Sheaf representations and locality of Riesz spaces with order unit

We present an algebraic study of Riesz spaces (ie, real vector lattices) with a (strong) order unit. We exploit a categorical equivalence between those structures and a variety of algebras called RMV–algebras. We prove two different sheaf representations for Riesz spaces with order unit: the first represents them as sheaves of linearly ordered Riesz spaces over a spectral space, the second represents them as sheaves of local Riesz spaces over a compact Hausdorff space. Motivated by the latter representation we study the class of local RMV–algebras. We study the algebraic properties of local RMV–algebra and provide a characterisation of them as special retracts of the real interval [0, 1]. Finally, we prove that the category of local RMV–algebras is equivalent to the category of all Riesz spaces. 2010 Mathematics Subject Classification 00Z99 (primary); 06D35 (secondary)


Introduction
The algebra of real-valued continuous functions C(X), for X a compact and Hausdorff space, has received great attention in all of its facets (see, eg, Gillman and Jerison [25] and references therein). Indeed, the rich structure of R lifts to C(X) in several ways, emanating a number of well known structures such as C * -algebras, Banach algebras, vector lattices, etc. In several of those cases the concept of norm is of crucial importance, yet norms elude the classical tools of universal algebra or first order logic.
A purely algebraic and basic structure that can be pulled back on C(X) is the structure of lattice ordered abelian group (for short -group). Since X is compact, by Weierstrass' Extreme Value Theorem the functions in C(X) are automatically bounded. This entails that any positively constant function u has the following property: for every f ∈ C(X) there exists an n ∈ N such that (n)u := n times u + · · · + u > f If G is an -group, an element u with the property described in Equation (1) is called order unit (or strong order unit). One of the reasons for which the concept of order unit is important is that it allows to define a semi-norm: if G is an -group and g ∈ G, g := inf p q ∈ Q | p, q ∈ N, q = 0 and q times |g| + · · · + |g|≤ p times u + · · · + u (2) where |g| := g ∨ −g. The operator is a norm if and only if G is Archimedean. In particular, if G = C(X) then is the uniform norm (also called sup norm). So, a single element of the algebra allows us to define the norm, but it can be readily seen by a compactness argument that order units are not first-order definable. Nevertheless, as next result shows, this obstruction is only due to the presentation of the structures, so a mere modification of the language dramatically simplifies the situation. Theorem 1.1 (Mundici [32,Theorem 3.9]) The category of -groups with order unit (u -groups, for short) and morphisms preserving the units is equivalent to a class of equationally defined algebras, called MV-algebras.
The importance of Theorem 1.1 cannot be overestimated, in that it allows the study of normed structures with the full paraphernalia of universal algebra.
Recall that a Riesz space is a vector space over the real numbers endowed with an order ≤ that is compatible with the vector space operations (ie, such that x ≤ y implies x + z ≤ y + z and λx ≤ λx) and ≤ is a lattice order (ie, every finite subset has a least upper bound and greatest lower bound). The real numbers with their usual operations and order form a Riesz space, also any L p -space with the almost everywhere point-wise partial order is a Riesz space. A classical reference for Riesz spaces is Luxemburg and Zaanen [29]. They are important structures in measure theory, but they also have shown to be of interest in economics (see eg Aliprantis and Burkinshaw [1]).
In Di Nola and Leuştean [18] it was noticed for the first time 1 that the equivalence of Theorem 1.1 can be extended to an equivalence between the category of Riesz spaces with order unit and the category of certain equationally-definable algebras called Riesz MV-algebras (RMV-algebras, for short).
In this work we exploit this equivalence to propose a universal algebraic study of these structures. This approach can be framed in a long list of successful attempts to use the tools of logic in functional analysis as for instance, the theory of approximate truth of The plan of the paper is as follows. In Section 2 we recall definitions and basic properties of MV-algebras and RMV-algebras that will be used in the rest of the paper. In Section 3 we give two sheaf representations of RMV-algebras: Corollary 3.11 represents them as sheaves of local RMV-algebras over a compact Hausdorff space; Corollary 3.13 represents them as sheaves of linearly ordered RMV-algebras over a spectral space. In Section 4 we study local RMV-algebras and prove the characterisations summarised below.
Theorem 1.2 Let A be an RMV-algebra. The following are equivalent: (i) A is local, (ii) A is isomorphic to an algebra of quasi-constant functions, (iii) A is generated by its radical, (iv) A/ Rad(A) ∼ = [0, 1], (v) A is isomorphic to Γ R (R − → × W, (1, 0)), for some Riesz space W , where − → × indicates the lexicographic product.
Proof The equivalence of (i) and (ii) is proved in Theorem 4.10. The equivalence of (i) and (iii) is proved in Theorem 4.14. The equivalence of (i) and (iv) is proved in Theorem 4.15. The equivalence of (i) and (v) is proved in Theorem 4.23.
Finally, in Section 4.4 we prove a categorial equivalence between local RMV-algebras and the full category of Riesz spaces-with or without order unit.

