. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Background The Frame Theoretic Analog Some Notes A Point-Free Version of the Alexandroff-Hausdorff Theorem Julio César Urenda Castañeda 1 , Francisco Ávila 1 , Ángel Zaldívar Corichi 2 1 University of Texas at El Paso, 2 Universidad de Guadalajara BLAST at NMSU June 12, 2021 Julio César Urenda Castañeda 1 , Francisco Ávila 1 , Ángel Zaldívar Corichi 2 UTEP
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Background The Frame Theoretic Analog Some Notes
A Point-Free Version of theAlexandroff-Hausdorff Theorem
Julio César Urenda Castañeda1, Francisco Ávila1,Ángel Zaldívar Corichi2
1University of Texas at El Paso, 2Universidad de Guadalajara
BLAST at NMSU
June 12, 2021
Julio César Urenda Castañeda1, Francisco Ávila1, Ángel Zaldívar Corichi2 UTEP
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Background The Frame Theoretic Analog Some Notes
Overview
1 BackgroundThe Cantor SetProperties of The Cantor Set
2 The Frame Theoretic AnalogSome Needed ElementsFraming Up
3 Some Notes
Julio César Urenda Castañeda1, Francisco Ávila1, Ángel Zaldívar Corichi2 UTEP
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Background The Frame Theoretic Analog Some Notes
The Cantor Set
Classical Construction
Start with the closed interval [0, 1]
Remove the open middle third to get [0, 1/3] ∪ [2/3, 1]Now remove the respective open middle thirds from [0, 1/3]and [2/3, 1] to get [0, 1/9]∪ [2/9, 3/9]∪ [6/9, 7/9]∪ [8/9, 9/9]Repeat ...Ad Infinitum
Julio César Urenda Castañeda1, Francisco Ávila1, Ángel Zaldívar Corichi2 UTEP
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Background The Frame Theoretic Analog Some Notes
The Cantor Set
Classical Construction
Start with the closed interval [0, 1]Remove the open middle third to get [0, 1/3] ∪ [2/3, 1]
Now remove the respective open middle thirds from [0, 1/3]and [2/3, 1] to get [0, 1/9]∪ [2/9, 3/9]∪ [6/9, 7/9]∪ [8/9, 9/9]Repeat ...Ad Infinitum
Julio César Urenda Castañeda1, Francisco Ávila1, Ángel Zaldívar Corichi2 UTEP
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Background The Frame Theoretic Analog Some Notes
The Cantor Set
Classical Construction
Start with the closed interval [0, 1]Remove the open middle third to get [0, 1/3] ∪ [2/3, 1]Now remove the respective open middle thirds from [0, 1/3]and [2/3, 1] to get [0, 1/9]∪ [2/9, 3/9]∪ [6/9, 7/9]∪ [8/9, 9/9]
Repeat ...Ad Infinitum
Julio César Urenda Castañeda1, Francisco Ávila1, Ángel Zaldívar Corichi2 UTEP
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Background The Frame Theoretic Analog Some Notes
The Cantor Set
Classical Construction
Start with the closed interval [0, 1]Remove the open middle third to get [0, 1/3] ∪ [2/3, 1]Now remove the respective open middle thirds from [0, 1/3]and [2/3, 1] to get [0, 1/9]∪ [2/9, 3/9]∪ [6/9, 7/9]∪ [8/9, 9/9]Repeat ...
