aa r X i v : . [ m a t h . L O ] A ug BETH DEFINABILITY AND THE STONE-WEIERSTRASS THEOREM
LUCA REGGIO
Abstract.
The Stone-Weierstrass Theorem for compact Hausdorff spaces is a basic resultof functional analysis with far-reaching consequences. We show that this theorem is a conse-quence of the Beth definability property of a certain infinitary equational logic, stating thatevery implicit definition can be made explicit. Introduction
Weierstrass’ Approximation Theorem states that every continuous real-valued functiondefined on a closed real interval can be uniformly approximated by polynomials. In 1937,Marshall Stone proved a vast generalisation of this theorem [34]; nowadays known as theStone-Weierstrass Theorem for compact Hausdorff spaces, it is a fundamental result of func-tional analysis with far-reaching consequences.Let X be a non-empty compact Hausdorff space and C( X, R ) the collection of all continuousfunctions X → R , where R is the set of real numbers with the usual Euclidean topology. Theset C( X, R ) is equipped with the uniform metric ̺ : C( X, R ) × C( X, R ) → [0 , ∞ ) given by ∀ f , g ∈ C( X, R ) , ̺ ( f , g ) = sup x ∈ X {| f ( x ) − g ( x ) |} . The Stone-Weierstrass Theorem provides sufficient conditions for a subset G ⊆ C( X, R ) tobe dense in the topology induced by the uniform metric. Recall that a subset G of C( X, R ) separates the points of X if, for any two distinct points x, y ∈ X , there is f ∈ G such that f ( x ) = f ( y ). The Stone-Weierstrass Theorem can be phrased as follows: Theorem 1 (Stone-Weierstrass Theorem) . Let X be a non-empty compact Hausdorff spaceand G a subset of C( X, R ) satisfying the following properties:(1) G separates the points of X ;(2) G contains the constant function of value ;(3) f ∈ G and α ∈ R imply α f ∈ G ;(4) f + g ∈ G and max { f , g } ∈ G whenever f , g ∈ G .Then G is dense in C( X, R ) in the topology induced by the uniform metric. Remark.
For a nice exposition of the previous result, along with a proof relying on the closureof the set G under lattice-theoretic operations, see [35]. The version stated above, involvingthe operations + and max, seems to be more widespread in analysis (cf. [15, Theorem 7.29]). This project has received funding from the European Union’s Horizon 2020 research and innovation pro-gramme under the Marie Sk lodowska-Curie grant agreement No 837724, and from the grant GA17-04630Sof the Czech Science Foundation.
On the other hand, Beth definability is a strong property that a logic may, or may not,satisfy. Informally, it states that every property which can be defined implicitly admits anexplicit definition. (For a precise definition, in our setting, see Section 4.) Beth’s theorem,proved in 1953, states that first-order logic has the Beth definability property [3]. This resultsheds light on a phenomenon which occurs frequently in the mathematical practice. Forinstance, a real closed field F (i.e., a field elementarily equivalent to R ) admits a unique totalorder which turns it into an ordered field. This order is given by ∀ x, y ∈ F ( x y ↔ ∃ z ( y − x = z )) . Beth’s theorem tells us that this is no coincidence: whenever a property can be described ina unique way, there is a first-order formula explicitly defining it.The aim of this paper is to show that the Stone-Weierstrass Theorem for compact Hausdorffspaces can be seen as a consequence of the Beth definability property for an appropriatelogic, thus obtaining a “logical proof” of the Stone-Weierstrass Theorem. (In fact, the Bethdefinability property can be regarded as an equivalent form of the Stone-Weierstrass Theorem,cf. Remark 23.) The logic in question is the equational consequence relation associated witha variety of algebras ∆ introduced in [25]. The latter is not a variety in the usual sense, `a la
Birkhoff, in that its signature contains an operation symbol of countably infinite arity. Allmembers of ∆ have reducts which are MV-algebras, the algebraic counterpart to Lukasiewiczinfinite-valued propositional logic [22, 23]. Therefore, the logic associated with ∆ is aninfinitary extension of Lukasiewicz logic (this extension is, in fact, conservative).The paper is organised as follows. In Section 2 we provide the necessary background onAbelian ℓ -groups, which capture the structure of the algebras of functions C( X, R ), MV-algebras and the infinitary variety ∆. Further, we recall the Cignoli-Dubuc-Mundici adjunc-tion between MV-algebras and the dual of compact Hausdorff spaces, and its restriction tothe variety ∆. The logic (cid:15) ∆ is introduced in Section 3, where some of its main propertiesare established. Finally, in Section 4, we prove that (cid:15) ∆ has the Beth definability propertyand show how the Stone-Weierstrass Theorem can be deduced from it. Related work.
An algebraic treatment of the Stone-Weierstrass Theorem was provided byBanaschewski [1], who showed that this result is ultimately a consequence of more generalproperties of the class of f -rings. Also, the relation between the Stone-Weierstrass Theoremand the surjectivity of epimorphisms (cf. Theorem 20), e.g. in the category of commutativeC ∗ -algebras, has long been known [31]. See also [5]. In the present article, we focus on thelogical gist of the Stone-Weierstrass Theorem and introduce an infinitary equational logic (cid:15) ∆ which may be thought of as a “logic for compact Hausdorff spaces” (in the same waythat classical propositional logic is a logic for zero-dimensional compact Hausdorff spaces byStone duality for Boolean algebras [33]). Recently, a modal calculus for compact Hausdorffspaces, based on de Vries duality, was introduced in [4]. For another approach, cf. [27].
Notation.