Preliminaries
We briefly recall the definition and some basic properties of MV-algebras needed in the paper, the standard references are Mundici [10] and Mundici [33].
Definition 2.1 An MV-algebra is a structure A, ⊕, * , 0 such that: A, ⊕, 0 is a commutative monoid (MV 1) Sheaf, locality and Riesz spaces Proof Since all properties from (1) to (9) are quasi-equations, by Theorem 2.4 it is enough to check that they hold in [0, 1]. We leave to the reader this easy exercise.
We now give the definition of Riesz MV-algebras (RMV-algebras, for short); they are MV-algebras endowed with a scalar multiplication by elements of the real interval [0, 1]. RMV-algebras were introduced in Di Nola and Leuştean [18] and further studied in Di Nola, Lapenta, and Leuştean [14,15]. Definition 2.6 (RMV-algebra) A Riesz MV-algebra (RMV-algebra for short) is an MV-algebra A endowed with an external multiplication f r for every real number r in [0, 1], satisfying the following conditions. For every x, y ∈ A and every r, s ∈ [0, 1]: where r · s indicates the product in [0, 1] and r s := max{0, r − s}. An RMV-algebra is said to be trivial if it satisfies 0 = 1.
Note that the MV-algebra [0, 1] of Example 2.3 is also an RMV-algebra where f r (x) = rx is the usual multiplication. Similarly to the case of MV-algebras: The only endomorphism of the RMV-algebra on [0, 1] is the identity.
Finally, if r is any real number in [0, 1] and m/n ≤ r ≤ p/q then, by Lemma 2.8(4), f m/n (x) ≤ f r (x) ≤ f p/q (x), so mx/n ≤ f r (x) ≤ px/q. Letting m/n and p/q tend to r, we have f r (x) = rx.
In the light of the previous result we will freely refer to [0, 1] as an RMV-algebra, for there is no confusion on the RMV-operations defined on it.
Theorem 2.11 Every non-trivial RMV-algebra A has a unique RMV-subalgebra isomorphic to [0, 1], given by the elements of the form f r (1). This subalgebra will henceforth be indicated by R(A).
Proof We start by noticing that in every RMV-algebra A, the subalgebra generated by the empty set is given by: Indeed, since 1 is a constant and f r are operations in the language of RMV-algebras, it is clear that those elements belong to the subalgebra of A generated by the empty set. The set R(A) is also a subalgebra of A; indeed, by Remark 2.2, it is enough to check that it is closed under the operations and f r for r ∈ [0, 1]: if s, t ∈ [0, 1], then f s (1) f t (1) = f s t (1) by (RMV 2) and f s (f t (1)) = f s·t (1) by (RMV 4). This ensures that R(A) is closed under RMV-operations.
Let us now prove that the map that sends f r (1) into r ∈ [0, 1] is well-defined. Let r = s be real numbers in [0, 1], then either s < r or r < s. Assuming without loss of generality that the first inequality holds we obtain r s = 0. The axiom RMV 2 gives f r (1) f s (1) = f r s (1). We claim that the right-hand side of last equation is different from 0. Indeed, suppose by way of contradiction that f r s (1) = 0. Since r s = 0 there exists m ∈ N such that 1/m < r s, and by Lemma 2.8(4) this entails f 1 m (1) = 0. Whence, by Lemma 2.9, f 1 (1) = 0 against the fact that A is non-trivial. So, f r (1) f s (1) = 0, therefore by (the contrapositive of) Proposition 2.5(1), we obtain that f r (1) = f s (1). It is straightforward to check that this map is an isomorphism of RMV-algebras from R(A) into [0, 1].
Suppose now that B is another subalgebra of A isomorphic to [0, 1]. Obviously R(A) is the smallest RMV-subalgebra of A, so R(A) ⊆ B. By way of contradiction, suppose there exists a ∈ B \ R(A). Let g : R(A) → [0, 1] and h : B → [0, 1] be the two given isomorphisms. Set b := g −1 (h(a)) ∈ R(A) ⊆ B, then in particular a = b and both belong to B. Now observe that h(a) = f h(a) (1) and thus, by Lemma 2.10, As h is injective, a = b, which contradicts our hypothesis. So a cannot exist and R(A) = B.
Our interest for RMV-algebras stems from the fact that the equivalence of Theorem 1.1 can be extended to an equivalence between RMV-algebras and Riesz spaces with order unit.