Ad Infinitum
Julio César Urenda Castañeda1, Francisco Ávila1, Ángel Zaldívar Corichi2 UTEP
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Background The Frame Theoretic Analog Some Notes
The Cantor Set
Classical Construction
Start with the closed interval [0, 1]Remove the open middle third to get [0, 1/3] ∪ [2/3, 1]Now remove the respective open middle thirds from [0, 1/3]and [2/3, 1] to get [0, 1/9]∪ [2/9, 3/9]∪ [6/9, 7/9]∪ [8/9, 9/9]Repeat ...Ad Infinitum
Julio César Urenda Castañeda1, Francisco Ávila1, Ángel Zaldívar Corichi2 UTEP
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Background The Frame Theoretic Analog Some Notes
The Cantor Set
In a More Pleasant Notation
Of course the above construction can be summarized bydeclaring the Cantor set C as the subset of the closed unitinterval
[0, 1]−∞⋃
n=0
3n−1⋃k=0
(3k + 13n+1 ,
3k + 23n+1
)
Equivalently C consists of real numbers whose ternaryexpansion contains only zeroes or twos, in other words
C =
{x ∈ [0, 1] | x =
∞∑n=0
bn3n+1 , bn = 0, 2
}
Julio César Urenda Castañeda1, Francisco Ávila1, Ángel Zaldívar Corichi2 UTEP
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Background The Frame Theoretic Analog Some Notes
The Cantor Set
In a More Pleasant Notation
Of course the above construction can be summarized bydeclaring the Cantor set C as the subset of the closed unitinterval
[0, 1]−∞⋃
n=0
3n−1⋃k=0
(3k + 13n+1 ,
3k + 23n+1
)
Equivalently C consists of real numbers whose ternaryexpansion contains only zeroes or twos, in other words
C =
{x ∈ [0, 1] | x =
∞∑n=0
bn3n+1 , bn = 0, 2
}
Julio César Urenda Castañeda1, Francisco Ávila1, Ángel Zaldívar Corichi2 UTEP
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Background The Frame Theoretic Analog Some Notes
Properties of The Cantor Set
Some Properties of C
The Cantor Set is
compact,metric,perfect, andtotally disconnected
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Background The Frame Theoretic Analog Some Notes
Properties of The Cantor Set
Some Properties of C
The Cantor Set iscompact,
metric,perfect, andtotally disconnected
Julio César Urenda Castañeda1, Francisco Ávila1, Ángel Zaldívar Corichi2 UTEP
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Background The Frame Theoretic Analog Some Notes
Properties of The Cantor Set
Some Properties of C
The Cantor Set iscompact,metric,
perfect, andtotally disconnected
Julio César Urenda Castañeda1, Francisco Ávila1, Ángel Zaldívar Corichi2 UTEP
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Background The Frame Theoretic Analog Some Notes
Properties of The Cantor Set
Some Properties of C
The Cantor Set iscompact,metric,perfect, and
totally disconnected
Julio César Urenda Castañeda1, Francisco Ávila1, Ángel Zaldívar Corichi2 UTEP
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Background The Frame Theoretic Analog Some Notes
Properties of The Cantor Set
Some Properties of C
The Cantor Set iscompact,metric,perfect, andtotally disconnected
Julio César Urenda Castañeda1, Francisco Ávila1, Ángel Zaldívar Corichi2 UTEP
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Background The Frame Theoretic Analog Some Notes
Properties of The Cantor Set
Classical Results
TheoremThe Cantor set is up to homeomorphism the only totallydisconnected, perfect compact metric space.
CorollaryThe Cantor set is homeomorphic to a coutable product of copies ofthe discrete space {0, 1}
CorollaryThe Cantor set is homeomorphic to a coutable product of copies ofitself.
Julio César Urenda Castañeda1, Francisco Ávila1, Ángel Zaldívar Corichi2 UTEP
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Background The Frame Theoretic Analog Some Notes
Properties of The Cantor Set
Classical Results
TheoremThe Cantor set is up to homeomorphism the only totallydisconnected, perfect compact metric space.
CorollaryThe Cantor set is homeomorphic to a coutable product of copies ofthe discrete space {0, 1}
CorollaryThe Cantor set is homeomorphic to a coutable product of copies ofitself.
Julio César Urenda Castañeda1, Francisco Ávila1, Ángel Zaldívar Corichi2 UTEP
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Background The Frame Theoretic Analog Some Notes
Properties of The Cantor Set
Classical Results
TheoremThe Cantor set is up to homeomorphism the only totallydisconnected, perfect compact metric space.