Continuous functions are denoted by f , g , h , while the symbols f, g, h are reservedfor variable assignments or algebra homomorphisms. MV- and δ -algebras, as well as their It is well known that the category KH of compact Hausdorff spaces and continuous maps is not duallyequivalent to a Birkhoff variety of algebras. Hence, there is no finitary equational logic for compact Hausdorffspaces. For more on the axiomatisability of the dual of KH , see e.g. [20, 25]. ETH DEFINABILITY AND THE STONE-WEIERSTRASS THEOREM 3 underlying sets, are denoted A , B . If is a lattice order, binary infima and binary supremaare denoted ∧ and ∨ , respectively.2. Preliminaries and background
Abelian ℓ -groups and MV-algebras. An Abelian ℓ -group is an Abelian group G ,written additively, equipped with a lattice order invariant under translations. That is, ∀ a , a , b ∈ G, a a = ⇒ a + b a + b. We say that G is unital if it is equipped with a distinguished element u ∈ G (the unit ) suchthat, for every a ∈ G , there is n ∈ N with nu > a . A prime example of unital Abelian ℓ -group is ( R , R is the additive group of real numbers equipped with the usualtotal order. More generally, for any topological space X , the set C( X, R ) of all continuous R -valued functions on X is an Abelian ℓ -group if the operations are defined pointwise. Theconstant function 1 X : X → R of value 1 is a unit for C( X, R ).We write ℓ - Ab for the category of unital Abelian ℓ -groups with unital ℓ -homomorphisms ,i.e. functions which are both lattice and group homomorphisms and preserve the units. Sub-objects in ℓ - Ab , i.e. sublattice subgroups containing the unit, are called unital ℓ -subgroups .Classical references for the theory of ℓ -groups include [6, 11, 13].Given an arbitrary unital Abelian ℓ -group ( G, u ), one can equip its unit interval Γ( G, u ) := { a ∈ G | a u } with the operations a ⊕ b := ( a + b ) ∧ u and ¬ a := u − a. (When the choice of the unit is clear from the context, we write Γ( G ) instead of Γ( G, u ).)The tuple (Γ( G ) , ⊕ , ¬ ,
0) carries the structure of an
MV-algebra .An
MV-algebra is an algebra ( A , ⊕ , ¬ , L MV = {⊕ , ¬ , } oftype (2 , , a, b ∈ A :(i) ( A , ⊕ ,
0) is a commutative monoid (iii) a ⊕ ¬ ¬ ¬ is an involution, i.e. ¬¬ a = a (iv) ¬ ( ¬ a ⊕ b ) ⊕ b = ¬ ( ¬ b ⊕ a ) ⊕ a While Boolean algebras are the algebraic counterpart to classical propositional logic, MV-algebras are the algebraic counterpart to Lukasiewicz infinite-valued propositional logic. Formore details, we refer the interested reader to [9, 29].The operation ⊕ should be regarded as a strong disjunction, and the involution ¬ playsthe rˆole of a negation which allows us to define a strong conjunction by a ⊙ b := ¬ ( ¬ a ⊕ ¬ b ) . It will be useful to introduce a further derived connective: a ⊖ b := a ⊙ ¬ b. We denote by MV the category of MV-algebras and MV-homomorphisms , i.e. the functionspreserving ⊕ , ¬ and 0. Any MV-algebra has an underlying structure of distributive latticebounded below by 0 and above by 1 := ¬
0. Binary joins (aka weak disjunctions) are givenby a ∨ b = ¬ ( ¬ a ⊕ b ) ⊕ b. LUCA REGGIO
Thus, item (iv) above states that a ∨ b = b ∨ a . Binary meets (aka weak conjunctions) aredefined by the De Morgan condition a ∧ b = ¬ ( ¬ a ∨ ¬ b ). Boolean algebras are precisely thoseMV-algebras in which the law of excluded middle a ∨ ¬ a = 1 holds.The standard MV-algebra is the real unit interval [0 ,
1] with neutral element 0 in whichthe operations ⊕ and ¬ are defined, respectively, by a ⊕ b := min { a + b, } (the connective ⊕ is frequently referred to as truncated addition ) and ¬ a := 1 − a. The derived operations ⊙ and ⊖ are then interpreted as a ⊙ b = max { , a + b − } and a ⊖ b = max { , a − b } ( ⊖ is often called truncated subtraction ). The underlying lattice orderof this MV-algebra coincides with the order that [0 ,
1] inherits from the real numbers. Whenwe refer to [0 ,
1] as an MV-algebra, we always mean the structure just described. Note thatthe standard MV-algebra [0 ,
1] coincides with Γ( R , ℓ -group ( R , X , the unit interval of the Abelian ℓ -group C( X, R ) with unit 1 X can be identified with the MV-algebraC( X, [0 , { f : X → [0 , | f is continuous } , where [0 ,
1] is equipped with the Euclidean topology and the MV-algebraic operations ofC( X, [0 , ℓ -homomorphism ( G, u ) → ( G ′ , u ′ ) between unital Abelian ℓ -groups restrictsto an MV-homomorphism Γ( G, u ) → Γ( G ′ , u ′ ). In fact, this assignment yields a functorΓ : ℓ - Ab → MV . In 1986, Mundici showed that Γ is an equivalence of categories. For aproof, and an explicit description of the quasi-inverse functor, see [28] or [9, 7.1.2, 7.1.7].
Theorem 2 (Mundici’s equivalence) . Γ : ℓ - Ab → MV is an equivalence of categories. Ideal theory in MV-algebras.
As in the case of rings, quotients (or congruences) ofMV-algebras can be described in terms of ideals. An ideal of an MV-algebra A is a subset of A which contains 0, is downward-closed in the lattice order of A and is closed under ⊕ . Theideal h S i generated by a non-empty subset S ⊆ A can be described as follows (cf. [9, 1.2.1]): h S i = { a ∈ A | a s ⊕ · · · ⊕ s n for some s , . . . , s n ∈ S } . (1)Given an ideal I ⊆ A , the corresponding congruence is ≡ I = { ( a, b ) ∈ A × A | d( a, b ) ∈ I } , where d is the derived binary operation(2) d( a, b ) := ( a ⊖ b ) ⊕ ( b ⊖ a ) . The binary operation d( - , - ) is known as Chang’s distance , and when interpreted in theMV-algebra [0 ,
1] it coincides with the usual Euclidean distance. We write A /I to denotethe quotient algebra A / ≡ I . Conversely, the ideal associated with a congruence ≡ on A is I ≡ = { a ∈ A | a ≡ } . We shall only need that Γ is full and faithful, hence it reflects isomorphisms. Cf. the proof of Theorem 1on page 18. This fact is easier to prove, and corresponds to Propositions 3.4-3.5 in [28].
ETH DEFINABILITY AND THE STONE-WEIERSTRASS THEOREM 5
This yields a bijective correspondence between ideals of A and congruences on A , see [9,1.2.6]. As a consequence of Zorn’s Lemma, an MV-algebra is non-trivial (i.e. it has twodistinct elements) precisely when it admits a maximal ideal [9, 1.2.15]. An MV-algebra A is simple if it has no non-trivial proper ideals. It is semisimple if it is a subdirect product ofsimple MV-algebras; equivalently, if the intersection of all maximal ideals is { } .Maximal ideals of an MV-algebra A can also be described in terms of MV-homomorphisms A → [0 , ,
1] is simple and h − (0) ⊆ A is a maximal ideal for every MV-homomorphism h : A → [0 , Theorem 3 (H¨older’s Theorem) . For every MV-algebra A and maximal ideal m ⊆ A , thereis a unique injective homomorphism of MV-algebras h m : A / m ֒ → [0 , . The assignments h h − (0) and m h m yield a bijection between hom MV ( A , [0 , andthe set of maximal ideals of A .Proof. See [16] and also [17, 18]. For the MV-algebraic version, cf. [9, 3.5.1]. (cid:3)
The Cignoli-Dubuc-Mundici adjunction.
We recall from [10] the Cignoli-Dubuc-Mundici adjunction between the category MV of MV-algebras with MV-homomorphismsand the dual of the category KH of compact Hausdorff spaces with continuous maps.This dual adjunction is natural (in the sense of [30]) and is induced by the dualising object[0 , A , equip the set of homomorphisms hom MV ( A , [0 , , A . The ensuing space, called the maximal spectrum of A anddenoted Max A , is compact and Hausdorff, see [29, Proposition 4.15] for a proof in terms ofmaximal ideals. The next lemma provides a characterisation of the closed subsets of Max A (for a proof, see [29, Section 4.4]). For any subset S ⊆ A , define V ( S ) := { h ∈ Max A | h ( a ) = 0 for all a ∈ S } . (3)If a ∈ A , we write V ( a ) as a shorthand for V ( { a } ). Lemma 4.
The following statements hold for every MV-algebra A and subsets S, S ′ ⊆ A .(1) V ( S ) = V ( h S i ) , where h S i is the ideal of A generated by S ;(2) V ( S ) ⊆ V ( S ′ ) if, and only if, h S i ⊇ h S ′ i ;(3) the closed subsets of Max A are precisely those of the form V ( I ) for an ideal I of A . Furthermore, for every MV-homomorphism k : A → B , the functionMax k : Max B → Max A , h h ◦ k is continuous because (Max k ) − ( V ( I )) = V ( h k ( I ) i ) for every ideal I of A . We get a functorMax : MV → KH op . The space Max A is akin to the prime spectrum of a commutative ring. However, in contrast to the caseof rings, Max A is Hausdorff because homomorphisms A → [0 ,
1] correspond to maximal ideals of A . LUCA REGGIO
Conversely, given a compact Hausdorff space X , recall the MV-algebra C( X, [0 , X → [0 , f : X → Y is a morphism in KH , it is easily seen thatthe induced map C f : C( Y, [0 , → C( X, [0 , , g g ◦ f is a morphism in MV . We can thus regard hom KH ( − , [0 , KH op → MV . For every compact Hausdorff space X there is a continuous map ε X : X → Max C( X, [0 , , x (C( X, [0 , ev x −−→ [0 , , f f ( x )) . Moreover, for every MV-algebra A , there is an MV-homomorphism η A : A → C(Max A , [0 , , a (Max A ev a −−→ [0 , , h h ( a )) . To improve readability, we will write b a instead of ev a . Thus, for all a ∈ A , b a : Max A → [0 , b a ( h ) = h ( a ).Denoting Id C the identity functor on a category C , it is not difficult to see that ε X and η A yield natural transformations ε : Max ◦ C → Id KH op and η : Id MV → C ◦ Max, respectively.