Theorem 2.12
There is a functor Γ R from the category of Riesz spaces with order unit to the category of RMV-algebras which is full, faithful and dense, hence it witnesses an equivalence of the aforementioned categories.
This result was originally stated in Di Nola and Leuştean [18]. However, the proof published there is incomplete because it relies on [18,Corollary 2], which is false as in turn it relies on the false Birkhoff [5,Corollary,page 349]. (For a counter-example see Di Nola, Lenzi, Marra, and Spada [16].) We remedy this gap by giving a full proof here.
Proof of Theorem 2.12 The functor Γ R is defined as follows. If R, +, −, ·, ∧, ∨, 0, u is a Riesz space with order unit, Γ R (R, u) is defined as the RMV-algebra whose elements are in [0, u] := {r ∈ R | 0 ≤ r ≤ u} and the operations are defined as: The verification that Γ R (R, u) satisfies axioms (RMV 1)-(RMV 4) is routine. If h : R → S is a unit preserving homomorphism of Riesz spaces, . Faithfulness: Let us prove that the functor is faithful, ie it is injective on morphisms. Consider two unit preserving morphisms of Riesz spaces f , f : (R, u) → (S, v) which coincide on [0, u] and let x be an arbitrary element of R. Since any element of R is the difference of two positive elements, there is no loss of generality is assuming that x ≥ 0. Then for some integer n we have 0 ≤ x ≤ nu.
We conclude that, f and f coincide on R. Fullness: Let us prove that the functor is full, ie it is surjective on morphisms. Let µ be a homomorphism of RMV-algebras, then µ is also a morphism of MV-algebras and, by Mundici [32,Proposition 3.5], it can be lifted to a morphism λ of the corresponding u -groups (R, u) and (S, v). We prove that µ is also a Riesz space homomorphism preserving the order unit. Let us write R + for the set {x ∈ R | x > 0} and similarly for R. We preliminary check that the property holds for positive elements, ie if x ∈ R + and r ∈ R + , then λ(rx) = r(λx). Indeed, under this assumption x = x 1 + . . . + x n , for suitable x i ∈ [0, u] and r = r 1 + 1 + . . . + 1, with r 1 ∈ [0, 1]. Recall that λ extends µ, and the latter is an RMV-homomorphism. Hence, λ(r 1 x i ) = r 1 λ(x i ) for every i ≤ n. From this, by distributivity of scalar multiplication in Riesz spaces, it follows λ(rx) = r(λx). Now we can check the general case. Let x ∈ R, r ∈ R. Then x = x 1 − x 2 where x 1 , x 2 ∈ R + ; r = r 1 − r 2 , where r 1 , r 2 ∈ R + ; and: .
Density: Finally, we prove that the functor is dense, ie, any RMV-algebra is isomorphic to an algebra in the range of Γ R . Let A be a RMV-algebra and let A be its MValgebraic reduct. Since the functor Γ of Theorem 1.1, from u -groups into MV-algebras, is an equivalence of categories, there is a u -group (R 0 , u) such that A = Γ(R 0 , u). We extend R 0 to a Riesz space R such that Γ R (R, u) = A. If s ∈ R, let z ∈ Z and r ∈ [0, 1) be such that s = z + r, we define, for any a ∈ A ⊆ R 0 : s · a := z times a + · · · + a +f r (a) By Mundici [10, Chapter 7] every element of R 0 is a finite sum of elements of A and their opposites, hence the scalar multiplication defined above extends to all R 0 . We leave to the reader the routine checking that R 0 endowed with this scalar multiplication is a Riesz space. This concludes the proof.