CorollaryThe Cantor set is homeomorphic to a coutable product of copies ofthe discrete space {0, 1}
CorollaryThe Cantor set is homeomorphic to a coutable product of copies ofitself.
Julio César Urenda Castañeda1, Francisco Ávila1, Ángel Zaldívar Corichi2 UTEP
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Background The Frame Theoretic Analog Some Notes
Properties of The Cantor Set
”The” Classical Result
The Alexandroff-Hausdorff TheoremEvery compact metric space X is a continuous image of the Cantorset.
Equivalently...For every compact metric space X there is a continuous ontofunction f : C → X
Julio César Urenda Castañeda1, Francisco Ávila1, Ángel Zaldívar Corichi2 UTEP
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Background The Frame Theoretic Analog Some Notes
Properties of The Cantor Set
”The” Classical Result
The Alexandroff-Hausdorff TheoremEvery compact metric space X is a continuous image of the Cantorset.
Equivalently...For every compact metric space X there is a continuous ontofunction f : C → X
Julio César Urenda Castañeda1, Francisco Ávila1, Ángel Zaldívar Corichi2 UTEP
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Background The Frame Theoretic Analog Some Notes
Some Needed Elements
A Particularization
Another Model of CThe set Z2 of the 2-adic integers is homeomorphic to the Cantorset, so there is continuous map from Z2 onto [0, 1]
A MapIn fact, the map ϕ : Z2 → [0, 1] where
ϕ
∑i≥0
bi2i
=∑i≥0
bi2i+1 .
is continuous and onto.
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Background The Frame Theoretic Analog Some Notes
Some Needed Elements
A Particularization
Another Model of CThe set Z2 of the 2-adic integers is homeomorphic to the Cantorset, so there is continuous map from Z2 onto [0, 1]
A MapIn fact, the map ϕ : Z2 → [0, 1] where
ϕ
∑i≥0
bi2i
=∑i≥0
bi2i+1 .
is continuous and onto.
Julio César Urenda Castañeda1, Francisco Ávila1, Ángel Zaldívar Corichi2 UTEP
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Background The Frame Theoretic Analog Some Notes
Some Needed Elements
Some Properties of ϕ
PropositionLet u be an integer coprime to 2, g a nonnegative integer where0 < u < 2g, then ϕ−1 (u/2g) has exactly two elements.
LemmaFor ϕ : Z2 → [0, 1] ⊂ R as above, we have
ϕ−1(−∞, u/2g) =⋃{
B(
a, 12g+k
)| ϕ(a) < u
2g − 12g+k , k ≥ 0
}
ϕ−1(u/2g,+∞) =⋃{
B(
a, 12g+k
)| ϕ(a) > u
2g +1
2g+k , k ≥ 0}
for every integer u and nonnegative integer g.
Julio César Urenda Castañeda1, Francisco Ávila1, Ángel Zaldívar Corichi2 UTEP
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Background The Frame Theoretic Analog Some Notes
Some Needed Elements
Some Properties of ϕ
PropositionLet u be an integer coprime to 2, g a nonnegative integer where0 < u < 2g, then ϕ−1 (u/2g) has exactly two elements.
LemmaFor ϕ : Z2 → [0, 1] ⊂ R as above, we have
ϕ−1(−∞, u/2g) =⋃{
B(
a, 12g+k
)| ϕ(a) < u
2g − 12g+k , k ≥ 0
}
ϕ−1(u/2g,+∞) =⋃{
B(
a, 12g+k
)| ϕ(a) > u
2g +1
2g+k , k ≥ 0}
for every integer u and nonnegative integer g.Julio César Urenda Castañeda1, Francisco Ávila1, Ángel Zaldívar Corichi2 UTEP
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Background The Frame Theoretic Analog Some Notes
Framing Up
A Couple of Free Frames
The Frame of RealsThe frame of the reals L(R) is generated by (p,−) and (−, q) withp, q ∈ D, for D a countably dense subset of R, subject to thefollowing relations.(1) (p,−) ∧ (−, q) = 0 whenever p ≥ q.(2) (p,−) ∨ (−, q) = 1 whenever p < q.(3) (p,−) =
∨{(r,−) | r > p}.
