Expanding Lattice Ordered Abelian Groups to Riesz Spaces


 First we give a necessary and sufficient condition for an abelian lattice ordered group to admit an expansion to a Riesz space (or vector lattice). Then we construct a totally ordered abelian group with two non-isomorphic Riesz space structures, thus improving a previous paper where the example was a non-totally ordered lattice ordered abelian group. This answers a question raised by Conrad in 1975. We give also a partial solution to another problem considered in the same paper. Finally, we apply our results to MV-algebras and Riesz MV-algebras.


Introduction
Our objects of interest are two kinds of algebraic structures: Riesz spaces (or vector lattices) and Riesz MV-algebras. See the preliminary section for the definitions.
Riesz spaces find applications in several fields like functional analysis, economy, etc. whereas Riesz MV-algebras find applications in many-valued logic, fuzzy logic, quantum mechanics, etc. For Riesz spaces, an example of monography is [10], whereas Riesz MV-algebras are more recent and we know no extensive treatment of the subject. These structures are enrichments of simpler structures: abelian l-groups and MV-algebras. For abelian l-groups see [1] or [2], whereas for MV-algebras see [4].
The relations between all these kinds of structures are very interesting and are studied, for instance, in [5,7] and [11] . This paper is devoted to the study of these relations. This paper is in a sense a continuation of [9], whereas [5] is our source of inspiration. Our main theme is the relation between abelian lattice ordered groups (abelian l-groups) and Riesz spaces. Each Riesz space is also, by definition, an abelian l-group. The problem now is: given an abelian l-group, can it be expanded to a Riesz space? and in how many ways?
First of all, one would like to have simple necessary and sufficient conditions for an l-group to admit a Riesz space structure. A necessary and sufficient condition is given in [5]. We propose another one. More generally we attempt to make a theory of l-groups, or MV-algebras, which admit at least one Riesz structure (we call these structures extendible).
For instance, a countable l-group cannot become a Riesz space, and Archimedean l-groups can have at most one structure. On the other hand, [5] gives examples of l-groups G with at least two Riesz space structures: for instance, G = R lex R, where R is the ordered group of the real numbers and lex denotes lexicographic product. More generally [5] proves that every totally ordered, non-Archimedean l-group has either no structure, or at least two. Another example of double structure is given by [12: Example 11.54].
In [5], several open problems are left. One of them (question II thereof) is whether all Riesz spaces over a given abelian l-group G are isomorphic. This problem is solved in the negative in [9] with an explicit counterexample of G. However, G is not totally ordered, whereas [5] asks for a totally ordered example. In this paper, we solve in the negative this problem, by exhibiting a totally ordered abelian group G with two non-isomorphic Riesz space structures. The construction of the example is similar to that of [9], but somewhat simpler. Another question of [5] is whether every l-group with exactly one Riesz space structure is Archimedean. We give a partial positive solution concerning the l-groups embedded in a product of totally ordered abelian groups which are closed under finite variant.
Finally, we turn to MV-algebras. As it often happens, we can use the Mundici functor of [11] to transfer information from l-groups to MV-algebras. This paper is no exception: the previous results on l-groups can be transferred to MV-algebras. In particular, via the Mundici functor, we can prove that there is a totally ordered MV-algebra with two non-isomorphic Riesz MV-algebra structures.

Preliminaries
In this preliminary section, we mostly follow [9]. We denote by N, Z, Q, R the sets of natural numbers (starting from 0), the integers, the rationals and the reals respectively.

l-groups
A lattice ordered abelian group (l-group) is a structure (G, +, ≤) such that: • (G, +) is an abelian group; The infimum and supremum of two elements x, y ∈ G will be denoted by x ∧ y and x ∨ y. A particular case is when the lattice order is total, in which case we say that (G, +, ≤) is a totally ordered abelian group.
A strong unit of an l-group G is an element u ∈ G such that for every x ∈ G there is n ∈ N such that x ≤ nu.
The absolute value of an element x ∈ G is |x| = x ∨ −x. When G is totally ordered, we simply have |x| = x if x ≥ 0 and |x| = −x if x < 0. Given x, y ∈ G, we say that x dominates y if there is n ∈ N such that |y| ≤ n|x|. We say that x, y are equidominant if they dominate each other. Note that equidominance is an equivalence relation (in literature the equidominance relation is often called Archimedean equivalence). We say that x strictly dominates y if x dominates y but y does not dominate x.
An l-group G is called Archimedean if any two non-zero elements of G are equidominant (id est Archimedean equivalent).
Note that l-groups are an equational class, so there is a natural notion of homomorphism of l-groups and a natural category of l-groups.