Definition 2.13 (MV and RMV ideals) Let
A be an RMV-algebra. A non-empty subset I of A is called MV-ideal if I is downward closed (ie, x ≤ y and y ∈ I imply x ∈ I ) and it is closed under ⊕. A non-empty subset J of A is called RMV-ideal if it is an MV-ideal and it is closed under the operations f r , ie, for every r Lemma 2.14 For any RMV-algebra A, there is an isomorphism from the lattice of MV-congruences of A and the lattice of its MV-ideals. Similarly, there is a lattice isomorphism between RMV-congruences of A and its RMV-ideals.
Proof Both isomorphisms are defined in the same way: if J is an MV-or RMV-ideal and θ is an MV-or RMV-congruence of A then: The fact that they are isomorphisms is well-known in the case of MV-algebras (see eg [10, Proposition 1.2.6]); the case of RMV-algebra is a straightforward adaptation of the argument for MV-algebras. See also Di Nola, Lenzi, Marra, and Spada [16] for a general proof that has both MV and RMV as special cases.
Proof By definition, every RMV-ideal is an MV-ideal. Vice versa, let A be an RMV-algebra and J ⊆ A be an MV-ideal. By Lemma 2.8(5), f r (x) ≤ x for every x ∈ A and for every r ∈ [0, 1], hence J is an RMV-ideal.
In the light of the previous lemma, we will simply speak of ideals.
Notation If A is an RMV-algebra, J is an ideal of A and a ∈ A, we henceforth write [a] J for the image of a under the natural epimorphism induced by the congruence θ J . If S ⊆ A, we write S for the ideal generated by the set S, ie the smallest ideal containing S. Finally, If I and J are ideals, we write I ∨ J for the ideal generated by the set I ∪ J . The following is a folklore result for MV-algebras. For sake of completeness we give here a proof for RMV-algebras.

Lemma 2.17
Let A be an RMV-algebra and a, b ∈ A. Let {a} , {b} be the principal ideals generated by a and b, respectively. Then We postpone to Section 4.1 a discussion on the aptness of the attribute local in the above definition.

Remark 2.21
Notice that local RMV-algebras do not form a subvariety, in fact no proper non-trivial subvariety of RMV-algebras exists as shown in Corollary 2.30 below. However, the class of local RMV-algebras is axiomatised, within the variety of RMV-algebras, by the first order formula: Indeed, an RMV-algebra is local if and only if its MV-algebra reduct is local and the above formula is known to axiomatise local MV-algebras Di Nola, Esposito, and Gerla [13, Theorem 8.1].

Lemma 2.22
Let A be RMV-algebra and I be an ideal of A, then: (1) I is maximal if and only if A/I is isomorphic to [0,1], in a unique way. (

2) I is prime if and only if the lattice order of A/I is linear. (3) I is primary if and only if A/I is local.
Proof (1) The quotient of an MV-algebra by a maximal ideal is isomorphic (in a unique way) to a subalgebra of [0, 1] (see eg Marra and Spada [31, Lemma 3.8]). As shown in the proof of Theorem 2.11, any non-trivial 0-generated RMV-subalgebra is isomorphic to [0, 1], so the quotient of an RMV-algebra by a maximal ideal must be isomorphic to [0, 1].
(3) Notice that the natural quotient map π I : A → A/I gives an isomorphism between the lattice of ideals of A containing I and the lattice of ideals of A/I (see eg Burris and Sankappanavar [6, Theorem 6.20, Chapter II]). Now suppose that I is primary and let M be the unique maximal ideal containing I . Then M/I must be the unique maximal ideal of A/I , so A/I is local. Conversely, suppose that A/I is local and let N, N be maximal ideals of A containing I . Then N/I and N /I are maximal ideals of A/I . Since A/I is local, we have N/I = N /I . Since N and N both contain I , this implies N = N . So I is primary.
We now recall a few results on RMV-algebras which are either already known or easy consequences of the fact that they hold in MV-algebras.
Definition 2.23 Let A be an RMV-algebra and P ⊆ A, we define We write y ⊥ as a shorthand for {y} ⊥ .
Proof For (1), x ∈ O(P) if and only if there exists y ∈ P such that x ∧ y = 0. This in turn is equivalent to ∃y ∈ P such that x ∈ y ⊥ , in other words x ∈ {y ⊥ | y ∈ P}.
(2) to (4)  One inclusion in (5) is obvious. For the other, suppose x ∈ y ⊥ , ie x ∧ y = 0. Let z ∈ {y} , then by Lemma 2.16, there exists n ∈ N such that z ≤ (n)y. By Proposition 2.5 (2) x ∧ z ≤ x ∧ (n)y and the latter is equal to 0 by (4) of the same proposition, so For (6), let I ⊆ A and take x, y ∈ I ⊥ , hence for all z ∈ I we have x ∧ z = y ∧ z = 0, by Proposition 2.5(3) we also have (x ⊕ y) ∧ z = 0, hence I ⊥ is closed under ⊕. The fact that I ⊥ is downward closed is an immediate consequence of the monotonicity of ∧, Proposition 2.5(2).
To prove (7), suppose h ∈ H , then by hypothesis To prove (8) we reason by contradiction. Suppose that there exists h ∈ H and h ∈ I ⊥ ; the latter implies that there exists some i ∈ I such that h ∧ i = 0, but H and I are downward closed, so h ∧ i belongs to both of them and is different from 0.