(4) (−, q) =∨{(−, s) | s < q}.
(5)∨{(p,−) | p ∈ D} = 1
(6)∨{(−, q) | q ∈ D} = 1
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Background The Frame Theoretic Analog Some Notes
Framing Up
A Couple of Free Frames (cont.)
The Frame of p-adic IntegersLet L(Zp) be the frame generated by the elements Br(a), wherea ∈ Z and r ∈ |Z| := {p−n+1 | n ∈ N}, subject to the followingrelations:(1) Br(a) ∧ Bs(b) = 0 whenever |a − b|p ≥ r and s ≤ r.(2) 1 =
∨{Br(a) : a ∈ Z, r ∈ |Z|
}.
(3) Br(a) =∨{
Bs(b) : |a − b|p < r, s < r, b ∈ Z, s ∈ |Z|}
.
NotationFor every generator B of L(Z2), there is a unique natural numbern with B = B1/2n(a) for some 2-adic integer a. For simplicity, wewrite B(a, n) instead of B1/2n(a).
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Background The Frame Theoretic Analog Some Notes
Framing Up
A Couple of Free Frames (cont.)
The Frame of p-adic IntegersLet L(Zp) be the frame generated by the elements Br(a), wherea ∈ Z and r ∈ |Z| := {p−n+1 | n ∈ N}, subject to the followingrelations:(1) Br(a) ∧ Bs(b) = 0 whenever |a − b|p ≥ r and s ≤ r.(2) 1 =
∨{Br(a) : a ∈ Z, r ∈ |Z|
}.
(3) Br(a) =∨{
Bs(b) : |a − b|p < r, s < r, b ∈ Z, s ∈ |Z|}
.
NotationFor every generator B of L(Z2), there is a unique natural numbern with B = B1/2n(a) for some 2-adic integer a. For simplicity, wewrite B(a, n) instead of B1/2n(a).
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Background The Frame Theoretic Analog Some Notes
Framing Up
A Frame Map
Defining on GeneratorsFollowing properties of ϕ : Z2 → [0, 1], we defineϕ̃ : L[0, 1] → L(Z2) on generators of L[0, 1] where
ϕ̃(−, u/2g) =∨{
B(a, g + k) | ϕ(a) < u2g − 1
2g+k and k ≥ 0}
ϕ̃(u/2g,−) =∨{
B(a, g + k) | ϕ(a) > u2g +
12g+k and k ≥ 0
}
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Background The Frame Theoretic Analog Some Notes
Framing Up
On ϕ̃
LemmaLet q be a dyadic rational, then ϕ̃(−, q) = 1 if and only if q > 1.
Lemma.For p and q dyadic rationals, ϕ̃(p, q) = 1 if and only if p < 0 andq > 1.
Cor.For p, q, r, and s dyadic rationals, ϕ̃ ((p, q) ∨ (r, s)) = 1 if and onlyif (p, q) ∨ (r, s) = 1.
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Background The Frame Theoretic Analog Some Notes
Framing Up
On ϕ̃
LemmaLet q be a dyadic rational, then ϕ̃(−, q) = 1 if and only if q > 1.
Lemma.For p and q dyadic rationals, ϕ̃(p, q) = 1 if and only if p < 0 andq > 1.
Cor.For p, q, r, and s dyadic rationals, ϕ̃ ((p, q) ∨ (r, s)) = 1 if and onlyif (p, q) ∨ (r, s) = 1.
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Background The Frame Theoretic Analog Some Notes
Framing Up
On ϕ̃
LemmaLet q be a dyadic rational, then ϕ̃(−, q) = 1 if and only if q > 1.
