On Meet-Continuous Dcpos

Kou, Hui; Liu, Yingming; Luo, Maokang

doi:10.1007/978-94-017-1291-0_5

Cited by 15 publications

(9 citation statements)

References 13 publications

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“…In this section, we show that a poset P is s 2 -continuous iff it is a meet s 2 -continuous and s 2 -quasicontinuous poset, generalizing the relevant characterization for dcpos (see [11,18]). [4,10].)…”

Section: Connections Between S 2 -Continuous Posets and S 2 -Quasiconmentioning

confidence: 98%

s2-Quasicontinuous posets

Zhang

2015

Theoretical Computer Science

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In this paper, we consider a common generalization of both s 2 -continuous posets and quasicontinuous domains, and we introduce new concepts of way below relations and s 2 -quasicontinuous posets. The main results are: (1) The way below relation on an s 2 -quasicontinuous poset has the interpolation property; (2) The λ 2 -topology on an s 2 -quasicontinuous poset is completely regular; (3) A poset is s 2 -continuous iff it is meet s 2 -continuous and s 2 -quasicontinuous.

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Section: Connections Between S 2 -Continuous Posets and S 2 -Quasiconmentioning

confidence: 98%

s2-Quasicontinuous posets

Zhang

2015

Theoretical Computer Science

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“…Every continuous dcpo is meet-continuous [12, Theorem III-2.11], meaning that if y ≤ sup D for any directed family D in Y , then y is in the Scott-closure of ↓ D ∩ ↓ y. (The theory of meet-continuous dcpos is due to Kou, Liu and Luo [26].) In the case at hand,…”

Section: The Baire Propertymentioning

confidence: 99%

Domain-complete and LCS-complete Spaces

Brecht

Goubault-Larrecq

Jia

et al. 2019

Electronic Notes in Theoretical Computer Science

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We study G δ subspaces of continuous dcpos, which we call domaincomplete spaces, and G δ subspaces of locally compact sober spaces, which we call LCS-complete spaces. Those include all locally compact sober spaces-in particular, all continuous dcpos-, all topologically complete spaces in the sense ofČech, and all quasi-Polish spaces-in particular, all Polish spaces. We show that LCS-complete spaces are sober, Wilker, compactly Choquet-complete, completely Baire, and -consonant-in particular, consonant; that the countably-based LCS-complete (resp., domaincomplete) spaces are the quasi-Polish spaces exactly; and that the metrizable LCS-complete (resp., domain-complete) spaces are the completely metrizable spaces. We include two applications: on LCS-complete spaces, all continuous valuations extend to measures, and sublinear previsions form a space homeomorphic to the convex Hoare powerdomain of the space of continuous valuations.Let us start with the following question: for which class of topological spaces X is it true that every (locally finite) continuous valuation on X extends to a measure on X, with its Borel σ-algebra? The question is well-studied, and Klaus Keimel and Jimmie Lawson have rounded it up nicely in [24]. A result by Mauricio Alvarez-Manilla et al.[2] (see also Theorem 5.3 of the paper by Keimel and Lawson) states that every locally compact sober space fits.Locally compact sober spaces are a pretty large class of spaces, including many non-Hausdorff spaces, and in particular all the continuous dcpos of domain theory. However, such a result will be of limited use to the ordinary measure theorist, who is used to working with Polish spaces, including such spaces as Baire space N N , which is definitely not a locally compact space.It is not too hard to extend the above theorem to the following larger class of spaces (and to drop the local finiteness assumption as well):Theorem 1.1 Let X be a (homeomorph of a) G δ subset of a locally compact sober space Y . Every continuous valuation ν on X extends to a measure on X with its Borel σ-algebra.We defer the proof of that result to Section 18. The point is that we do have a measure extension theorem on a class of spaces that contains both the continuous dcpos of domain theory and the Polish spaces of topological measure theory. We will call such spaces LCS-complete, and we are aware that this is probably not an optimal name. Topologically complete would have been a better name, if it had not been taken already [5].Another remarkable class of spaces is the class of quasi-Polish spaces, discovered and studied by the first author [7]. This one generalizes both ω-continuous dcpos and Polish spaces, and we will see in Section 5 that the class of LCScomplete spaces is a proper superclass. We will also see that there is no countably-based LCS-complete space that would fail to be quasi-Polish. Hence LCS-complete spaces can be seen as an extension of the notion of quasi-Polish spaces, and the extension is strict only for non-countably based spaces.Generally, our pur...

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“…Recall that a complete lattice L is called meet continuous if it satisfies the distributive law that binary meets distribute over directed joins. Kou, Liu and Luo extended the definition of meet continuity to general dcpos and presented a purely topological characterization (Kou et al 2003). A further generalization of meet continuity from dcpos to the setting of posets has been studied in the literature (Mao and Xu 2009).…”

Section: Introductionmentioning

confidence: 99%

θ-continuity and D_θ-completion of posets

Zhang¹,

Li²,

Jia³

2017

Math. Struct. Comp. Sci.

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We introduce a new concept of continuity of posets, called θ-continuity. Topological characterizations of θ-continuous posets are put forward. We also present two types of dcpo-completion of posets which are Dθ-completion and Ds2-completion. Connections between these notions of continuity and dcpo-completions of posets are investigated. The main results are (1) a poset P is θ-continuous iff its θ-topology lattice is completely distributive iff it is a quasi θ-continuous and meet θ-continuous poset iff its Dθ-completion is a domain; (2) the Dθ-completion of a poset B is isomorphic to a domain L iff B is a θ-embedded basis of L; (3) if a poset P is θ-continuous, then the Dθ-completion Dθ(P) is isomorphic to the round ideal completion RI(P, ≪θ).

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On Meet-Continuous Dcpos

Cited by 15 publications

References 13 publications

s2-Quasicontinuous posets

s2-Quasicontinuous posets

Domain-complete and LCS-complete Spaces

θ-continuity and D_θ-completion of posets

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On Meet-Continuous Dcpos

Cited by 15 publications

References 13 publications

s2-Quasicontinuous posets

s2-Quasicontinuous posets

Domain-complete and LCS-complete Spaces

θ-continuity and Dθ-completion of posets

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θ-continuity and D_θ-completion of posets