Lemma 2.26
In every RMV-algebra A: where X -the base space-and F -theétalé spaceare topological spaces, and π : F → X is a local homeomorphism, ie, for each a ∈ F , there exist open sets A a and A π(a), such that π is a homeomorphism from A into A .
If x ∈ X , we denote by F x the set π −1 (x) and we call it the stalk at x. If Y ⊆ X , then F(Y) denotes the set of continuous maps σ : Y → F , such that σ(y) ∈ F y for all y ∈ Y . The elements of F(Y) are called (local) sections over Y , the elements of F(X) are called global sections. Let K be a class of algebras in some language L, if (F, X, π) is a sheaf space of algebras of type L such that conditions in (1) and (2) above hold with "algebra of type L" replaced by "algebra in K", then (F, X, π) is a sheaf space of K-algebras.
Remark 3.3 If in the above definition K is a variety (=equational class of algebras), then the conditions of (1) and (2) are equivalent (see [12,Lemma 1.2] and the ensuing discussion).
It is shown in [12,Theorem 1.5] that, in the context of varieties, the definition above of sheaf space of K-algebras is equivalent to the classical definition of sheaves as contravariant functors with values in K satisfying the gluing axioms.

Compact sheaf representation
Given an algebra A, an element a ∈ A, and a congruence Θ on A, we indicate by [a] Θ the equivalence class of a in A/Θ. If {Θ x | x ∈ X} is a family of congruences on an algebra A and U ⊆ X , we define Θ U = x∈U Θ x . Since we will often work with ideals rather than congruences, if I is an ideal of A and a ∈ A, we will write [a] I for [a] θ I , where θ I is the congruence associated to I by Lemma 2.14.
Definition 3.4 (The sheaf associated to an algebra) Given an algebra A, a topological space (X, τ ), and a family {Θ x | x ∈ X} of congruences on A, one can define the triple (F A , X, π) as follows: i) F A is the disjoint union of the quotients A/Θ x as x varies in X ; in symbols: The function π is defined as the projection from F A into X , ie, π([a] Θx ) := x. iii) Upon defining, for any a ∈ A, the functionâ : X → F A asâ(x) := [a] Θx , the set F A is endowed with the topology generated by the sub-basis {â[U] | U ∈ τ, a ∈ A}.
In general, the above-defined triple may fail to be a sheaf of algebras. The following result, due to Davey, characterises in terms of the topology on X the cases in which the triple is indeed a sheaf .
Additionally, if the above condition holds, and A is a subdirect product of the algebras {A/Θ x | x ∈ X}, then the assignment a →â from the algebra A into the algebra of global sections of (F A , X, π) is an injective L-homomorphism.  Proof In order to prove that (F A , Prim O (A), π) is a sheaf of algebras we use Theorem 3.5 (1). To this end we first need to prove: one has that {b 1 } ∨ · · · ∨ {b n } = A. Let us define: where the latter equality holds because of Lemma 2.17. By way of contradiction suppose that x ∈ H ik ∩ I k but x ∈ I ⊥ i . This means that there exists a ∈ I i such that x ∧ a = 0. By Lemma 2.26 there exists N ∈ Prim O (A) such that x ∧ a ∈ N . But x ∧ a ∈ I i ∩ I k , because x ∈ I k and a ∈ I i and they are both downward closed. So, N ∈ U(I i ∩ I k ) and since x ∈ H ik , we have x ∈ N . In turn this would imply x ∧ a ∈ N , which contradicts our hypothesis. So, in both cases we have H ik ∩ I k ⊆ I ⊥ i .
As an immediate consequence of the claim and Lemma 2.24 (6) we obtain that J i ⊆ I ⊥ i for every i ≤ n. In addition . Therefore d(a i , a) ∈ O(N), for each N ∈ U(b i ) and i = 1, . . . , n. It follows thatâ | U(bi) =â i | U(bi) = σ |Ui for every i = 1, 2, . . . , n, that is σ =â.
Corollary 3.11 Every RMV-algebra is isomorphic to the algebra of global sections of a sheaf of local RMV-algebras over a compact Hausdorff space.
Proof By combining Theorem 3.8 with Lemma 3.10.