Lemma.For p and q dyadic rationals, ϕ̃(p, q) = 1 if and only if p < 0 andq > 1.
Cor.For p, q, r, and s dyadic rationals, ϕ̃ ((p, q) ∨ (r, s)) = 1 if and onlyif (p, q) ∨ (r, s) = 1.
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Background The Frame Theoretic Analog Some Notes
Framing Up
On ϕ̃ (cont.)
First Conclusionϕ̃ is a frame morphism.
LemmaIf ϕ̃(u/2g, v/2h) = 0 then (u/2g, v/2h) = 0.
Second Conclusionϕ̃ is an injective frame morphism.
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Background The Frame Theoretic Analog Some Notes
Framing Up
On ϕ̃ (cont.)
First Conclusionϕ̃ is a frame morphism.
LemmaIf ϕ̃(u/2g, v/2h) = 0 then (u/2g, v/2h) = 0.
Second Conclusionϕ̃ is an injective frame morphism.
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Background The Frame Theoretic Analog Some Notes
Framing Up
On ϕ̃ (cont.)
First Conclusionϕ̃ is a frame morphism.
LemmaIf ϕ̃(u/2g, v/2h) = 0 then (u/2g, v/2h) = 0.
Second Conclusionϕ̃ is an injective frame morphism.
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Background The Frame Theoretic Analog Some Notes
Framing Up
On ϕ̃ (cont.)
First Conclusionϕ̃ is a frame morphism.
LemmaIf ϕ̃(u/2g, v/2h) = 0 then (u/2g, v/2h) = 0.
Second Conclusionϕ̃ is an injective frame morphism.
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Background The Frame Theoretic Analog Some Notes
Framing Up
Coproducts
An ExtensionLet φ : L → M a dense morphism between compact regular framesthen φ induces an injective frame morphism
φ♭ : KL → KM
Cor.There is an injective frame morphism
ϕ̄ : KL[0, 1] → KL(Z2)
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Background The Frame Theoretic Analog Some Notes
Framing Up
Coproducts
An ExtensionLet φ : L → M a dense morphism between compact regular framesthen φ induces an injective frame morphism
φ♭ : KL → KM
Cor.There is an injective frame morphism
ϕ̄ : KL[0, 1] → KL(Z2)
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Background The Frame Theoretic Analog Some Notes
Framing Up
Cantor and Hilbert
LemmaAny countable coproduct of L(Z2) is isomorphic to L(Z2).
DefinitionThe Hilbert cube frame, H, is NL[0, 1].
Prop.There is an injective frame morphism η : H → L(Z2)
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Background The Frame Theoretic Analog Some Notes
Framing Up
Cantor and Hilbert
LemmaAny countable coproduct of L(Z2) is isomorphic to L(Z2).
DefinitionThe Hilbert cube frame, H, is NL[0, 1].
Prop.There is an injective frame morphism η : H → L(Z2)
Julio César Urenda Castañeda1, Francisco Ávila1, Ángel Zaldívar Corichi2 UTEP
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Background The Frame Theoretic Analog Some Notes
Framing Up
Cantor and Hilbert
LemmaAny countable coproduct of L(Z2) is isomorphic to L(Z2).
DefinitionThe Hilbert cube frame, H, is NL[0, 1].
Prop.There is an injective frame morphism η : H → L(Z2)
Julio César Urenda Castañeda1, Francisco Ávila1, Ángel Zaldívar Corichi2 UTEP
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Background The Frame Theoretic Analog Some Notes
Framing Up
Urysohn and Retracts
TheoremThe following are equivalent for a frame L:
a) L is regular and has a countable basis.b) L is a quotient of a H.
DefinitionLet L and M frames, we say that L is retract of M if, there is asurjective frame morphism ρ : M → L such that there exists amorphism ι : L → M such that ρι = idL.