Theorem 5 (Cignoli-Dubuc-Mundici adjunction) . The natural transformations η and ε arethe unit and counit, respectively, of an adjunction Max ⊣ C : KH op → MV . Furthermore,(1) for every X in KH , the component ε X : X → Max C( X, [0 , of the counit is ahomeomorphism, i.e. the functor C is full and faithful;(2) for every A in MV , the component η A : A → C(Max A , [0 , of the unit is injectiveif, and only if, A is semisimple. The adjunction Max ⊣ C was introduced in [10]. For a proof of items 1-2, cf. [29, 4.16].2.4.
The variety ∆ . Any adjunction restricts to an equivalence between the full subcate-gories defined by the fixed objects, i.e. those objects for which the components of the unit andcounit, respectively, are isomorphisms. In the case of the Cignoli-Dubuc-Mundici adjunctionMax ⊣ C : KH op → MV , because the counit is a natural isomorphism, we know that there is a full subcategory of MV dually equivalent to KH . In [25], the variety ∆ of δ -algebras was defined and shown to beisomorphic to a full subcategory of MV dually equivalent to KH . However, the proof ofthe duality between ∆ and KH relies on the Stone-Weierstrass Theorem. In this section werecall some facts about δ -algebras which do not depend on the Stone-Weierstrass Theorem.Consider the algebraic language L ∆ = { δ, ⊕ , ¬ , } of type ( ω, , , δ takes as argument a countable sequence of terms, we write x, y and 0 as shorthands for thesequences x , x , . . . , y , y , . . . , and 0 , , . . . , respectively. It will be convenient to introduce aderived unary operation f , to be thought of as multiplication by : f ( x ) := δ ( x, . Moreover, recall from equation (2) that d denotes Chang’s distance.
ETH DEFINABILITY AND THE STONE-WEIERSTRASS THEOREM 7
Definition 6. A δ -algebra is an L ∆ -algebra ( A , δ, ⊕ , ¬ ,
0) such that ( A , ⊕ , ¬ ,
0) is an MV-algebra and the following identities are satisfied for every x, y ∈ A and x, y ∈ A ω :(i) d (cid:0) δ ( x ) , δ ( x , (cid:1) = δ (0 , x , x , . . . ) (iv) δ (0 , x ) = f ( δ ( x ))(ii) f ( δ ( x )) = δ ( f ( x ) , f ( x ) , . . . ) (v) δ ( x ⊕ y , x ⊕ y , . . . ) > δ ( x , x , . . . )(iii) δ ( x, x, . . . ) = x (vi) f ( x ⊖ y ) = f ( x ) ⊖ f ( y )A homomorphism of δ -algebras, or δ -homomorphism for short, is a homomorphism of theunderlying MV-algebras which preserves the operation δ . In fact, it will follow from item 2in Theorem 7 below that all MV-homomorphisms between δ -algebras are δ -homomorphisms.We write ∆ for the category (as well as for the variety) of δ -algebras and δ -homomorphisms. Remark.
Since the operation δ has infinite arity, ∆ is not a variety of Birkhoff algebras.Thus, we rely on the theory of varieties of infinitary algebras as developed by S lomi´nskiin [32], see also [21]. In the following sections we will not need to exploit the axiomatisationin Definition 6. Instead, we will make use of the properties of ∆ summarised in Theorem 7.The operation δ , and its semantic interpretation which we now recall, were introduced byIsbell in [19]. In the unit interval [0 , x ∈ [0 , ω , set δ ( x ) = ∞ X i =1 x i i . It is not difficult to see that the standard MV-algebra [0 , δ , is a δ -algebra in which the unary operation f coincides with the multiplication by .More generally, for every compact Hausdorff space X , the MV-algebra C( X, [0 , δ -algebra if, for all g ∈ C( X, [0 , ω , δ ( g ) is defined as the uniformly convergent series(5) δ ( g ) = ∞ X i =1 g i i . Throughout, whenever we regard C( X, [0 , δ -algebra, we assume that the interpretationof the operation δ is the one above. In fact, in [25, Corollary 6.4] it is shown that there is noother structure of δ -algebra expanding the pointwise MV-algebraic structure of C( X, [0 , KH op → MV factors through the forgetful functor ∆ → MV .Hence, the Cignoli-Dubuc-Mundici adjunction restricts to an adjunctionMax ⊣ C : KH op → ∆ . Theorem 7.
The following statements hold:(1) the underlying MV-algebra of any δ -algebra is semisimple;(2) the forgetful functor ∆ → MV is full;(3) for every A ∈ ∆ , the map η A : A → C(Max A , [0 , is an injective δ -homomorphism.Proof. The first two items are Theorems 5.5 and 6.3, respectively, in [25]. The third itemfollows at once from the first two, along with Theorem 5. (cid:3)
Remark 8.
By item 2 in Theorem 7, if we compose the contravariant functor Max : ∆ → KH with the forgetful functor KH → Set , we get the functor hom ∆ ( − , [0 , → Set . Thelatter is representable, so it sends colimits to limits. Since the forgetful functor KH → Set
LUCA REGGIO reflects limits, Max : ∆ → KH sends colimits in ∆ to limits in KH . For instance, Maxsends epis to monos. In particular, if h : A ։ B is a surjective homomorphism in ∆, Max h identifies Max B with a closed subspace of Max A .3. The logic (cid:15) ∆ The purpose of this section is to introduce the infinitary equational logic (cid:15) ∆ and prove itsmain properties: the compactness and local deduction theorems, and some of their conse-quences. Every Birkhoff variety V of finitary algebras comes with an associated logic, namelythe equational consequence (cid:15) V . See, e.g., [26, Section 2]. This concept makes sense also forvarieties of infinitary algebras in the sense of S lomi´nski’s work [32]. We shall spell this outin the case of the variety ∆ introduced in Section 2.4.To improve readability, write L instead of L ∆ for the language of δ -algebras. Given a(possibly infinite) set of propositional variables x , T ( x ) denotes the algebra of L -terms over x .In other words, T ( x ) is the absolutely free L -algebra over x . In varieties of Birkhoff algebras,the algebra F ( x ) free over x can be constructed in a canonical way as a quotient ρ : T ( x ) ։ F ( x ) . This is true also for varieties of infinitary algebras, provided there is a cardinal κ such thatall operations have arity smaller than κ , cf. [32, Ch. III]. If A is an L -algebra and f : x → A is a function, also called an assignment of the variables x in A , we denote by e f : T ( x ) → A the unique L -homomorphism extending f .An L -equation in the variables x is a pair ( α, β ) ∈ T ( x ) × T ( x ) of L -terms; we shall usethe more suggestive notation α ≈ β for the equation ( α, β ). Arbitrary L -equations will bedenoted by ε , and sets of L -equations by Σ , Γ or Π. To emphasize that the variables of an L -term, L -equation, or set of L -equations, are contained in x , we write α ( x ) , ε ( x ), or Σ( x ).Further, to improve readability, we drop reference to the language L when speaking of L -terms, L -equations and L -homomorphisms. Given a set of equations Σ( x ), an L -algebra A and an assignment f : x → A , set A , f | = Σ if, and only if, Σ ⊆ ker e f . If A , f | = Σ, we say that Σ is satisfied in A with respect to the variable assignment f .Further, we write A | = Σ provided A , f | = Σ for all assignments f : x → A . For any set ofequations Σ( x ) ∪ { ε ( x ) } , defineΣ (cid:15) ∆ ε ⇐⇒ for every A ∈ ∆ and assignment f : x → A , A , f | = Σ entails A , f | = ε. Finally, given a set of equations Σ( x ) ∪ Γ( x ), the equational consequence relation (cid:15) ∆ is definedby Σ (cid:15) ∆ Γ if, and only if, Σ (cid:15) ∆ ε for every ε ∈ Γ.As with groups, every equation α ≈ β in the language of δ -algebras is equivalent to one ofthe form α ′ ≈