Spectral sheaf representation
In this section we give an application of the following result by Cornish that provides sufficient conditions to represent an algebra using the spectrum of prime ideals as base space, topologised with the co-Zariski topology. Recall that the co-Zariski topology is defined on a set of prime ideals exactly as the Zariski topology, but the basic open sets and the basic closed sets are swapped. Recall also that a congruence θ is called prime, if θ = θ 1 ∩ θ 2 implies θ = θ 1 or θ = θ 2 . It is known, and easy to check, that prime congruences correspond to prime ideals by the isomorphisms of Lemma 2.14.   Proof Consider an RMV-algebra A and take as D in Theorem 3.12 above the whole set of congruences on A. Then (1) to (3) and (6) are obvious. (4) and (5) hold because of Lemma 2.28 (although they are well-known to hold for MV-algebras, and the results obviously extend to RMV-algebras). Finally, to see that (7) holds, notice that the compact elements in the lattice of congruences of A are the finitely-generated congruences, which correspond to principal ideals. So an application of Lemma 2.17 concludes the proof.
Remark 3.14 As one of the referees pointed out the two sheaf representations of this section could alternatively be derived by the recent results contained in Gehrke and van Gool [23] and in particular from Corollary 3.12 contained therein. Indeed, as noted after Theorem 3.12, the variety of RMV-algebras is both congruence permutable and congruence distributive. In addition RMV-algebras have the Compact Intersection Property, by Lemma 2.17. In contrast with the injectivity of Theorem 3.5(1), [23,Corollary 3.12] guarantees without further work that the algebra is isomorphic to the algebra of global sections of its representing sheaf, therefore our Lemma 3.10 would not be necessary. However, the results in [23] require extra work to characterise precisely the stalks in the representation, while in our approach they are clearly described by the construction itself.

Local RMV-algebras 4.1 Localisation of RMV-algebras
We begin this section motivating the name local RMV-algebras. The reasons are essentially similar to the ones that motivated the name local MV-algebra (see Belluce, Di Nola, and Gerla [2, Section 5]).
In modern terms one could define the abstract concept of localisation as follows.
Definition 4.1 Assume that A is an algebraic structure in which it make sense to speak about prime and maximal ideal. Let P be a prime ideal of A. A localization of A at the prime P is an algebra B of the same type of A such that: Here by Spec(A) we mean the space of prime ideals of A with the Zariski topology.
We will show that one can perform a localisation of an RMV-algebra at an arbitrary prime ideal. Unfortunately the localisation is not unique: this is the reason why we had to include the possibility of a subalgebra in (1) above; however, we will see that there is a canonical construction.
Thorough this section A is assumed to be a non-trivial RMV-algebra. We fix some notation: Notation Given an RMV-algebra A and an ideal P of A, we set where A ≤ A means that A is an RMV-subalgebra of A. We further set: Finally, if X ⊆ A we set Ralg(X) to be the RMV-subalgebra of A generated by X . Proof The support of the algebra Ralg(P) can be described as the set: {t(p 1 , . . . , p k ) | t is an RMV-term and p 1 , . . . , p k ∈ P} .
Note that P is not necessarily the unique maximal ideal of Ralg(P). In fact, consider A = [0, 1] 2 and P the ideal generated by the element (0, 1). Then it is readily seen that (0, 1) generates the full RMV-algebra A, hence Ralg(P) = A but A is obviously not local.

