signed-bit representations of real numbers and the constructive stone-yosida theorem robert s....
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Signed-Bit Representations of Real Numbers and the Constructive
Stone-Yosida Theorem
Robert S. Lubarsky
and Fred Richman
Florida Atlantic University
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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.)
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Proof idea For R ⊆ ℝW, w W induces w� : R → ℝ via w�(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. █
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Proof idea For R ⊆ ℝW, w W induces w� : R → ℝ via w�(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…
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(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?
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(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 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.
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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) ) .
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Dedekind real r is located:
if p<q, then p<r or q>r.
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Dedekind real r is located:
if p<q, then p<r or q>r.
Signed-bit representation:
●(-2, 2)
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Dedekind real r is located:
if p<q, then p<r or q>r.
Signed-bit representation:
(-2, 1) ● (-1, 2) ●
●(-2, 2)
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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)
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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)
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● ● ●(-2, -1) (-3/2, -1/2) (-1, 0)
(-2, 0) ● ● (0, ●2)
(-1, 1)
●(-2, 2)
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● ● ● ● ●(-2, -1) (-3/2, -1/2) (-1, 0) (-1/2, 1/2) (0, 1)
(-2, 0) ● ● (0, ●2)
(-1, 1)
●(-2, 2)
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● ● ● ● ● ● ●(-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
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● ● ● ● ● ● ●
(-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)
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● ● ● ● ● ● ●
(-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)
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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.
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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.
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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 ideal Ir through Tr is a (non-empty) subset closed downwards and under join and with no terminal element.
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Definition An ideal through TR is an ideal Ir through each Tr such that finite products stay in TR: ΠrX Ir ⊆ TR (X finite).
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Definition An ideal through TR is an ideal Ir through each Tr such that finite products stay in TR: ΠrX Ir ⊆ TR (X finite).Theorem There is a canonical bijection between ideals through TR and Σ. (Recall Σ is Hom(R, ℝ).)
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Definition An ideal through TR is an ideal Ir through each Tr such that finite products stay in TR: ΠrX 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.)
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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.
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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.
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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.