0. Just observe that, for any A ∈ ∆ and elements a, b ∈ A , a = b ⇐⇒ d( a, b ) = 0where d is the Chang’s distance, cf. equation (2). By applying the involution ¬ , we can alsotransform the equation α ≈ β into an equivalent one of the form α ′′ ≈
1. Furthermore, since
ETH DEFINABILITY AND THE STONE-WEIERSTRASS THEOREM 9 every member of ∆ embeds into a power of [0 ,
1] by item 3 in Theorem 7, it follows that(6) Σ (cid:15) ∆ ε ⇐⇒ for every assignment f : x → [0 , , [0 , , f | = Σ entails [0 , , f | = ε. In other words, the logic (cid:15) ∆ is complete with respect to the δ -algebra [0 ,
1] (algebraically:[0 ,
1] generates the variety ∆).An important observation is that the logic (cid:15) ∆ is compact because the corresponding maxi-mal spectra are topologically compact. We call a set of equations Σ( x ) a theory , and say thatΣ( x ) is consistent if there are a non-trivial algebra A ∈ ∆ (i.e., A has at least two distinctelements) and an assignment f : x → A such that A , f | = Σ. Recalling from Section 2.2 that A is non-trivial if, and only if, it admits a maximal ideal, we see that Σ is consistent preciselywhen there is an assignment f : x → [0 ,
1] satisfying [0 , , f | = Σ. Here and throughout thepaper, by an ideal of a δ -algebra we mean an ideal of the underlying MV-algebra. Lemma 9 (Compactness) . A theory Σ( x ) is consistent if, and only if, each of its finitesubsets is consistent.Proof. For the non-trivial direction, suppose that every finite subset of Σ( x ) is consistent.We can assume without loss of generality thatΣ( x ) = { α i ( x ) ≈ | i ∈ I } for a set of terms { α i | i ∈ I } . With the notation in (3), each term α i yields a closed subset V ( ρ ( α i )) of Max F ( x ), where ρ : T ( x ) ։ F ( x ) is the canonical quotient. We claim that { V ( ρ ( α i )) | i ∈ I } has the finite intersection property. For any finite subset I ⊆ I the theoryΣ = { α i ( x ) ≈ | i ∈ I } is consistent, whence there is an assignment f : x → [0 ,
1] satisfying [0 , , f | = Σ . Denotingby h : F ( x ) → [0 ,
1] the unique homomorphism extending f , we have h ( ρ ( α i )) = e f ( α i ) = 0 for every i ∈ I , i.e. h ∈ T { V ( ρ ( α i )) | i ∈ I } . We conclude that { V ( ρ ( α i )) | i ∈ I } has the finite intersectionproperty and so, as the space Max F ( x ) is compact, T { V ( ρ ( α i )) | i ∈ I } 6 = ∅ . Therefore,there is a homomorphism k : F ( x ) → [0 ,
1] such that k ( ρ ( α i )) = 0 for every i ∈ I . We get[0 , , k ◦ ρ ↾ x | = Σ, showing that the theory Σ( x ) is consistent. (cid:3) Remark 10.
The logic (cid:15) ∆ is compact but not strongly compact, i.e. it is not the case thatΣ (cid:15) ∆ ε entails Σ (cid:15) ∆ ε for some finite subset Σ ⊆ Σ. For instance, let Σ = { x i ≈ | i ∈ N } and ε = { δ ( x ) ≈ } , where x = x , x , x , . . . . By item (iii) in Definition 6, we have δ (0) = 0,hence Σ (cid:15) ∆ ε . However, it is not difficult to see that Σ (cid:15) ∆ ε for every finite subset Σ ⊆ Σ.To see this, let Σ be a finite subset of Σ and pick j ∈ N such that the equation x j ≈ . If x = { x i | i ∈ N } , the assignment f : x → [0 ,
1] which is 1 on x j and 0elsewhere satisfies [0 , , f | = Σ but [0 , , f = ε .Next, we prove a local deduction theorem for (cid:15) ∆ , analogous to the one for Lukasiewiczinfinite-valued logic. To this end, recall that the operation ⊙ admits an upper adjoint → , i.e. a ⊙ b c ⇔ a b → c . Explicitly, b → c = ¬ b ⊕ c . For every k ∈ N and term α , write kα := α ⊕ · · · ⊕ α | {z } k times and α k := α ⊙ · · · ⊙ α | {z } k times . An elementary computation shows that k ( ¬ α ) = ¬ ( α k ). Lemma 11 (Local Deduction Theorem) . Let Σ( x ) be a theory, and α ( x ) , β ( x ) two termssuch that Σ ∪ { α ≈ } (cid:15) ∆ β ≈ . Then there is k ∈ N such that Σ (cid:15) ∆ ( α k → β ) ≈ .Proof. The same proof as for Lukasiewicz logic, hinging on the ideal theory in MV-algebras,applies here mutatis mutandis. We recall the proof for the sake of completeness. Assumewithout loss of generality that Σ( x ) = { γ i ≈ | i ∈ I } for some set of terms { γ i ( x ) | i ∈ I } and let h : F ( x ) → [0 ,
1] be a homomorphism suchthat h ( ρ ( α )) = 1 and h ( ρ ( γ i )) = 1 for every i ∈ I , where ρ : T ( x ) ։ F ( x ) is the canonicalquotient. Then, [0 , , h ◦ ρ ↾ x | = Σ ∪ { α ≈ } and, because Σ ∪ { α ≈ } (cid:15) ∆ β ≈
1, we conclude that [0 , , h ◦ ρ ↾ x | = β ≈
1. This meansthat h ( ρ ( β )) = 1. Applying the involution ¬ , we obtain V ( {¬ ρ ( γ i ) | i ∈ I } ∪ {¬ ρ ( α ) } ) ⊆ V ( ¬ ρ ( β )) . In view of item 2 in Lemma 4, ¬ ρ ( β ) belongs to the ideal of F ( x ) generated by the set {¬ ρ ( γ i ) | i ∈ I } ∪ {¬ ρ ( α ) } . By equation (1), there are ϕ , . . . , ϕ n ∈ {¬ ρ ( γ i ) | i ∈ I } and k ∈ N such that ¬ ρ ( β ) ϕ ⊕ · · · ⊕ ϕ n ⊕ k ( ¬ ρ ( α ))= ϕ ⊕ · · · ⊕ ϕ n ⊕ ¬ ( ρ ( α ) k )= ρ ( α ) k → ( ϕ ⊕ · · · ⊕ ϕ n ) , which yields ρ ( α ) k ⊙ ¬ ρ ( β ) ϕ ⊕ · · · ⊕ ϕ n . So, ρ ( α ) k ⊙ ¬ ρ ( β ) belongs to the ideal of F ( x )generated by {¬ ρ ( γ i ) | i ∈ I } . We claim thatΣ (cid:15) ∆ ( α k → β ) ≈ . Let A ∈ ∆, and f : x → A an assignment satisfying A , f | = Σ. If g : F ( x ) → A is theunique homomorphism extending f , then g − (0) is an ideal of F ( x ) containing ¬ ρ ( γ i ) forevery i ∈ I . Since ρ ( α ) k ⊙ ¬ ρ ( β ) belongs to the ideal generated by {¬ ρ ( γ i ) | i ∈ I } , we get ρ ( α ) k ⊙ ¬ ρ ( β ) ∈ g − (0). Therefore, ρ ( α ) k → ρ ( β ) = ¬ ρ ( α ) k ⊕ ρ ( β ) = ¬ ( ρ ( α ) k ⊙ ¬ ρ ( β )) ∈ g − (1) . We conclude that e f ( α k → β ) = ( g ◦ ρ )( α k → β ) = g ( ρ ( α ) k → ρ ( β )) = 1 , i.e. A , f | = ( α k → β ) ≈
1. This settles the lemma. (cid:3)
The following proposition is a special case of a Robinson’s Joint Consistency Theorem forthe logic (cid:15) ∆ , where the two theories share the set of propositional variables. It will allow usto prove a useful interpolation result, namely Corollary 13. ETH DEFINABILITY AND THE STONE-WEIERSTRASS THEOREM 11
Proposition 12.