Proposition 4.3 Let
A be an RMV-algebra, P any ideal of A, and A ∈ L A (P).
Then A /Ω A (P) is a local RMV-algebra with maximal ideal P/Ω A (P). In particular Ralg(P)/Ω A (P) is a local RMV-algebra with maximal ideal P/Ω A (P).
Proof This was proved for MV-algebras in [2,Proposition 12]. Since it is a statement about ideals, in the light of Lemma 2.15, it also holds for RMV-algebras. Summing up, for every A ∈ L A (P), and every P ∈ Spec(A), the algebra A /O(P) satisfies Definition 4.1. In particular Ralg(P)/O(P) can be considered the canonical localization of the RMV-algebra A at the prime ideal P.
The rest of this section is devoted to the study of local RMV-algebras.  Proof Since st is surjective,st is surjective as well. Moreover we have:

Definition 4.7 If
A is an RMV-algebra and x, y ∈ A, we say that x and y are infinitely close if d(x, y) ∈ Rad(A).
Definition 4.8 A function g from a set X into some non-trivial, linearly ordered RMV-algebra C is called quasi-constant if it is infinitely close to some constant function; in symbols, if for all x, y ∈ X , d(g(x), g(y)) ∈ Rad(C). Since P and P are arbitrary prime ideals, g a is quasi-constant, as required. Proof If an RMV-algebra A is local, then by Theorem 4.10 it is an algebra of quasi-constant functions into an ultrapower [0, 1] * , and the latter is linearly ordered. Conversely, if A can be represented as an algebra of quasi-constant functions with values in a linearly ordered RMV-algebra C, then by Theorem 2.29, C can be embedded into some ultrapower of [0,1]. Notice that the composition of a quasi-constant function into C with this embedding gives a quasi-constant function with values in the ultrapower, because the image under the embedding of the radical of C is included in the radical of the ultrapower. Hence we can again apply Theorem 4.10 to obtain that A is local.

Radical retractions
Lemma 4.13 Let A be an MV-algebra and suppose a, b ∈ A are infinitely close. Then: (2). If a, b ∈ A are infinitely close then a b, b a ∈ Rad(A). Notice that: by Equation (3) = (a * (a * ∧ b * )) ⊕ 0 = a * (a * ∧ b * ) by Equation (5) and by Equation (3) and Equation (6) = (a * a) ∨ (a * b) = a * b by Proposition 2.5(1) and (5) = b a ∈ Rad(A) by Equation (3) Therefore, a * and a * ∧ b * are infinitely close. By absorption a * = (a * ∧ b * ) ∨ a * , hence we can apply (1) to obtain a τ ∈ Rad(A) such that a * = (a * ∧ b * ) ⊕ τ . Now, by Equation (MV 3) and Equation (3) we obtain where ε, τ ∈ Rad(A), and this concludes the proof. Conversely, suppose that A is local. By Theorem 4.10 A is isomorphic to an algebra of quasi-constant functions from a set X into some ultrapower [0, 1] * . So, for any a ∈ A there exists r ∈ [0, 1] such for all i ∈ X , st(a(i)) = r. So, for all i ∈ X , d(a(i), f r (1)) ∈ Rad([0, 1] * ). Hence, by Lemma 4.13 (2), a can be written as a polynomial with coefficients in Rad(A) ∪ b, where b ∈ R(A). Since R(A) is the 0-generated subalgebra of A, the algebra A is generated by its radical. To prove uniqueness, suppose s, t are sections of π M . Since s, t factor the identity they must be injective, so s   (1) A is Rad-retractive. In fact, it is semisimple so the radical is zero, so trivially the algebra is radical-retractive (because it is zero-retractive). (2) A is not local. In fact, the ideal of pairs with the first component zero is maximal, and the one with the second component zero is also maximal. So A shows that the implication of Remark 4.18 cannot be inverted. (3) A is retracts onto [0, 1]. In fact, as a projection we can take (for instance) the projection of A on the first component, and as a section the diagonal map sending x ∈ [0, 1] to (x, x) ∈ A. So A shows that Theorem 4.15 cannot be trivially generalised.
One could conjecture that Rad-retractiveness is a property shared by all RMV-algebras. In fact, the conjecture is not easily disproved as all natural examples of RMV-algebras have this property. However, one can build a counter-example to the conjecture.

Proposition 4.20
There is an RMV-algebra which is not Rad-retractive.

