) For a constant symbol c ∈ L, define c i∈I Mi := (cMi )i∈I . Remark 3.61. Notice that Theorem 3.52 guarantees that the interpretations of the symbols of L given above have the same modulus of uniform continuity given by the language. Theorem 3.62. (Łoś Theorem; c.f. [BBHU08] Thrm 5.4). Let (V, dV ) be a co-divisible V-domain. If (Mi )i∈I is a sequence of L-structures, then for any L-formula φ(x1 , ..., xn) (if φ has quantifiers, we require that its interpretations in any L-structure Mi satisfyQthe hypothesis of Proposition 3.42) and any tuple ((a1i)i∈I , ..., (ani)i∈I ) ∈ ( i∈I Mi )n , we have that φMD ((a1i)i∈I , ..., (ani)i∈I ) = limi,D φMi (a1i , ..., ani) Proof. We proceed by induction on L-formulae. (1) φ : d(x1 , x2 ) (d(x1 , x2 ))M ((a1i)i∈I , (a2i)i∈I ) := dM ((a1i )i∈I , (a2i)i∈I ) = limi,D dMi (a1i , a2i) (definition of a distance in a product) = limi,D (d(x1 , x2 ))Mi (a1i, a2i )) CO-QUANTALE VALUED LOGICS 31 (2) φ : R(x1 , ..., xn), where R is a predicate symbol in L. (R(x1 , ..., xn)((a1i)i∈I , ..., (ani)i∈I ))M := (R((a1i)i∈I , ..., (ani)i∈I ))M = T ′ ◦ θ((RMi (a1i , ..., ani))i∈I ) (by Definition 3.60 (1)) = limi,D RMi ((a1i, ..., ani)) (by definition of θ and T ′ ) (3) φ : u(σ1 , ..., σm)(x1 , ..., xn), where u : Vm → V is a uniformly continuous mapping and σ1 , · · · σm are L-formulae such that Mi 1 n 1 n σM k ((ai )i∈I , ..., (ai )i∈I ) = limi,D σk (ai , ..., ai ) for all k ∈ {1, ..., m} (induction hypothesis). For the sake of simplicity, denote ((a1i)i∈I , ..., (ani)i∈I ) =: a and (a1i , ..., ani) = ai . M (u(σ1 , ...σm))M (a) := u(σM 1 (a), ..., σm (a)) Mi i = u(limi,D σM 1 (ai ), ..., limi,D σm (ai )) (induction hypothesis) i Define bi,k := σM k (ai ) for any k ∈ {1, · · · , m} and i ∈ I. Notice that {i ∈ I : dV (u(limi,D bi,1 , ..., limi,D bi,m ), u(bi,1, ..., bi,m)) ≤ ǫ} contains the set {i ∈ I : dV n ((limi,D bi,1 ..., limi,D bi,m ), (bi,1, ..., bi,m)) ≤ ∆(ǫ)}   n _ = i ∈ I : {dV (limi,D bi,k , bi,k )} ≤ ∆(ǫ) , k=1 whenever ∆ is a modulus of uniform continuity for u. Notice that this previous set belongs to D, because it contains Tn {i ∈ I|dV (limi,D bi,k , bi,k ) ≤ ∆(ǫ)}, which belongs to D by k=1 definition of a D-limit and since D is an ultrafilter. Therefore, Mi Mi Mi i u(limi,D σM 1 (ai ), ..., limi,D σm (ai )) = limi,D u(σ1 (ai ), · · · σm (ai )) limi,D (u(σ1 , · · · , σm ))Mi (ai ). W (4) φ : x ϕ(x, x1 , ..., xn) Let ϕ(x, x1 , ..., xn) be an L-formula such that ϕM ((bi )i∈I , (a1i)i∈I , ..., (ani)i∈I ) = limi∈I ϕMi (bi , a1i , ..., ani) 32 D. REYES AND P. ZAMBRANO Q for all (bi )i∈I ∈ i∈I Mi (induction hypothesis). For the sake of simplicity, denote a := ((a1i)i∈I , ..., (ani)i∈I ), ai := (a1i , ..., ani) and b := (bi )i∈I . So, _ x ϕ(x, x1 , ..., xn) !M (a) = _ x ϕ(x, a) !M _ = {ϕM (b, a) : b ∈ M} (by Definition 3.24 (4)) _ = {limi,D ϕMi (bi , ai ) : b ∈ M} (induction hypothesis) _ ≤ limi,D {ϕMi (bi , ai ) : b ∈ M} (by Proposition 3.44) By Proposition 3.44, given ǫ ∈ V+ there exists a sequence Q (cj)j∈I of tuples cj := (cji )i∈I ∈ i∈I Mi such that limi,D ( _ {ϕMi (bi , ai ) : b ∈ M}) ≤ limi,D ϕMi (cii , ai ) + ǫ _ ≤ {limi,D ϕMi (bi , a(i)) : b ∈ M} + ǫ where the last inequality follows from monotonicity of ≤ Q i and since (ci )i∈I ∈ i∈I Mi . Since ǫ ∈ V+ is arbitrary, we have W W that limi,D {ϕMi (bi , ai ) : b ∈ M} ≤ limi,D {ϕMi (bi , ai ) : b ∈ M}. by antisymmetry W Therefore, W of ≤ weMmay say that Mi limi,D {ϕ (b(i), aiW ) : b ∈ M} = {limi,D ϕW i (b(i), a(i)) : Mi b ∈ M}. Notice W that {ϕ (biM, ai ) : b ∈ M} =W {ϕ(c, ai ) : c ∈ Mi }, then ( xWϕ(x, x1 , ..., xn)) (a) = limi,D {ϕ(c, ai ) : c ∈ Mi } = limi,D ( x ϕ(x, x1 , ..., xn))Mi (ai ), as desired. V (5) φ : x ϕ(x, x1 , ..., xn) Let ϕ(x, x1 , ..., xn) be an L-formula such that ϕM ((bi )i∈I , (a1i)i∈I , ..., (ani)i∈I ) = limi∈I ϕMi (bi , a1i , ..., ani) CO-QUANTALE VALUED LOGICS 33 Q for all (bi )i∈I ∈ i∈I Mi . For the sake of simplicity, denote 1 n a := ((ai )i∈I , ..., (ai )i∈I ), ai := (a1i, ..., ani) and b := (bi )i∈I . So, !M !M ^ ^ ϕ(x, x1 , ..., xn) (a) = ϕ(x, a) x x ^ = {ϕM (b, a) : b ∈ M} (by Definition 3.24 (5)) ^ = {limi,D ϕMi (bi , ai ) : b ∈ M} (induction hypothesis) ^ ≥ limi,D {ϕMi (bi , ai ) : b ∈ M} (by Proposition 3.44) By Proposition 3.44 and by hypothesis, given ǫ ∈ V+ there Q exists a sequence (bj)j∈I of tuples bj := (bji )i∈I ∈ i∈I Mi such that limi,D ( ^ {ϕMi (bi , ai ) : b ∈ M}) + ǫ ≥ limi,D ϕMi (bii , ai ) ^ ≥ {limi,D ϕMi (bi , ai ) : b ∈ M} follows from the fact that (bii )i∈I ∈ Qwhere the last inequality + ǫ ∈ V was takenVarbitrarily, we have that i∈I M Vi . Since Mi limi,D {ϕ V (bi , ai ) : b ∈ M} ≥ {limi,D ϕMi (bi , ai ) : b ∈ M} ≥ limi,D {ϕMi (b(i), ai) : b ∈ M}. By antisymmetry of ≤, we may say that ^ ^ limi,D {ϕMi (bi , ai ) : b ∈ M} = {limi,D ϕMi (bi , ai ) : b ∈ M}. V M V i Since, V {ϕMi (bi , ai ) : b ∈ M} = {ϕ V (c, ai ) : c ∈ Mi }, M then ( Vx ϕ(x, x1 , ..., xn)) (a) = limi,D {ϕ(c, ai ) : c ∈ Mi } = limi,D ( x ϕ(x, x1 , ..., xn))Mi (ai ), as desired.  Note 3.63. In case that we want to work in symmetric spaces, the same argument as above
to prove a version of Łoś Theorem for the DQ works ultraproduct i∈I Mi ∼ . 3.4.4. Some consequences of Łoś Theorem. In this setting, Łoś Theorem implies a version of Compactness Theorem and the existence of some kind of ω1 -saturated models, as it holds in both first order and Continuous logics. 34 D. REYES AND P. ZAMBRANO First, we provide a proof of Compactness Theorem, up to Łoś Theorem. 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− theory which 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 := {E1 , ..., En}, there exists some Mλ ∈ C such that Mλ |= Ek for all k ∈ {1, ..., n}. Fixed an L-condition E ∈ T, define S(E) := {λ ∈ Λ : E ∈ λ}. Notice that S(E1) ∩ ... ∩ S(En ) 6= ∅ ( λ := {E1 , · · · , En} ∈ S(E1 ) ∩ ... ∩ S(En )), therefore {S(E) : E ∈ T} satisfies the Finite Intersection Property. Q Let D be an ultrafilter over Λ extending {S(E) : E ∈ T}. Let M := λ∈Λ Mλ be the respective D-product of the sequence of Lstructures (Mλ )λ∈Λ . Given E ∈ T, where E : ψ = 0 (ψ an L-sentence). Notice that for all λ ∈ S(E) we have that Mλ |= E; i.e., ψMλ = 0. Since by construction S(E) ∈ D, by Łoś Theorem (Theorem 3.62) it follows that limλ,D ψMλ = 0. Notice that ψM = limλ,D ψMλ = 0, so M |= E. Therefore, M |= T.  Keisler showed in [Kei64] the existence of saturated structures by using ultraproducts. In [BBHU08], there is a proof of an analogous result to Keisler’s construction by using metric ultraproducts. In the following lines, we provide a proof of this result in the logic propposed in this paper, supposing that V is provided with the symmetric distance (abusing of the notation, we will denote by dV ) and co-divisible value co-quantale satistying the SAFA Property (Definition 2.27). Definition 3.65. Let L be a language based on a continuous structure, Γ (x1 , ..., xn) be a set of L-conditions and M be an L-structure. We say that Γ (x1 , ..., xn) is satisfiable in M, if and only if, there exist a1 , ..., an ∈ M such that M |= E(a1 , ..., an) for all E(x1 , · · · , xn ) ∈ Γ (x1 , ..., xn). 