For any two theories Σ ( x ) , Σ ( x ) , the union Σ ∪ Σ is consistent if, andonly if, there is no term α ( x ) such that Σ (cid:15) ∆ α ≈ and Σ (cid:15) ∆ α ≈ .Proof. If either Σ or Σ are inconsistent, there is nothing to prove. Hence, assume Σ , Σ are consistent. Clearly, if the union Σ ∪ Σ is consistent, there is no term α ( x ) satisfyingΣ (cid:15) ∆ α ≈ (cid:15) ∆ α ≈
1. On the other hand, if Σ ∪ Σ is inconsistent, by Lemma 9there is a finite subset { ε , . . . , ε n } ⊆ Σ such that Σ ∪ { ε , . . . , ε n } is inconsistent. Assumingthat each equation ε i is of the form α i ≈ α i ( x ), define the term γ ( x ) := α ∧ · · · ∧ α n . Then, Σ ∪ { γ ≈ } (cid:15) ∆ ≈
1. By Lemma 11, there is k ∈ N such that Σ (cid:15) ∆ ( γ k → ≈ (cid:15) ∆ ¬ ( γ k ) ≈
1. Therefore, α := γ k satisfies Σ (cid:15) ∆ α ≈
0. Further, Σ (cid:15) ∆ γ ≈ (cid:15) ∆ α ≈ (cid:3) In order to prove the following corollary let us observe that, for every term α ( x ) ∈ T ( x )and real number r ∈ [0 , rα ∈ T ( x ) such that(7) ∀ h ∈ Max F ( x ) , \ ρ ( rα )( h ) = r · (cid:0) d ρ ( α )( h ) (cid:1) , where ρ : T ( x ) ։ F ( x ) is the canonical quotient and \ ρ ( rα ) , d ρ ( α ) : Max F ( x ) → [0 ,
1] are thecontinuous functions defined in (4). In other words, the multiplication by real scalars in [0 , δ -algebras. If α ( x ) is the constant 1, we write r instead of r rα , consider a binary expansion r ∈ { , } ω of r and let s = ( s i ) i ∈ N ∈ T ( x ) ω , where s i := ( α if r i = 10 otherwise.It follows from equation (5) that the term rα := δ ( s ) satisfies the desired property.Furthermore, note that with any subset J of F ( x ) we can associate a theory(8) Σ[ J ] := { α ≈ | α ( x ) ∈ ρ − ( J ) } . The theories of the form Σ[ J ], for J an ideal of F ( x ), will play a crucial rˆole in the following. Corollary 13.
Let A ∈ ∆ , C , . . . , C n pairwise disjoint closed subsets of Max A , and r , . . . , r n ∈ [0 , . There is an element a ∈ A satisfying b a ↾ C i = r i for every i ∈ { , . . . , n } .Proof. Let x be a set such that there is a surjective homomorphism f : F ( x ) ։ A . By Re-mark 8, the maximal spectrum Max A can be identified with a closed subspace of Max F ( x ).Whence, by item 3 in Lemma 4, each closed set C i is of the form V ( J i ) ∩ Max A for someideal J i of F ( x ). With the notation of (8), for each 1 i n we consider the theory Σ[ J i ].We have ∅ = C ∩ · · · ∩ C n = V ( J ) ∩ · · · ∩ V ( J n ) ∩ Max A = V ( h J ∪ · · · ∪ J n ∪ f − (0) i ) , i.e. h J ∪ · · · ∪ J n ∪ f − (0) i is the improper ideal of F ( x ). Hence, the theoryΣ[ J ] ∪ · · · ∪ Σ[ J n ] ∪ Γ is inconsistent, where Γ( x ) := ker( f ◦ ρ ). By Proposition 12, for each 1 i n there is aterm α i such that Σ[ J i ] ∪ Γ (cid:15) ∆ α i ≈ [ j = i Σ[ J j ] ∪ Γ (cid:15) ∆ α i ≈ . That is, [ ρ ( α i ) ↾ C i = 1 and [ ρ ( α i ) ↾ C j = 0 whenever j = i . If a ∈ A is the image of the term r α ∨ · · · ∨ r n α n under f ◦ ρ : T ( x ) ։ A , using the fact that C , . . . , C n are pairwise disjoint,we conclude that b a ↾ C i = r i for every 1 i n . (cid:3) The proof of the previous corollary relies on the observation that the multiplication byreal numbers in [0 ,
1] is definable in the language of δ -algebras. In particular, the elementsof [0 ,
1] are definable constants. This yields an elementary proof of the following fact:
Lemma 14.
For every set x there is a homeomorphism Max F ( x ) ∼ = [0 , x , where the Ty-chonoff cube [0 , x is equipped with the product topology.Proof. Since the functor Max sends coproducts in ∆ to products in KH by Remark 8, and F ( x ) is the coproduct of x copies of the algebra F ( x ) free on one generator, it suffices to showthat Max F ( x ) ∼ = [0 , ν : Max F ( x ) → [0 ,
1] be the function sending a homomorphism h : F ( x ) → [0 ,
1] to h ( ρ ( x )). The map ν is clearly a bijection. Further, for any ε, ε ′ ∈ [0 , ν − ([ ε, ε ′ ]) = { h ∈ Max F ( x ) | ε h ( ρ ( x )) ε ′ } = { h ∈ Max F ( x ) | ε ⊖ h ( ρ ( x )) = 0 = h ( ρ ( x )) ⊖ ε ′ } = { h ∈ Max F ( x ) | h ( ρ ( ε ) ⊖ ρ ( x )) = 0 = h ( ρ ( x ) ⊖ ρ ( ε ′ )) } = V ( ρ ( ε ) ⊖ ρ ( x )) ∩ V ( ρ ( x ) ⊖ ρ ( ε ′ ))which is a closed subset of Max F ( x ). Hence, ν is a continuous bijection. As every continuousbijection between compact Hausdorff spaces is a homeomorphism, the statement follows. (cid:3) By the previous lemma, along with item 3 in Lemma 4, we deduce that (up to homeomor-phism) the closed subsets of a Tychonoff space [0 , x are precisely those of the form V ( J )for some ideal J of F ( x ). Translating from ideals to theories, cf. equation (8), we see thatProposition 12 yields the following Urysohn’s Lemma for Tychonoff cubes: Given disjointclosed subsets C , C ⊆ [0 , x , there is a continuous function f : [0 , x → [0 ,
1] satisfying f ↾ C = 0 and f ↾ C = 1. Just observe that, for any two ideals J , J of F ( x ), V ( J ) ∩ V ( J ) = ∅ precisely when the theory Σ[ J ] ∪ Σ[ J ] is inconsistent.4. The Beth definability property
In this section we prove that the logic (cid:15) ∆ has the Beth definability property, assertingthat implicit definability is equivalent to explicit definability. We then derive the Stone-Weierstrass Theorem from the Beth definability property of (cid:15) ∆ . Definition 15.