We preliminarily show:
Claim 3 For every a ∈ A, if a is infinitely close to g, then there is a unary RMV-term u such that u(a) > 0 and u(a) is infinitesimal.
Proof of Claim 3 Let a ∈ A. Since A is generated by g there exists a unary RMV-term t such that a = t(g). By Di Nola and Leuştean [18, Theorem 10] the interpretation of t in the RMV-algebra [0, 1] is a continuous piecewise linear function, let ρ be its right derivative in 0. Since there exists an open neighbourhood of 0 on which the interpretation of t in [0, 1] is linear, ρ is a real number and there exists 0 = m ∈ N such that for any n ≥ m, t 1 n + ε = ρ · 1 n + ε . Since t (g (n)) is infinitely close to g (n) for every n ∈ N, it must be ρ = 1. So, for any n ≥ m, t 1 n + ε = 1 n + ε.
Set u 0 (x) := ((m − 1) x) x. Straightforward calculations show that Equation (11) implies that u 0 t 1 m + ε is a positive infinitesimal, whereas for n > m, u 0 t 1 n + ε = 0. It follows that the set I ⊆ N such that for i ∈ I , u 0 t 1 i + ε is not infinitesimal and different form 0 is finite. So there must exist k ∈ N such that 1 k < u 0 t 1 i + ε for all i ∈ I . Set u 1 := (k) u 0 . By previous calculations u 1 t 1 m + ε = 0 is a positive infinitesimal, if n > m then u 1 t 1 n + ε = 0, and if n < m then u 1 t 1 n + ε is either equal to 1 or infinitesimal. Finally let u (x) := (u 1 (x)) ∧ (u 1 (x)) * ; we have u t 1 n + ε = 0 is a positive infinitesimal, if n > m then u t 1 n + ε = 0, and if n < m then u t 1 n + ε is either equal to 0 or infinitesimal. Summing up, u (t (g)) is a positive infinitesimal.
To conclude the proof of Proposition 4.20 we reason by way of contradiction. So, suppose that the quotient map π : A → A/ Rad(A) is a retraction, ie there is a section s : A/ Rad(A) → A such that π • s is the identity. Notice that s must be injective. Let e := s • π , so e • e = s • π • s • π = e. The injectivity of s amounts to saying that e(x) = e(y) if and only if π(x) = π(y), therefore form e(x) = e(e(x)) we deduce π(x) = π(e(x)); in other words e(x) is infinitely close to x for every x ∈ A (intuitively, the map e "chooses" a representative in each equivalence class modulo the radical). So, in particular, e(g) is infinitely close to g, hence by Claim 3, there is a term u such that u(e(g)) is infinitesimal and non-zero. It follows that e(u(e(g))) = 0. Since e is a homomorphism and u is a term, we have e(u(e(g))) = u(e(e(g))) = u(e(g)) = 0 and this contradiction concludes the proof.
It should be noticed that an MV-algebra may be endowed with several non-isomorphic RMV-algebra structures, we refer the reader to Di Nola, Lenzi, Marra, and Spada [16] for more details on this and to Lenzi [28], which solves the corresponding problem posed by Conrad in 1975.

An equivalence between Local RMV-algebras and Riesz spaces
Theorem 4.23 An RMV-algebra is local if and only if it is isomorphic to Γ R (R − → × W, (1, 0 W )) where W is a Riesz space.
Proof Let W be an arbitrary Riesz space and let A ∼ = Γ R (R − → × W, (1, 0 W )). We show that A is local. Consider the set M = {a ∈ A | a = (0, w) for w ∈ W}. It is easy to see that M is an ideal of A. Furthermore, every element of A \ M has the form (r, w) with 0 < r ≤ 1, so there exists a positive integer n such that r > 1/n, whence (n)(r, w) = 1 A . We conclude that no element of A \ M can lie in a proper ideal of A. Summing up, M is the greatest ideal of A, hence A is local.
Conversely, let A be a local RMV-algebra, hence by Theorem 4.14 A is generated by its radical. Now let (V, u) be the Riesz space with order unit, given by Theorem 2.12, such that A = Γ R (V, u). Then V , as a group, is generated by A. Moreover, since A can be generated by Rad(A) by the RMV algebra operations, A can also be generated by Rad(A) by using the Riesz space operations and u. Summing up, the Riesz space V is generated by Rad(A) ∪ {u}. Let W be the Riesz subspace of V generated, as a Riesz space, by Rad(A). Note that every element w of W is infinitesimal with respect to u, in the sense that for every n ∈ N we have n|w| ≤ u.
So, every element v of V can be written as xu + w, where x is a real number and w ∈ W . The pair (x, w) is unique, because if xu + w = x u + w then (x − x )u = w − w , and the right hand side is infinitesimal with respect to u, whereas u is not infinitesimal with respect to itself, so we must have x − x = 0 and therefore w = w . The map h sending v to (x, w) gives a map from V to R − → × W which is a vector space isomorphism. We