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 κsaturated, if and only if, given A ⊆ M such that |A| < κ and Γ (x1 , ..., xn) a set of L(A)-conditions with parameters in A, it holds that if Γ (x1 , ..., xn) is finitely satisfiable in M then Γ (x1 , ..., xn) is satisfiable in M. Definition 3.67. An ultrafilter D over K 6= ∅ is said to be countablyincomplete, if and only if, there exists some {An : n ∈ N} ⊆ D such that T n∈N An = ∅. CO-QUANTALE VALUED LOGICS 35 Proposition 3.68. Let D be an coutably-incomplete ultrafilter over K 6= ∅, then there exists a countableTsubcollection {Jn : n ∈ N} of D such that Jn+1 ⊆ Jn for all n ∈ N and n∈N Jn = ∅. Proof. By hypothesis, there exists a subcollection {An : n ∈ N} ⊆ D T of D tal que n∈N An = ∅. Define J0 := A0 and Jn+1 := Jn ∩ An+1 for any n ∈ N \ {0}.  Proposition 3.69. (c.f. [BBHU08]; Prop. 7.6) Let V be a compact, Hausdorff, co-divisible value co-quantale satisfying SAFA, provided with the symmetric distance dV . Let L be a countable language based on a continuous structure and D be a countably-incomplete ultrafilter over a non empty set Λ. Given any Λ-sequence of L-structures (Mλ )λ∈Λ , its D-product MD is ω1 -saturated, assuming that all L-formulae satisfy the hypothesis in Łoś Theorem (Theorem 3.62). Proof. For the sake of Qsimplicity, let us analyze L-conditions with one variable x. Let A ⊆ λ∈Λ Mλ be countable and Γ (x) be a set of L(A)conditions with parameters in A which is finitely satisfiable in MD . We will prove that Γ (x) is satisfiable in MD . Since L is countable, let Γ (x) := {ψn (x) : n < ω} be an enumeration of Γ (x). Since D is contably-incomplete, by Proposition 3.68 there exists a sequence (Jn )n∈N of elements in D such that Jn+1 ⊆ Jn for all T n ∈ N and n∈N Jn = ∅. By Q hypothesis, for all k ∈ N the set {ψ1 (x), ..., ψk(x)} is satisfiable in λ∈Λ MλQ . By Łoś Theorem (Theorem 3.62), there exists some a := (aλ )λ∈Λ ∈ λ∈Λ Mλ such that for all n ∈ {1, · · · , k} we have that MD + λ ψn Mλ (a) = limλ,D ψM n (aλ ) = 0. Therefore, given ǫ ∈ V we may λ say that {λ ∈ Λ : dV (0, ψM n (aλ )) ≤ ǫ} ∈ D. Since dV is the symmetric λ distance, by Remark 2.36 we have that {λ ∈ Λ : ψM n (aλ ) ≤ ǫ} ∈ D. Since V satisfies SAFA Property (Definition 2.27), there exists a sequence (uk )k∈N in V such that V (1) n∈N un = 0. (2) for all n ∈ N, 0 ≺ un (3) for all n ∈ N, un+1 ≤ un λ Therefore, {λ ∈ Λ : ψM u } ∈ D for all l ∈ N. This implies that n V(aλ ) ≤ Wk l Ak := {λ ∈ Λ : Mλ |= x∈Mλ n=1 ψn (x) ≤ uk+1 } ∈ D whenever k ∈ N. Define the sequence (Xn )n∈N of elements in D as follows: X0 := Λ V W and Xk := Jk ∩ Ak if k ∈ N \ {0}. Notice that x∈Mλ kn=1 ψn (x) ≤ W Vk+1 ⊆ An for all n ∈ N and x∈Mλ n=1 ψn (x) ≤ uk+2 ≤ uk+1 , then An+1 T then Xn+1 ⊆ Xn for all n ∈ N. Notice that n∈N Xn = ∅, therefore if λ ∈ Λ there exists kλ ∈ N such that kλ := max{n ∈ N : λ ∈ Xn }. 36 D. REYES AND P. ZAMBRANO Q Define a := (aλ )λ∈Λ ∈ λ∈Λ Mλ as follows: In case that kλ = 0 , take a as any element in Mλ , otherwise take aλ ∈ Mλ such that W Mλ λ {ψn (aλ ) : n ≤ kλ } ≤ ukλ . So, if k ∈ N then for any n ∈ N such λ that k ≤ n and λ ∈ Xn we have that n ≤ kλ , hence ψM k (aλ ) ≤ ukλ ≤ unQ. 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