Consider a set of variables x , a variable y not in x and a theory Σ( x, y ). Forany variable z , write Σ( x, z ) for the theory obtained from Σ( x, y ) by replacing y by z . Wesay that Σ implicitly defines y over x if, for every variable z ,Σ( x, y ) ∪ Σ( x, z ) (cid:15) ∆ y ≈ z. ETH DEFINABILITY AND THE STONE-WEIERSTRASS THEOREM 13
Further, Σ explicitly defines y over x if there is a term α y ( x ) such thatΣ( x, y ) (cid:15) ∆ y ≈ α y . The meaning of implicit definability is that every assignment f : x → A into an algebra A ∈ ∆ can be extended to at most one assignment g : x, y → A satisfying A , g | = Σ. On theother hand, an explicit definition α y of y witnesses the fact that the interpretation of y in amodel of Σ( x, y ) is completely determined by the interpretation of x . Clearly, if Σ explicitlydefines y over x , then it implicitly defines y over x . The Beth definability property statesthat the converse holds as well. Definition 16.
The logic (cid:15) ∆ has the Beth definability property if a theory Σ( x, y ) explicitlydefines y over x whenever it implicitly defines y over x .Throughout this section, in view of Lemma 14, we identify a Tychonoff cube [0 , x withthe maximal spectrum Max F ( x ). In fact, it follows from the proof of the latter lemma thatthe map Max F ( x ) → [0 , x sending a homomorphism h : F ( x ) → [0 ,
1] to h ◦ ρ ↾ x : x → [0 , x , a closed subset X ⊆ [0 , x and a continuous function f : X → [0 , y is a variable not in x , the graph of f can be identified with a closed subset of [0 , x,y ∼ = Max F ( x, y ). Hence, by Lemma 4, it ishomeomorphic to V ( J f ) for some ideal J f ⊆ F ( x, y ). Define the theory(9) Σ f ( x, y ) := { α ≈ | α ( x, y ) ∈ ρ − ( J f ) } , i.e. Σ f = Σ[ J f ] with the notation in (8). Note that an assignment g : x, y → [0 ,
1] satisfies[0 , , g | = Σ f if, and only if, it lies on the graph of f when regarded as a point of [0 , x,y .Thus, Σ f implicitly defines y over x because the graph of f is a functional relation: Lemma 17.
The theory Σ f implicitly defines y over x .Proof. Consider an assignment f : x → A with A ∈ ∆, and assume g : x, y → A is anassignment extending f and satisfying A , g | = Σ f . We show that g is the only assignment ofthe variables x, y with these properties.If A is the trivial algebra, this is clearly true. Hence, let us suppose A is non-trivial. Wecan assume without loss of generality that A = [0 , g ′ : x, y → [0 ,
1] is another assignmentextending f and satisfying [0 , , g ′ | = Σ f , and π : [0 , x,y ։ [0 , x is the projection map, weget π ( g ) = π ( g ′ ) because g and g ′ both extend f . Therefore, since g and g ′ belong to thegraph of f , which is a functional relation, it must be g = g ′ . (cid:3) By definition, the theory Σ f ( x, y ) explicitly defines y over x if there is a term α y ( x ) suchthat Σ f (cid:15) ∆ y ≈ α y . By equation (6), this is equivalent to saying that, for every assignment g : x, y → [0 , , , g | = Σ f entails [0 , , g | = y ≈ α y . We already observed that [0 , , g | = Σ f if, and only if, g belongs to the graph of f . In turn,if h : F ( x, y ) → [0 ,
1] is the unique homomorphism extending g , π : [0 , x,y ։ [0 , x is the projection on the x -coordinates, and π y : [0 , x,y ։ [0 ,
1] is the projection on the y -coordinate,[0 , , g | = y ≈ α y ⇐⇒ h ( ρ ( y )) = h ( ρ ( α y )) ⇐⇒ d ρ ( y )( g ) = \ ρ ( α y )( π ( g )) ⇐⇒ π y ( g ) = \ ρ ( α y )( π ( g ))because d ρ ( y ) : [0 , x,y → [0 ,
1] coincides with π y . Whence,(10) Σ f explicitly defines y over x ⇐⇒ ∃ β ∈ F ( x ) such that b β ↾ X = f . Note that, by the previous discussion, for the left-to-right direction we can take β = ρ ( α y ).Then, for all assignments g ∈ [0 , x,y lying on the graph of f , π y ( g ) = f ( π ( g )). Hence, π y ( g ) = \ ρ ( α y )( π ( g )) for all assignments g with [0 , , g | = Σ f ⇐⇒ f ( w ) = \ ρ ( α y )( w ) ∀ w ∈ X. Remark 18.
By considering all theories of the form Σ f , the right-hand condition in (10)implies the following form of the Tietze-Urysohn Extension Theorem: Every continuousfunction f : X → [0 ,
1] defined on a closed subset X of a Tychonoff space [0 , x can beextended to a continuous function on [0 , x .In Theorem 20, we will see that the existence of explicit definitions of the type (10) isenough to conclude that all implicit definitions can be made explicit, i.e. that (cid:15) ∆ has theBeth definability property. We start by proving the following useful fact: Lemma 19.
For any δ -algebra A , η A : A → C(Max A , [0 , is an epimorphism in ∆ .Proof. Consider distinct homomorphisms h , h : C(Max A , [0 , ⇒ B , for some B in ∆.We must prove that h ◦ η A = h ◦ η A . The map η B is injective by item 3 in Theorem 7,whence the latter condition is equivalent to η B ◦ h ◦ η A = η B ◦ h ◦ η A . A C(Max A , [0 , B , [0 , η A η B ◦ h η B ◦ h By item 1 in Theorem 5, the functor C : KH op → MV is full, therefore there are continuousfunctions f , f : Max B ⇒ Max A satisfying C f = η B ◦ h and C f = η B ◦ h . Since η B ◦ h = η B ◦ h , there exists h ∈ C(Max A , [0 , h ◦ f = C f ( h ) = C f ( h ) = h ◦ f . Let x ∈ Max B be such that h ( f ( x )) = h ( f ( x )). It is enough to find a ∈ A satisfying b a ( f ( x )) = h ( f ( x )) and b a ( f ( x )) = h ( f ( x )), for then we will have(( η B ◦ h ◦ η A )( a ))( x ) = (C f ( b a ))( x ) = b a ( f ( x )) = h ( f ( x ))and similarly (( η B ◦ h ◦ η A )( a ))( x ) = h ( f ( x )), showing that η B ◦ h ◦ η A = η B ◦ h ◦ η A . Theexistence of such a ∈ A follows from Corollary 13 and the fact that f ( x ) = f ( x ), by setting C = { f ( x ) } , C = { f ( x ) } , r = h ( f ( x )) and r = h ( f ( x )). (cid:3) Theorem 20.