Riesz spaces
A Riesz space is a structure (G, +, ≤, ρ) which is an l-group with a structure of vector space over R, formally a map + : G × G → G and a map ρ : R × G → G, satisfying the usual vector space axioms, that is (letting rv = ρ(r, v)): and such that if v ≥ 0 and r is a positive real, then rv ≥ 0.
Like l-groups, Riesz spaces form an equational class, so there is a natural notion of homomorphism of Riesz spaces and of the category of Riesz spaces.

MV-algebras
An MV-algebra is a structure (A, ⊕, 0, 1, ¬) where: • (A, ⊕, 0) is a commutative monoid where 0 is the neutral element; Intuitively, ⊕ is a kind of sum, and ¬ is a kind of negation.
The most important, and motivating, example of an MV-algebra is based on the unit real interval, where A = [0, 1], x ⊕ y = min(x + y, 1) and ¬x = 1 − x.
Other derived connectives in MV-algebras are x y = ¬(¬x ⊕ ¬y) (a kind of a product, dual to the sum) and x y = x ¬y (a kind of difference).
Once again we have an equational class, a natural notion of homomorphism and category.

A categorial equivalence
In [5], a necessary and sufficient condition is given for an abelian l-group to admit an expansion to a Riesz space structure. In this paper, we give another one, and we build a category of "expanded l-groups" equivalent to the category of Riesz spaces.
We will say that an expanding family of a divisible group G is a family G(b) b∈G + of subgroups of G such that: , y ∈ G(c) and for every rational q, x < qb if and only if y < qc, then x+y ∈ G(b+c) and x ∧ y ∈ G(b ∧ c).
We will say that an abelian l-group G is Riesz expandable if G is divisible and admits an expanding family. Conversely, suppose all conditions are met. If b ∈ G + , and r ∈ R, we let rb be the image of r in the unique ordered group isomorphism between R and G(b) If g is any element of G, Note that the decomposition of g as a difference of two positive elements is not unique, but the definition of rg is independent of the decomposition This gives a Riesz space structure on G. In fact, from the last item, it follows for every r ∈ R and for every b, c > 0 that rb + rc = r(b + c) and rb ∧ rc = r(b ∧ c). The properties extend from G + to G.
We note that expanding families are not unique.
There is an l-group G with at least two different expanding families.
P r o o f. By the examples given first (to our knowledge) in [5], there is an l-group G with at least two Riesz space structures for every q ∈ Q, and for every real r, ρ(r, b) is the unique element of G(b) such that ρ(q, b) < ρ(r, b) < ρ(q , b) for every pair of rationals q < r < q , and ρ (r, b) is the unique element of G (b) such that ρ(q, b) < ρ(r, b) < ρ(q , b) for every pair of rationals q < r < q . So ρ(r, b) = ρ (r, b) for every real r and b ∈ G + , so ρ = ρ .
Let us call expanded l-group a structure (G, G(b) b∈G + ) where G is a divisible l-group and G(b) is an expanding family of subgroups of G.
We can make a category of expanded l-groups by taking as morphisms between (G, G(b)) and (H, H(c)) the homomorphism of groups f :

On convex expanded families
P r o o f. Suppose for a contradiction that G is not Archimedean. Then there are b, ∈ G + such that n ≤ b for every n ∈ N . Now b ≤ b + ≤ 2b, so b + ∈ G(b) and ∈ G(b). But every nonzero element of G(b) dominates b, whereas dominates b. This contradiction concludes the proof.
Note that if G is Archimedean then G has at most one Riesz space structure, hence at most one expanding family.
Conversely, suppose G is an Archimedean l-group with an expanding family G(b). Then G(b) is not necessarily convex for every b, for example:

