Oz in pointfree topology

Banaschewski, Bernhard; Dube, Themba; Gilmour, Christopher; Walters-Wayland, Joanne

doi:10.2989/qm.2009.32.2.4.797

Cited by 14 publications

(8 citation statements)

References 6 publications

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“…Thus, if X is a Tychonoff space and U a cozero-set of X, then βX cl βX U = r OX (U * ). Now it should be clear from this that indeed our condition (7) generalizes condition (5) in [8,Theorem 2.5] because β(OX) ∼ = O(βX).…”

Section: Sub(βl)mentioning

confidence: 75%

“…It is however not immediate that our item (7) is a generalization of item (5) in [8,Theorem 2.4]. Let us show that it is.…”

Section: Sub(βl)mentioning

confidence: 94%

“…We recall from [5] that an Oz-frame is a frame in which every pseudocomplement is a cozero element. In [19], a frame L is called perfectly normal in case it is normal and Coz L = L. Every perfectly normal frame (and hence every metrizable frame) is an Oz-frame.…”

Section: Sub(βl)mentioning

confidence: 99%

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Covering maximal ideals with minimal primes

Dube

Ighedo

2015

Algebra Univers.

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A ring is a UMP-ring if every maximal ideal in the ring is the union of the minimal prime ideals it contains. Banerjee, Ghosh and Henriksen have characterized Tychonoff spaces X for which C(X) is a UMP-ring. One of the characterizations is that every singleton of βX is what is called a nearly round subset. In this article, we define nearly round quotient maps, and use them to characterize completely regular frames L for which RL is a UMP-ring. All such frames are almost P -frames, and an Oz-frame is of this kind precisely when it is an almost P -frame. If L is perfectly normal (and hence if L is metrizable), then RL is a UMP-ring if and only if L is Boolean. If A is a UMP-ring which is a Q-algebra, then every ideal of A, when viewed as a ring in its own right, is a UMP-ring.

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Section: Sub(βl)mentioning

confidence: 75%

“…It is however not immediate that our item (7) is a generalization of item (5) in [8,Theorem 2.4]. Let us show that it is.…”

Section: Sub(βl)mentioning

confidence: 94%

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Covering maximal ideals with minimal primes

Dube

Ighedo

2015

Algebra Univers.

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“…Locales where each regular element is a cozero element are called Oz locales and are the natural pointfree counterpart of Oz spaces. They were introduced in [2] and further studied in [1]. Recall that, by Proposition 2.3 of [1], a locale is Oz if and only if every element of the form ∨ n∈N (a n ∧ b n ) with all a n and b n being regular is a countable union of elements well inside it.…”

Section: Variants Of Normality and Oz Localesmentioning

confidence: 99%

“…One interesting observation is that perfect and mildly normal locales coincide with the so-called Oz locales extensively studied in the last decade (cf. [1,2,5,6]).…”

Section: Introductionmentioning

confidence: 99%

Perfectness in locales

García

Kubiak

Picado

2017

Quaestiones Mathematicae

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This paper makes a comparison between two notions of perfectness for locales which come as direct reformulations of the two equivalent topological definitions of perfectness. These reformulations are no longer equivalent. It will be documented that a locale may appropriately be called perfect if each of its open sublocales is a join of countably many closed sublocales. Certain circumstances are exhibited in which both reformulations coincide. This paper also studies perfectness in mildly normal locales. It is shown that perfect and mildly normal locales coincide with the Oz locales extensively studied in the last decade.

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Picado

Pultr

2021

Separation in Point-Free Topology

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Oz in pointfree topology

Cited by 14 publications

References 6 publications

Covering maximal ideals with minimal primes

Covering maximal ideals with minimal primes

Perfectness in locales

More on Normality and Related Properties

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