The following statements are equivalent:(1) (cid:15) ∆ has the Beth definability property;(2) for every set x and continuous function f : X → [0 , defined on a closed subset X ⊆ [0 , x , the theory Σ f ( x, y ) from equation (9) explicitly defines y in terms of x ; ETH DEFINABILITY AND THE STONE-WEIERSTRASS THEOREM 15 (3) for every A ∈ ∆ , the homomorphism η A : A → C(Max A , [0 , is an isomorphism;(4) every epimorphism in ∆ is surjective.Proof. ⇒
2. This is an immediate consequence of Lemma 17.2 ⇒
3. Since η A : A → C(Max A , [0 , f : Max A → [0 ,
1] andlet x be a set such that there exists a surjective homomorphism p : F ( x ) ։ A . By Remark 8,the space Max A can be identified with a closed subspace of Max F ( x ) ∼ = [0 , x . If the theoryΣ f ( x, y ) explicitly defines y in terms of x , in view of equation (10) there is β ∈ F ( x ) suchthat b β ↾ Max A = f . But b β ↾ Max A = d p ( β ) = η A ( p ( β )) , thus η A is surjective.3 ⇒
4. It is enough to show that every homomorphism in ∆ which is both epi and monois an isomorphism. For any homomorphism h : A → B in ∆, by naturality of η , there is acommutative diagram as follows. A C(Max A , [0 , B C(Max B , [0 , h η A C(Max h ) η B If h is epi, then Max h : Max B → Max A is injective by Remark 8. We claim that Max h issurjective provided h is mono, i.e. injective.Suppose h is mono and identify A with a subalgebra of B . By the congruence extensionproperty for MV-algebras, see e.g. [12, Proposition 8.2], for any homomorphism k : A → [0 , k − (0) of A generates a proper ideal of B . The latter canthen be extended to a maximal ideal m of B by Zorn’s Lemma. Composing the quotient map B ։ B / m with the unique injective homomorphism h m : B / m ֒ → [0 ,
1] provided by H¨older’sTheorem, we get a homomorphism B → [0 ,
1] which extends k . This shows that Max h issurjective whenever h is mono.Therefore, if h is both epi and mono, Max h is a continuous bijection between compactHausdorff spaces, hence a homeomorphism. We conclude that C(Max h ) is an isomorphism in∆. Since the square above commutes and η A , η B are isomorphisms, h is also an isomorphism.4 ⇒
1. The following argument, essentially due to Makkai [24, Section 1], exploits thefact that the category KH is regular. Suppose that a theory Σ( x, y ), with y a variablenot in x , implicitly defines y over x . Let z be a variable which is distinct from y and notcontained in x , and consider the following diagram in ∆ (we write e.g. F ( x, y ) / Σ( x, y ) for thequotient of F ( x, y ) with respect to the congruence generated by the image of Σ( x, y ) under A category is regular if it has finite limits, every morphism factors as a regular epi followed by a mono,and regular epis are stable under pullbacks. See [2, 8]. In KH , the (regular epi, mono) factorisation of acontinuous map is the usual factorisation through its set-theoretic image endowed with the subspace topology. ρ × ρ : T ( x, y ) → F ( x, y ) ) F ( x ) F ( x, y ) / Σ( x, y ) F ( x, y ) / Σ( x, y ) F ( x, y, z ) / Σ( x, y ) ∪ Σ( x, z ) gg h h where • g is the composition of F ( x ) ֒ → F ( x, y ) with the quotient F ( x, y ) ։ F ( x, y ) / Σ( x, y ), • h is the composition of the inclusion F ( x, y ) / Σ( x, y ) ֒ → F ( x, y, z ) / Σ( x, y ) with the quo-tient map F ( x, y, z ) / Σ( x, y ) ։ F ( x, y, z ) / Σ( x, y ) ∪ Σ( x, z ), • h is obtained by first applying the isomorphism F ( x, y ) / Σ( x, y ) → F ( x, z ) / Σ( x, z ), whichreplaces y by z , then the inclusion F ( x, z ) / Σ( x, z ) ֒ → F ( x, y, z ) / Σ( x, z ), and finally thequotient F ( x, y, z ) / Σ( x, z ) ։ F ( x, y, z ) / Σ( x, y ) ∪ Σ( x, z ).It is not difficult to see that the diagram above is a pushout square in ∆. Now, consider thefollowing equaliser diagram in the category ∆. B F ( x, y ) / Σ( x, y ) F ( x, y, z ) / Σ( x, y ) ∪ Σ( x, z ) i h h Since h ◦ g = h ◦ g , by the universal property of B there is a homomorphism j : F ( x ) → B such that g = i ◦ j . Claim. ( j, i ) is the (epi, regular mono) factorisation of g : F ( x ) → F ( x, y ) / Σ( x, y ) .Proof of Claim. If all epimorphisms in ∆ are surjective, for every A ∈ ∆ the embedding η A : A → C(Max A , [0 , ⊣ C : KH op → ∆ yields an equivalence ∆ ∼ = KH op . Let f : X → Y be the continuousmap in KH dual to g . Since KH is a regular category, the (regular epi, mono) factorisationof f is ( e, m ), where e : X ։ Z is the coequaliser of the kernel pair of f , and m : Z ֒ → Y isthe unique morphism provided by the universal property of Z . See e.g. [2, p. 7] for a proof.Recall that the kernel pair of f is obtained by taking the pullback of f along itself. Thus, byconstruction, the dual of e is i and the dual of m is j . We conclude that ( j, i ) is the (epi,regular mono) factorisation of g . (cid:3) Since all epimorphisms in ∆ are surjective, j must be a surjection. Because Σ( x, y ) im-plicitly defines y in terms of x , we haveΣ( x, y ) ∪ Σ( x, z ) (cid:15) ∆ y ≈ z. Thus, the homomorphisms h and h coincide on the equivalence class of y in F ( x, y ) / Σ( x, y ).Let γ denote this equivalence class. Then γ ∈ B and, by surjectivity of j , there is β ∈ F ( x )such that j ( β ) = γ . If α y is any element of T ( x ) whose image under ρ : T ( x ) ։ F ( x ) is β ,we get g ( ρ ( α y )) = i ( j ( β )) = γ . Whence, Σ( x, y ) (cid:15) ∆ y ≈ α y as was to be proved. (cid:3) Remark 21.
The equivalence between items 1 and 4 in Theorem 20 is known, in the frame-work of abstract algebraic logic, as the Blok-Hoogland Theorem [7]. In the particular caseof the equational consequence (cid:15) V associated with a Birkhoff variety V , the Blok-HooglandTheorem states that all epimorphisms in V are surjective if, and only if, (cid:15) V has the so-called ETH DEFINABILITY AND THE STONE-WEIERSTRASS THEOREM 17 infinite Beth property . This result does not apply in our setting because ∆ is not a Birkhoffvariety of algebras. However, a lengthy but rather straightforward verification shows thatthe Blok-Hoogland Theorem can be generalised to all varieties of infinitary algebras in thesense of S lomi´nski [32]. Here we have opted for a more direct proof, specific to the variety∆, which emphasises the rˆole of the theories Σ f .We are now in a position to prove that (cid:15) ∆ satisfies the Beth definability property. Theorem 22.