On l-groups closed under finite variant
An open question of [5] is whether a non-Archimedean Riesz space can have only one Riesz space structure (compatible with its l-group structure). The following is a partial answer (recall that every l-group is embeddable in a product of totally ordered l-groups).
Theorem 5.1. Let G be a non-Archimedean l-group embedded in a product of totally ordered l-groups and closed under finite variant. Then G admits either zero or more than one Riesz space structure.
P r o o f. Let G ⊆ Π i∈I G i , where each G i is a totally ordered abelian group. Since G is closed under finite variant, for every i ∈I, G contains the vector u i consisting of 1 in position i and 0 elsewhere. Let ρ be a Riesz space structure on G. Let r ∈ R + and q, q ∈ Q such that 0 < q < r < q . Then ρ(r, u i ) is between ρ(q, u i ) and ρ(q , u i ), hence ρ(r, u i ) must have zero in all components different from i. Moreover, suppose any v ∈ G has v i = 0 for some i ∈ I. Then v is orthogonal to u i (i.e. v ∧ u i = 0), and by definition of Riesz structure, ρ(v) ∧ ρ(u i ) = 0. That is, ρ(v) has i-th coordinate equal to zero. By additivity, if v and w have the same i-th component, then ρ(r, v) and ρ(r, w) have the same i-th component. So, for every i ∈ I, there is a map ρ i : and ρ i is a Riesz space structure on G i . In other words, all the Riesz space structures on G are products of Riesz space structures on G i . Now suppose G has a Riesz space structure ρ. By the argument above, every G i has a Riesz space structure ρ i . Since G is non-Archimedean, some G i must be non-Archimedean. Any such G i must have another Riesz space structure ρ i . Now, let ρ : R × G → G be the map such that ρ (r, v) = w if and only if w i = ρ i (r, v i ) and w j = ρ i (r, v j ) for every j = i. ρ is well defined because ρ (r, v) is a finite variant of ρ(r, v), and is a Riesz space structure on G. Since ρ i = ρ i , we conclude ρ = ρ .
Like in [9], we call atom of an l-group G an element a ∈ G + such that for every b, c ∈ G + with b, c ≤ a we have b ∧ c = 0.
Note that every l-group closed under finite variant is atomic (i.e. below every positive element there is a positive atom). In fact, every positive real multiple of u i is an atom, and every positive element is above some positive real multiple of u i for some i ∈ I. We conjecture that the previous theorem generalizes to atomic l-groups.

A totally ordered example
We have said that [9] gives a construction of a unital l-group with two non-isomorphic Riesz space structures. In this section we adapt the construction of [9] to the case of totally ordered abelian groups, and we obtain: There is a totally ordered abelian group G with a strong unit u, such that G has two non-isomorphic Riesz space structures ρ 1 and ρ 2 .
P r o o f. Like in [9], the idea is to build a group with two "asymmetric" Riesz structures.
Let R a be the field of the real algebraic numbers. R and R a are real closed fields, so they are elementarily equivalent. By Frayne's Theorem there is an embedding j 1 : R → * R a , where * R a = (R a ) I /U is an ultrapower of R a (so I is a set and U is an ultrafilter over I).
Let K 0 be the set of finite sums Σ i j 1 (r i )j 2 (s i ) where r i , s i ∈ R. K 0 is a Riesz subspace of K in both Riesz structures ρ 1 and ρ 2 , and has a strong unit j 1 (1) (note that j 1 (1) = j 2 (1)).
Note that K 0 has the cardinality of the continuum, so K 0 is included in at most 2 ℵ0 Archimedean classes (to our knowledge the exact number of Archimedean classes of K 0 is not known, note that K 0 is defined in an indirect way by an ultraproduct construction).
The idea is to consider certain sequences of elements of K 0 indexed by a regular cardinal Λ sufficiently large. More precisely, we fix two regular cardinals η, Λ such that 2 ℵ0 < η < Λ.
Note that any two elements of K 0 have Archimedean distance less than η. Let us equip the group K Λ 0 with the lexicographic ordering. That is, we let g < h if and only if the first nonzero component of h − g is positive. In this way K Λ 0 is a totally ordered abelian group. Let G ⊆ K Λ 0 be the set of all sequences g ∈ K Λ 0 such that for every α < Λ, g(α) can be written, in the vector space (K 0 , ρ 1 ), as a real linear combination of some finite set F ⊆ K 0 independent of α. In other words, the range of g has finite dimension in (K 0 , ρ 1 ). In symbols, Note that G inherits from K Λ 0 (and from K) the two vector space structures above, which we will still call ρ 1 and ρ 2 .
An example of strong unit of G is simply u = (j 1 (1), 0, 0, . . .). Note that in order to have a strong unit, we do not need the condition (present in [9]) that the components of the elements of G are bounded.
Since G is totally ordered, the absolute value of an element g of G, written |g|, is simply g if g ≥ 0, and −g if g < 0; and the Archimedean classes (id est equidominance classes) of G are also totally ordered in the natural way.
Let us call Archimedean distance between g, h ∈ G the number, possiby infinite, of Archimedean classes between g and h.
P r o o f. Γ ⊆ G because we can take F = {1} and k 1 = 1.
For the second point, consider g ∈ G. Then g(α) = Σ i∈F j 1 (r i,α )k i , where F is finite and Let γ ij the Λ-sequence such that γ ij (α) = j 1 (r i,α )j 1 (r ij ). Then γ ij ∈ Γ. Moreover and, letting α range over Λ, we have that is, g is a linear combination of Γ in the vector space (G, ρ 2 ).
The following corollary, instead, is new.

