Signed-Bit Representations of Real Numbers and the Constructive Stone-Yosida Theorem Robert S. Lubarsky and Fred Richman Florida Atlantic University

Def A Riesz space R is a lattice ordered vector space. Canonical example A (natural) collection of functions from some domain into the reals ℝ -- meet and join computed pointwise. Representation Theorem (Stone, Yosida, …) (classical) Every Riesz space is embeddable into a function space. (Extensions: Preserving certain structure; domain a quotient of a function space, etc.)

Proof idea For R ⊆ ℝ W, w  W induces ŵ : R → ℝ via ŵ(f) = f(w). So for a general Riesz space R, the desired domain is a subset of Σ = Hom(R, ℝ). So embed R into ℝ Σ. █

Proof idea For R ⊆ ℝ W, w  W induces ŵ : R → ℝ via ŵ(f) = f(w). So for a general Riesz space R, the desired domain is a subset of Σ = Hom(R, ℝ). So embed R into ℝ Σ. █ Why does Σ have non-trivial elements? Classically, the Axiom of Choice. Constructively…

(Coquand-Spitters) (DC) If R is separable, and every element is normable, and r  R is (sufficiently) different from 0 then there is a σ  Σ such that σ(r) ≠ 0. (C-S) Is DC necessary? What choice principle is involved?

(Coquand-Spitters) (DC) If R is separable, and every element is normable, and r  R is (sufficiently) different from 0 then there is a σ  Σ such that σ(r) ≠ 0. (C-S) Is DC necessary? What choice principle is involved? Example Let R a be generated by the projections onto the real and complex lines of the solutions to x 2 = a. If you cannot decide whether a = 0, then you cannot in general find the roots.

Let 1  R be distinguished. Def (r  R) r is normable if glb {q  ℚ | q > r} (i.e. sup(r)) exists. Def Pos(r) if sup(r) > 0. Def r  (p, q) = (r – p) ⋀ (q – r) Note For r a function, Pos( r  (p, q) ) iff rng(r) ⋂ (p, q) is non-empty. Hence r can be identified with those intervals (p, q) such that Pos( r  (p, q) ).

Dedekind real r is located: if p r.

Dedekind real r is located: if p r. Signed-bit representation: ● (-2, 2)

Dedekind real r is located: if p r. Signed-bit representation: (-2, 1) ● ● (-1, 2) ● (-2, 2)

Dedekind real r is located: if p r. Signed-bit representation: (-2, 0) ● ● ● (0, 2) (-1, 1) (-2, 1) ● ● (-1, 2) ● (-2, 2)

Dedekind real r is located: if p r. Signed-bit representation: (-2, 0) ● ● ● (0, 2) (-1, 1) ● (-2, 2)

● ●● (-2, -1) (-3/2, -1/2) (-1, 0) (-2, 0) ● ● ● (0, 2) (-1, 1) ● (-2, 2)

● ●● ● ● (-2, -1) (-3/2, -1/2) (-1, 0) (-1/2, 1/2) (0, 1) (-2, 0) ● ● ● (0, 2) (-1, 1) ● (-2, 2)

● ●● ● ● ● ● (-2, -1) (-3/2, -1/2) (-1, 0) (-1/2, 1/2) (0, 1) (1/2, 3/2) (1, 2) (-2, 0) ● ● ● (0, 2) (-1, 1) ● (-2, 2) (the pseudo-tree) T

● ●● ● ● ● ● (-2, -1) (-3/2, -1/2) (-1, 0) (-1/2, 1/2) (0, 1) (1/2, 3/2) (1, 2) (-2, 0) ● ● ● (0, 2) (-1, 1) ● (-2, 2)

● ●● ● ● ● ● (-2, -1) (-3/2, -1/2) (-1, 0) (-1/2, 1/2) (0, 1) (1/2, 3/2) (1, 2) (-2, 0) ● ● ● (0, 2) (-1, 1) ● (-2, 2)

Recall that r  R can be identified with { (p, q) | Pos(r  (p, q) ) }. That in turn is a sub-tree T r of T with the extendibility property: no terminal nodes. For r, s  R and intervals I, J, (I, J) is in the signed-bit representation of R iff Pos( r  I ⋀ s  J ). The signed-bit representation T R of R is the set of all such finite sequences of intervals, indexed by members of R.

Theorem For every set T X of finite sequences (indexed by X) of intervals from T with the extendibility property, there is a canonical Riesz space R ⊇ X such that T X is the signed- bit representation of X.

Definition An ideal I r through T r is a (non- empty) subset closed downwards and under join and with no terminal element.

Definition An ideal through T R is an ideal I r through each T r such that finite products stay in T R : Π r  X I r ⊆ T R (X finite).

Theorem There is a canonical bijection between ideals through T R and Σ. (Recall Σ is Hom(R, ℝ).)

Definition An ideal through T R is an ideal I r through each T r such that finite products stay in T R : Π r  X I r ⊆ T R (X finite). Theorem There is a canonical bijection between ideals through T R and Σ. (Recall Σ is Hom(R, ℝ).) (Note: This can be extended to account for ideals through T X and homomorphisms of the Riesz space generated by X.)

So the existence of homomorphisms from Riesz spaces with dense subsets of size  is equivalent to the existence of ideals through subsets of  -sized products of T with the extendibility property. This is a Martin’s Axiom-like property of set theory.

Let ℱ be the topological space of all ideals through T X, where X has two elements. An open set is given by finitely many pieces of positive and negative information. Positive information is a pair of intervals, i.e. a member of T × T. Negative information is a pair of such closed intervals.

Claim The (full) topological model over ℱ is the (canonical) generic model for a Riesz space with two generators, and that space has no homomorphisms to ℝ.