The logic (cid:15) ∆ has the Beth definability property.Proof. Let x be an arbitrary set, X ⊆ [0 , x a closed subset and f : X → [0 ,
1] a continuousfunction. In view of Theorem 20, it suffices to show that the theory Σ f ( x, y ) explicitly defines y in terms of x .By item 3 in Lemma 4, there is an ideal J of F ( x ) such that X ∼ = V ( J ). Let A ∈ ∆ be F ( x ) /J , with quotient map p : F ( x ) ։ A . We have Max A ∼ = X , and so A can be identifiedwith a subalgebra of C( X, [0 , η A : A ֒ → C(Max A , [0 , A , [0 , ∼ = C( X, [0 , u, v : x → [0 ,
1] which belong to X , there is a u,v ∈ A such that d a u,v ( u ) = f ( u ) and d a u,v ( v ) = f ( v ) . For each a u,v , pick a term α u,v ∈ T ( x ) whose image under the composition p ◦ ρ : T ( x ) → A is a u,v . Fix an arbitrary ε ∈ (0 ,
1] and define the theoryΓ u,v ( x, y ) := Σ f ( x, y ) ∪ { ( α u,v ⊖ y ) ∧ ε ≈ ε } , where ⊖ is the truncated subtraction (see Section 2.1) and ε is the definable constant corre-sponding to ε , cf. (7). We claim that, for all u ∈ X , the theory S v ∈ X Γ u,v is inconsistent.Assume towards a contradiction that S v ∈ X Γ u,v is consistent. Then, by (6), there is anassignment f : x, y → [0 ,
1] satisfying [0 , , f | = Γ u,v for every v ∈ X . If v ′ : x → [0 ,
1] is therestriction of f to x , note that v ′ ∈ X because [0 , , f | = Σ f . Thus,[0 , , f | = Γ u,v ′ = ⇒ [0 , , f | = Σ f ( x, y ) ∪ { ( α u,v ′ ⊖ y ) ∧ ε ≈ ε } = ⇒ [0 , , f | = { y ≈ f ( v ′ ) } ∪ { ( α u,v ′ ⊖ y ) ∧ ε ≈ ε } = ⇒ [0 , , f | = ( α u,v ′ ⊖ f ( v ′ )) ∧ ε ≈ ε, contradicting the fact that d a u,v ′ ( v ′ ) = f ( v ′ ), i.e. d a u,v ′ ( v ′ ) ⊖ f ( v ′ ) = 0.By compactness of (cid:15) ∆ (Lemma 9), there are v , . . . , v m ∈ X such that S mi =1 Γ u,v i is incon-sistent. That is, for every w ∈ X there is 1 i m such that d a u,v i ( w ) ⊖ f ( w ) < ε , whence d a u,v i ( w ) < f ( w ) + ε . Therefore, the term λ u := α u,v ∧ · · · ∧ α u,v m satisfies ∀ w ∈ X, [ ρ ( λ u )( w ) < f ( w ) + ε. Now, for any u ∈ X , define the theoryΓ ′ u ( x, y ) := Σ f ( x, y ) ∪ { ( y ⊖ λ u ) ∧ ε ≈ ε } . Reasoning as before, it is not difficult to see that the theory S u ∈ X Γ ′ u is inconsistent. Bycompactness of (cid:15) ∆ , there are u , . . . , u n ∈ X such that, for every w ∈ X , f ( w ) ⊖ \ ρ ( λ u j )( w ) < ε for some 1 j n , whence \ ρ ( λ u j )( w ) > f ( w ) − ε . The term µ := λ u ∨ · · · ∨ λ u n satisfies(11) ∀ w ∈ X, f ( w ) − ε < d ρ ( µ )( w ) < f ( w ) + ε. Since ε ∈ (0 ,
1] was arbitrary, equation (11) entails that f belongs to the closure of A in thetopology induced by the uniform metric of C( X, [0 , f ∈ A . (The followingargument is already implicit in [25]; we briefly recall it for the sake of completeness.)Suppose f is the uniform limit of a sequence ( f i ) i ∈ N ∈ A ω . We can assume without lossof generality that this sequence is increasing, cf. the proof of [25, Lemma 7.5]. Since themultiplication by any real number in [0 ,
1] is definable in the language of δ -algebras, we seethat ( f i ) i ∈ N ∈ A ω . Extract a subsequence ( s i ) i ∈ N of ( f i ) i ∈ N satisfying, for every i ∈ N ,sup u ′ ∈ X {| s i ( u ′ ) − s i − ( u ′ ) |} i . Then an elementary computation shows that f = δ (2 s , ( s ⊖ s ) , . . . , i ( s i ⊖ s i − ) , . . . ) ∈ A .For a proof, see [25, Lemma 7.6]. We conclude that f = f ⊕ f ∈ A .To settle the theorem, pick β ∈ F ( x ) such that p ( β ) = f , where p : F ( x ) ։ A is thequotient map. Then, b β ↾ X = f . By equation (10), Σ f explicitly defines y over x . (cid:3) To conclude, we show how to derive the Stone-Weierstrass Theorem for compact Hausdorffspaces from the Beth definability property of (cid:15) ∆ . Proof of Theorem 1.
Suppose X is a non-empty compact Hausdorff space. If G ⊆ C( X, R )satisfies the assumptions in the statement of Theorem 1, then it is a unital ℓ -subgroup ofC( X, R ) which is divisible (as an Abelian group) and separates the points of X . Write G forthe closure of G in the topology induced by the uniform metric and observe that this is againa divisible unital ℓ -subgroup of C( X, R ). We must prove that G = C( X, R ). By Theorem 2,it suffices to show that the inclusion Γ( G ) ֒ → C( X, [0 , G ) is a δ -algebra and it separates the points of X .The Abelian ℓ -group G separates the points of X and is divisible, hence its unit intervalseparates the points of X . A fortiori, the MV-subalgebra Γ( G ) of C( X, [0 , X . Next, we show that Γ( G ) is a δ -subalgebra of C( X, [0 , δ in C( X, [0 , f ∈ Γ( G ) ω . In thealgebra C( X, [0 , δ ( f ) = ∞ X i =1 f i i = lim n →∞ n X i =1 f i i where the latter limit is uniform. Since G is divisible and closed under uniform limits, δ ( f ) ∈ G . Because each f i belongs to the unit interval of G , so does δ ( f ).To improve readability, write A for the δ -algebra Γ( G ). Since A separates the pointsof X we have Max A ∼ = X , cf. [29, Theorem 4.16], so the inclusion A ֒ → C( X, [0 , η A : A ֒ → C(Max A , [0 , A , [0 , ∼ = C( X, [0 , (cid:15) ∆ has the Beth definabilityproperty. It follows by Theorem 20 that the homomorphism η A is an isomorphism, thereforethe inclusion A ֒ → C( X, [0 , (cid:3) Remark 23.
The previous proof exploits the Beth definability property of (cid:15) ∆ to derive theStone-Weierstrass Theorem. In turn, Theorem 8.1 in [25] shows that an application of anappropriate version of the Stone-Weierstrass Theorem yields item 3 in Theorem 20, whence ETH DEFINABILITY AND THE STONE-WEIERSTRASS THEOREM 19 the Beth definability property of (cid:15) ∆ . In this sense, the Beth definability property of (cid:15) ∆ isequivalent to the Stone-Weierstrass Theorem for compact Hausdorff spaces.5. Conclusion
In [34, p. 467], Stone observes that the proof of Weierstrass’ Approximation Theorem canbe divided into two parts. The first part, of which he provides a generalisation, can beregarded, in his words, as “algebraico-topological”, while he refers to the second part as the“analytical kernel” of the proof. On the other hand, Banaschewski showed in [1] that eventhis analytical kernel can be proved by algebraic means in the setting of f -rings. In a sense,Banaschewski fully brought out the algebraic content of the Stone-Weierstrass Theorem, thusconcluding a process started by Stone himself.In a similar vein, we exposed the logical content of the Stone-Weierstrass Theorem byrelating it to the Beth definability property. The reader will recognise in the first part of theproof of Theorem 22 a standard argument used to prove the Stone-Weierstrass Theorem: thisbrings out those classical analytical arguments which admit a purely logical reformulation.As observed in Remark 23, the Beth definability property of the logic (cid:15) ∆ can be ultimatelyregarded as an equivalent form of the Stone-Weierstrass Theorem for compact Hausdorffspaces. Moreover, we saw at the end of Section 3 that a special case of a Robinson’s JointConsistency Theorem for the logic (cid:15) ∆ (Proposition 12) yields a Urysohn’s Lemma for Ty-chonoff cubes. Also, it is not difficult to see that the fact that any continuous map betweencompact Hausdorff spaces is closed implies the deductive interpolation property of (cid:15) ∆ (infact, even the right uniform deductive interpolation property [14]).These observations relate properties of the equational logic (cid:15) ∆ and basic results of func-tional analysis. We leave for future research a systematic study of this logic and its topologicalcounterpart, as well as a possible extension of these results beyond compact Hausdorff spaces. Acknowledgements.
I am grateful to Nick Bezhanishvili for several useful comments onan earlier draft of this article.
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