EXPANDING LATTICE ORDERED ABELIAN GROUPS TO RIESZ SPACES
Corollary 6.1. Every element g ∈ G is generated in (G, ρ 2 ) by positive elements of Γ which have Archimedean distance less than η from g.
Then h i ∈ Γ, h i still generate g and either h i (α) = 0, or h i has Archimedean distance less than η from g. Since the h i generate g, there must be some index i 1 such that h i1 (α) = 0. So, for every i such that h i (α) = 0, we replace h i with h i1 + h i .
Let g n be a sequence of elements of G. A weak sum of g n , if it exists, is an element s ∈ G such that, for every n, s − g 1 − g 2 − · · · − g n is dominated by g n at a distance at least η. Note that a weak sum is not necessarily unique, because the components α < Λ of s beyond the first nonzero components of all g n are not specified (and such components exist because Λ is an uncountable regular cardinal).
We have the following key lemma.
Lemma 6.2. Every positive decreasing sequence g n of elements of Γ with distances at least η admits a weak sum s.
Like in [9] we say that an enriched Riesz space is a triple (G, ρ, B), where (G, ρ) is a Riesz space and B is a subset of G. An isomorphism of enriched Riesz spaces (G, ρ, B) and (G , ρ , B ) is an isomorphism between the Riesz spaces (G, ρ) and (G , ρ ) which sends B bijectively onto B . Now suppose by contradiction that (G, ρ 1 ) is isomorphic to (G, ρ 2 ). Then for some subset ∆ of G, the enriched Ries space (G, ρ 2 , Γ) is isomorphic to (G, ρ 1 , ∆). So ∆ must satisfy Corollary 6.1 and Lemma 6.2 (up to replacing ρ 2 with ρ 1 ). Let us choose a sequence (t n ) of real transcendental numbers linearly independent over the subfield R a of R. Then already in [9] it was observed: ). The sequence (j 2 (t n )) is linearly independent in the vector space (K 0 , ρ 1 ).
P r o o f. In fact, let us suppose that Σ n∈F j 1 (r n )j2(t n ) = 0, where F = ∅ is finite and r n = 0 for every n ∈ F .
Note that j 1 (r n ) ∈ * R a and that * R a is the ultrapower (R a ) I /U. Instead, j 2 : R → R I /U is the diagonal embedding, so j 2 (t n ) is the U-class of the constant sequence t n .
Suppose the U-class j 1 (r n ) contains a tuple of real algebraic numbers (r n,i ) i∈I . By Loś's Theorem on ultraproducts, we obtain Σ n∈F r n,i t n = 0 and r n,i = 0 for every n ∈ F and for almost all i ∈ I with respect to U. So, for some i ∈ U, we have Σ n∈F r n,i t n = 0 and r n,i ∈ R a {0}. But this is not possible since the sequence (t n ) is linearly independent over R a .
The idea of the following lemma (which gives the main construction) is to use the sequence j 2 (t n ) and define a sequence δ n of positive elements of ∆ such that Lemma 6.2 may be applied to δ n .
P r o o f. The proof goes by complete induction. As a base step, we let k 1 = 1. Let f 1 be the corresponding quasiconstant element of G. By Corollary 6.1, f 1 is generated in (G, ρ 1 ) by a finite set ∆ 1 of positive elements of ∆, which cannot be empty. Let δ 1 be any element of ∆ 1 .
The inductive step n + 1 is as follows. We have ∆ i , δ i , k i for 1 ≤ i ≤ n. By definition of G, the components of every element of ∆ 1 ∪ · · · ∪ ∆ n have finite dimension in (K 0 , ρ 1 ), whereas the sequence j 2 (t n ) has infinite dimension. So we can find a number k n+1 ∈ N so high that j 2 (t kn+1 ) is not generated by ∆ 1 ∪ ∆ 2 ∪ · · · ∪ ∆ n .
The inductive construction is thus completed. Now by Lemma 6.2 the positive sequence δ n ∈ ∆ constructed in the previous lemma admits a weak sum s ∈ G. By definition of weak sum, for every n ∈ N , we have s(nη) = δ 1 (nη) + · · · + δ n (nη).