Julio César Urenda Castañeda1, Francisco Ávila1, Ángel Zaldívar Corichi2 UTEP
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Background The Frame Theoretic Analog Some Notes
Framing Up
Urysohn and Retracts
TheoremThe following are equivalent for a frame L:
a) L is regular and has a countable basis.b) L is a quotient of a H.
DefinitionLet L and M frames, we say that L is retract of M if, there is asurjective frame morphism ρ : M → L such that there exists amorphism ι : L → M such that ρι = idL.
Julio César Urenda Castañeda1, Francisco Ávila1, Ángel Zaldívar Corichi2 UTEP
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Background The Frame Theoretic Analog Some Notes
Framing Up
On L(Z2)
LemmaLet b = B(a, n) and x be in L(Z2) where b is a basic element andx < 1. If b ̸≤ x, then there is a basic element b′ = B(a′, n + 1) < bsuch that b′ ̸≤ x.
LemmaLet B the set of basic elements of L(Z2) and x < 1, then there is afunction f : B → B such that f(b) ̸≤ x for all b ∈ B. Moreover, fcan be chosen so that f preserves the “radius” of each ball in B.
Julio César Urenda Castañeda1, Francisco Ávila1, Ángel Zaldívar Corichi2 UTEP
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Background The Frame Theoretic Analog Some Notes
Framing Up
On L(Z2)
LemmaLet b = B(a, n) and x be in L(Z2) where b is a basic element andx < 1. If b ̸≤ x, then there is a basic element b′ = B(a′, n + 1) < bsuch that b′ ̸≤ x.
LemmaLet B the set of basic elements of L(Z2) and x < 1, then there is afunction f : B → B such that f(b) ̸≤ x for all b ∈ B. Moreover, fcan be chosen so that f preserves the “radius” of each ball in B.
Julio César Urenda Castañeda1, Francisco Ávila1, Ángel Zaldívar Corichi2 UTEP
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Background The Frame Theoretic Analog Some Notes
Framing Up
Closed Quotient
Prop.Let x be in L(Z2) where x < 1, and let B the set of basic elementsof L(Z2). The function h : B → L(Z2) given by
h(b) =∨
f−1({b})
where f is as above extends to a frame endomorphism H on L(Z2)with x =
∨H−1({0}).
Cor.Every non-trivial closed quotient of L(Z2) is a retraction of L(Z2).
Julio César Urenda Castañeda1, Francisco Ávila1, Ángel Zaldívar Corichi2 UTEP
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Background The Frame Theoretic Analog Some Notes
Framing Up
Closed Quotient
Prop.Let x be in L(Z2) where x < 1, and let B the set of basic elementsof L(Z2). The function h : B → L(Z2) given by
h(b) =∨
f−1({b})
where f is as above extends to a frame endomorphism H on L(Z2)with x =
∨H−1({0}).
Cor.Every non-trivial closed quotient of L(Z2) is a retraction of L(Z2).
Julio César Urenda Castañeda1, Francisco Ávila1, Ángel Zaldívar Corichi2 UTEP
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Background The Frame Theoretic Analog Some Notes
Framing Up
Our Main Result
TheoremFor every compact metrizable frame L there is an embedding
ϑ : L → L(Z2).
In other words every compact metrizable frame can be identified asa subframe of the Cantor frame L(Z2).
