Embeddings into P(N)/fin and extension of automorphisms

Bella, Angelo; Dow, Alan; Hart, Klaas Pieter; Hrušák, Michael; Mill, Jan van; Ursino, Pietro

doi:10.4064/fm174-3-7

Cited by 6 publications

(6 citation statements)

References 9 publications

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“…Using this containment, it is clear that the inductive hypotheses (2) and (3) for n = ℓ follow from the inductive hypotheses (2) and (3)…”

Section: A Proof Of the Main Theoremmentioning

confidence: 91%

“…Notice that in the previous lemma, we actually showed a bit more than was stated: every countable intersection of dense open subsets of Iso(σ) contains a dense set of trivial maps. Let us point out that, by modifying the proof of Theorem 2.3 in [3], one may show that every non-empty G δ subset of H(ω * ) contains a trivial map. Thus it is not entirely surprising that the statement of Lemma 4.1 can be strengthened in this way.…”

Section: But Then Condition (Cmentioning

confidence: 99%

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The isomorphism class of the shift map

Brian¹

2019

Preprint

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The shift map is the self-homeomorphism of ω * = βω \ ω induced by the successor function n → n + 1 on ω. We prove that the isomorphism classes of σ and σ −1 cannot be separated by a Borel set in H(ω * ), the space of all self-homeomorphisms of ω * equipped with the compact-open topology.Van Douwen proved it is consistent for σ and σ −1 not to be isomorphic. Whether it is also consistent for them to be isomorphic is an open problem. The theorem stated above can be thought of as a counterpoint to van Douwen's result: while σ and σ −1 may not be isomorphic, there is no simple topological property that distinguishes them.As a relatively straightforward consequence of the main theorem, we deduce that OCA + MA implies the set of continuous images of σ fails to be Borel in H(ω * ). (Here a "continuous image" of σ is meant in the sense of topological dynamics: any h ∈ H(ω * ) such that q • σ = h • q for some continuous surjection q : ω * → ω * .) This contrasts starkly with a recent theorem of the author showing that under CH, the continuous images of σ form a closed subset of H(ω * ).

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“…Using this containment, it is clear that the inductive hypotheses (2) and (3) for n = ℓ follow from the inductive hypotheses (2) and (3)…”

Section: A Proof Of the Main Theoremmentioning

confidence: 91%

Section: But Then Condition (Cmentioning

confidence: 99%

The isomorphism class of the shift map

Brian¹

2019

Preprint

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“…A well-known theorem of Parovičenko (see e.g. [2]) says that, every Boolean algebra of size ≤ 1 embeds into P( )/Fin. So assuming CH (the continuum hypothesis), every Boolean algebra of size at most continuum can be embedded into P( )/Fin, therefore every partially ordered set of size at most continuum can be embedded into P( )/Fin.…”

Section: Zhi Yinmentioning

confidence: 99%

EMBEDDINGS OF P(ω)/Fin INTO BOREL EQUIVALENCE RELATIONS BETWEEN ℓ_p AND ℓ_q

Yin¹

2015

J. symb. log.

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“…The name hydration caves or swelling caves (Quellungshölen in German) was proposed for them [15,[21][22][23][24]. Caves of this type are very rare in the world [25][26][27].…”

Section: Introductionmentioning

confidence: 99%

Morphology of Dome- and Tepee-Like Landforms Generated by Expansive Hydration of Weathering Anhydrite: A Case Study at Dingwall, Nova Scotia, Canada

et al. 2022

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The gypsum-anhydrite rocks in the abandoned quarry at Dingwall (Nova Scotia, Canada) are subjected to physical and chemical weathering, including hydration of the anhydrite, i.e., its transformation into secondary gypsum under the influence of water. This process is known to lead to the localized volume increase of the rock and the formation of spectacular hydration landforms: domes, tepees and ridges. Cavities appearing in the interior of these domes are often unique hydration caves (Quellungshöhlen in German). For the first time, this paper gives detailed geomorphometric characteristics of the 77 dome- and tepee-like hydration landforms growing today at Dingwall based on their digital surface models and orthophotomaps, made with the method of photogrammetry integrated with direct measurements. The length of hydration landforms varies from 1.86 to 23.05 m and the relative height varies from 0.33 to 2.09 m. Their approximate shape in a plan view varies from nearly circular, through oval, to elongated with a length-to-width ratio rarely exceeding 5:2. Length, width and relative height are characterized by moderate mutual correlation with proportional relations expressed by linear equations, testifying that the hydration landforms generally preserve the same or very similar shape independent of their sizes. The averaged thickness of the detached rock layer ranges from 6 to 46 cm. The size of the forms seems to depend on this thickness—the forms larger in extent (longer) generally have a thicker detached rock layer. Master (and other) joints and, to a lesser extent, layering in the bedrock influence the development of hydration landforms, particularly by controlling the place where the entrances are open to internal cavities or caves. Three structural types of the bedrock influencing the growth of hydration forms were recognized: with master joints, with layering and with both of them. The latter type of bedrock has the most complex impact on the morphology of hydration landforms because it depends on the number of master joint sets and the mutual orientation of joints and layering, which are changeable across the quarry. The durability of the hydration forms over time depends, among others, on the density of fractures in the detached rock layer.

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Embeddings into P(N)/fin and extension of automorphisms

Cited by 6 publications

References 9 publications

The isomorphism class of the shift map

The isomorphism class of the shift map

EMBEDDINGS OF P(ω)/Fin INTO BOREL EQUIVALENCE RELATIONS BETWEEN ℓ_p AND ℓ_q

Morphology of Dome- and Tepee-Like Landforms Generated by Expansive Hydration of Weathering Anhydrite: A Case Study at Dingwall, Nova Scotia, Canada

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Embeddings into P(N)/fin and extension of automorphisms

Cited by 6 publications

References 9 publications

The isomorphism class of the shift map

The isomorphism class of the shift map

EMBEDDINGS OF P(ω)/Fin INTO BOREL EQUIVALENCE RELATIONS BETWEEN ℓp AND ℓq

Morphology of Dome- and Tepee-Like Landforms Generated by Expansive Hydration of Weathering Anhydrite: A Case Study at Dingwall, Nova Scotia, Canada

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EMBEDDINGS OF P(ω)/Fin INTO BOREL EQUIVALENCE RELATIONS BETWEEN ℓ_p AND ℓ_q