Applications to MV-algebras
We note that a condition for the MV-algebra reducts of Riesz MV-algebras can be inferred from Theorem 3.1 by applying the results of [7], where the Mundici equivalence (Γ, Ξ) of [11] between MV-algebras and abelian unital l-groups is specialized to an equivalence (Γ , Ξ ) between Riesz MV-algebras and unital Riesz spaces. In fact, we have: P r o o f. Suppose A is the reduct of a Riesz MV-algebra R. Then by [7], Ξ(A) is the l-group reduct of the Riesz space Ξ (R).
Conversely, if the l-group Ξ(A) is the reduct of a Riesz space S = Ξ (R), then A is the reduct of the Riesz MV-algebra R.

EXPANDING LATTICE ORDERED ABELIAN GROUPS TO RIESZ SPACES
However we have also a direct characterization in terms of MV-algebras. For this aim we call difference structure a structure (A, ) where is a binary operation on A. For instance, every MV-algebra is a difference structure with respect to its usual truncated difference operation.
What we call difference structures are related to the D-posets of [3]. We will say that an MV-algebra A is Riesz-extendable if A is divisible and there is a family of difference substructures of A, R(a) a∈A such that: • R(0) = 0; • if a > 0 then R(a) is isomorphic to [0, 1] as a difference structure, and the (unique) isomorphism sends a to 1; • if a ∈ R(a ) then R(a) ⊆ R(a ); • if x ∈ R(a), x ∈ R(a ) and for every rational q ∈ [0, 1], x < qa if and only if x < qa , then x x ∈ R(a a ). Conversely, suppose all conditions are met. Let r ∈ [0, 1] and a ∈ A. If a = 0 then we let ra = 0. If a > 0 then we let ra be the image of r in the unique difference isomorphism from [0, 1] to R(a). This is a Riesz MV-algebra structure on A.
We note that, by [9], the Riesz MV-algebra structure on an MV-algebra, when it exists, is not necessarily unique, not even up to isomorphism. This means that the family R(a) is not uniquely determined by A.
As a corollary of Theorem 6.1, we obtain: There is a totally ordered MV-algebra with two non-isomorphic Riesz MV-algebra structures.
P r o o f. Let (G, u) be a totally ordered abelian group with a strong unit u such that G has two non-isomorphic Riesz space structures (such a group exists by Theorem 6.1). Consider the MValgebra A = Γ M (G, u), where Γ M is the Mundici functor of [11]. So the universe of A is the set {x ∈ G|0 ≤ x ≤ u} and the MV-algebra operations are x ⊕ y = min(x + y, u) and ¬x = u − x.
We note (see [7: Theorem 3]) that every Riesz space structure on G gives a Riesz MV-algebra structure on A. Actually, in [7] there is an equivalence Γ DL between the category of Riesz spaces with strong unit and Riesz MV-algebras, which coincides with Γ M when restricted to the MV-algebra reducts of Riesz MV-algebras and the abelian l-group reducts of Riesz spaces.
So, the structures ρ 1 and ρ 2 on A cannot be isomorphic, otherwise by the functor Γ DL we should have an isomorphism between (G, ρ 1 ) and (G, ρ 2 ), contrary to Theorem 6.1.