Julio César Urenda Castañeda1, Francisco Ávila1, Ángel Zaldívar Corichi2 UTEP
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Background The Frame Theoretic Analog Some Notes
Framing Up
Proof
Start with a compact metrizable frame L
We have a surjective frame morphism ρ : H → LAlso an embedding θ : H → L(Z2)
Diagrammatically
H L(Z2)
L
θ
ρ
Julio César Urenda Castañeda1, Francisco Ávila1, Ángel Zaldívar Corichi2 UTEP
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Background The Frame Theoretic Analog Some Notes
Framing Up
Proof
Start with a compact metrizable frame LWe have a surjective frame morphism ρ : H → L
Also an embedding θ : H → L(Z2)
Diagrammatically
H L(Z2)
L
θ
ρ
Julio César Urenda Castañeda1, Francisco Ávila1, Ángel Zaldívar Corichi2 UTEP
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Background The Frame Theoretic Analog Some Notes
Framing Up
Proof
Start with a compact metrizable frame LWe have a surjective frame morphism ρ : H → LAlso an embedding θ : H → L(Z2)
Diagrammatically
H L(Z2)
L
θ
ρ
Julio César Urenda Castañeda1, Francisco Ávila1, Ángel Zaldívar Corichi2 UTEP
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Background The Frame Theoretic Analog Some Notes
Framing Up
Proof
Start with a compact metrizable frame LWe have a surjective frame morphism ρ : H → LAlso an embedding θ : H → L(Z2)
Diagrammatically
H L(Z2)
L
θ
ρ
Julio César Urenda Castañeda1, Francisco Ávila1, Ángel Zaldívar Corichi2 UTEP
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Background The Frame Theoretic Analog Some Notes
Framing Up
Proof (cont.)
We identify L ∼=↑ x for some x ∈ H
Thus under θ, we have θ(↑ x) is closed in L(Z2), that is,θ(↑ x) =↑ θ(x)We have a commutative diagram
H L(Z2
L ↑ θ(x)
θ
ρ u∼=
We thus have an injective morphism ↑ θ(x) → L(Z2)
Julio César Urenda Castañeda1, Francisco Ávila1, Ángel Zaldívar Corichi2 UTEP
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Background The Frame Theoretic Analog Some Notes
Framing Up
Proof (cont.)
We identify L ∼=↑ x for some x ∈ H
Thus under θ, we have θ(↑ x) is closed in L(Z2), that is,θ(↑ x) =↑ θ(x)
We have a commutative diagram
H L(Z2
L ↑ θ(x)
θ
ρ u∼=
We thus have an injective morphism ↑ θ(x) → L(Z2)
Julio César Urenda Castañeda1, Francisco Ávila1, Ángel Zaldívar Corichi2 UTEP
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Background The Frame Theoretic Analog Some Notes
Framing Up
Proof (cont.)
We identify L ∼=↑ x for some x ∈ H
Thus under θ, we have θ(↑ x) is closed in L(Z2), that is,θ(↑ x) =↑ θ(x)We have a commutative diagram
H L(Z2
L ↑ θ(x)
θ
ρ u∼=
We thus have an injective morphism ↑ θ(x) → L(Z2)
Julio César Urenda Castañeda1, Francisco Ávila1, Ángel Zaldívar Corichi2 UTEP
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Background The Frame Theoretic Analog Some Notes
Framing Up
Proof (cont.)
We identify L ∼=↑ x for some x ∈ H
Thus under θ, we have θ(↑ x) is closed in L(Z2), that is,θ(↑ x) =↑ θ(x)We have a commutative diagram
H L(Z2
L ↑ θ(x)
θ
ρ u∼=
We thus have an injective morphism ↑ θ(x) → L(Z2)
Julio César Urenda Castañeda1, Francisco Ávila1, Ángel Zaldívar Corichi2 UTEP
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Background The Frame Theoretic Analog Some Notes
L(Zp) Generated by Words
Many of the above results were inspired by identifying generatorsas members a language. In fact, the basic elements of L(Zp) canbe finite words over an alphabet with p symbols subject to a prefixrelation:
w1 ≤ w2 if and only if w1 = w2 ◦ s for some finite word s,in this case the empty word acts as the top of the frame.
Example0011, 0010 ≤ 001, but 00 ̸≤ 01. In fact 00 ∧ 01 = ⊥ and00 ∨ 01 = 0
So at least, computations of finite suprema and infima are notterribly complicated.
