Signed-Bit Representations of Real Numbers and the Constructive Stone-Yosida Theorem

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

2. 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.)

3. Proof idea For R ⊆ ℝW, 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 ℝΣ. █

4. Proof idea For R ⊆ ℝW, 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 ℝΣ. █ Why does Σ have non-trivial elements? Classically, the Axiom of Choice. Constructively…

5. (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?

6. (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? ExampleLet Ra be generated by the projections onto the real and complex lines of the solutions to x2 = a. If you cannot decide whether a = 0, then you cannot in general find the roots.

7. 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. Defr  (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) ) .

8. Dedekind real r is located: if p<q, then p<r or q>r.

9. Dedekind real r is located: if p<q, then p<r or q>r. Signed-bit representation: ● (-2, 2)

10. Dedekind real r is located: if p<q, then p<r or q>r. Signed-bit representation: (-2, 1) ● ● (-1, 2) ● (-2, 2)

11. Dedekind real r is located: if p<q, then p<r or q>r. Signed-bit representation: (-2, 0) ● ● ● (0, 2) (-1, 1) (-2, 1) ● ● (-1, 2) ● (-2, 2)

12. Dedekind real r is located: if p<q, then p<r or q>r. Signed-bit representation: (-2, 0) ● ● ● (0, 2) (-1, 1) ● (-2, 2)

13. ● ● ● (-2, -1) (-3/2, -1/2) (-1, 0) (-2, 0) ● ● ● (0, 2) (-1, 1) ● (-2, 2)

14. ● ● ● ● ● (-2, -1) (-3/2, -1/2) (-1, 0)(-1/2, 1/2) (0, 1) (-2, 0) ● ● ● (0, 2) (-1, 1) ● (-2, 2)

15. ● ● ● ● ● ● ● (-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

16. ● ● ● ● ● ● ● (-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)

17. ● ● ●● ● ● ● (-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)

18. Recall that r R can be identified with { (p, q) | Pos(r (p, q) ) }. That in turn is a sub-tree Tr 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 TR of R is the set of all such finite sequences of intervals, indexed by members of R.

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

20. Theorem For every set TX of finite sequences (indexed by X) of intervals from T with the extendibility property, there is a canonical Riesz space R ⊇ X such that TX is the signed-bit representation of X. Definition An idealIr through Tr is a (non-empty) subset closed downwards and under join and with no terminal element.

21. Definition An idealthrough TR is an ideal Ir through each Tr such that finite products stay in TR: ΠrX Ir⊆ TR (X finite).

22. Definition An idealthrough TR is an ideal Ir through each Tr such that finite products stay in TR: ΠrX Ir⊆ TR (X finite). Theorem There is a canonical bijection between ideals through TR and Σ. (Recall Σ is Hom(R, ℝ).)

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

24. 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.

25. Let ℱ be the topological space of all ideals through TX, 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.

26. Let ℱ be the topological space of all ideals through TX, 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.