Julio César Urenda Castañeda1, Francisco Ávila1, Ángel Zaldívar Corichi2 UTEP
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Background The Frame Theoretic Analog Some Notes
L(Zp) Generated by Words
Many of the above results were inspired by identifying generatorsas members a language. In fact, the basic elements of L(Zp) canbe finite words over an alphabet with p symbols subject to a prefixrelation: w1 ≤ w2 if and only if w1 = w2 ◦ s for some finite word s,in this case the empty word acts as the top of the frame.
Example0011, 0010 ≤ 001, but 00 ̸≤ 01. In fact 00 ∧ 01 = ⊥ and00 ∨ 01 = 0
So at least, computations of finite suprema and infima are notterribly complicated.
Julio César Urenda Castañeda1, Francisco Ávila1, Ángel Zaldívar Corichi2 UTEP
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Background The Frame Theoretic Analog Some Notes
L(Zp) Generated by Words
Many of the above results were inspired by identifying generatorsas members a language. In fact, the basic elements of L(Zp) canbe finite words over an alphabet with p symbols subject to a prefixrelation: w1 ≤ w2 if and only if w1 = w2 ◦ s for some finite word s,in this case the empty word acts as the top of the frame.
Example0011, 0010 ≤ 001, but 00 ̸≤ 01. In fact 00 ∧ 01 = ⊥ and00 ∨ 01 = 0
So at least, computations of finite suprema and infima are notterribly complicated.
Julio César Urenda Castañeda1, Francisco Ávila1, Ángel Zaldívar Corichi2 UTEP
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Background The Frame Theoretic Analog Some Notes
L(Zp) Generated by Words
Many of the above results were inspired by identifying generatorsas members a language. In fact, the basic elements of L(Zp) canbe finite words over an alphabet with p symbols subject to a prefixrelation: w1 ≤ w2 if and only if w1 = w2 ◦ s for some finite word s,in this case the empty word acts as the top of the frame.
Example0011, 0010 ≤ 001, but 00 ̸≤ 01. In fact 00 ∧ 01 = ⊥ and00 ∨ 01 = 0
So at least, computations of finite suprema and infima are notterribly complicated.
Julio César Urenda Castañeda1, Francisco Ávila1, Ángel Zaldívar Corichi2 UTEP
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Background The Frame Theoretic Analog Some Notes
A Concrete Map
Since L(Z2) is generated by finite binary words, say on {0, 1}, andL(Z3) by finite ternary words, say on {a, b, c}.
Now for eachternary word replace every instance a by 00, those of b by 01, andeach of c by 1. This extends to a map from L(Z3) to L(Z2). Infact, it is a frame isomorphism.
Julio César Urenda Castañeda1, Francisco Ávila1, Ángel Zaldívar Corichi2 UTEP
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Background The Frame Theoretic Analog Some Notes
A Concrete Map
Since L(Z2) is generated by finite binary words, say on {0, 1}, andL(Z3) by finite ternary words, say on {a, b, c}. Now for eachternary word replace every instance a by 00, those of b by 01, andeach of c by 1. This extends to a map from L(Z3) to L(Z2).
Infact, it is a frame isomorphism.
Julio César Urenda Castañeda1, Francisco Ávila1, Ángel Zaldívar Corichi2 UTEP
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Background The Frame Theoretic Analog Some Notes
A Concrete Map
Since L(Z2) is generated by finite binary words, say on {0, 1}, andL(Z3) by finite ternary words, say on {a, b, c}. Now for eachternary word replace every instance a by 00, those of b by 01, andeach of c by 1. This extends to a map from L(Z3) to L(Z2). Infact, it is a frame isomorphism.
Julio César Urenda Castañeda1, Francisco Ávila1, Ángel Zaldívar Corichi2 UTEP
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Background The Frame Theoretic Analog Some Notes
Julio César Urenda Castañeda1, Francisco Ávila1, Ángel Zaldívar Corichi2 UTEP
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Background The Frame Theoretic Analog Some Notes
Thank You!
Julio César Urenda Castañeda1, Francisco Ávila1, Ángel Zaldívar Corichi2 UTEP
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