The P-frame reflection of a completely regular frame

Ball, Richard N.; Walters-Wayland, Joanne; Zenk, Eric Richard

doi:10.1016/j.topol.2011.06.013

Cited by 16 publications

(11 citation statements)

References 12 publications

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“…The equivalence of (1) and (2) is an immediate application of 1.4.2. So suppose these conditions hold in L, and consider a b c for which no element d can be found to satisfy (3). We claim that A a, where A a pb Ò cq.…”

Section: Proposition (Proposition 110 [2]mentioning

confidence: 99%

“…For the claim could fail only if a d ¤ A for some d L, in which case b Ò c a Ò d, contrary to assumption. But a is not the exact meet of A, for A b Ò c while a b Ò c. This cannot happen by (2), so we are forced to conclude that an element d can be found satisfying (3).…”

Section: Proposition (Proposition 110 [2]mentioning

confidence: 99%

“…Indeed, this is taken as the frame definition: a completely regular frame L is said to be a P -frame if each cozero element is complemented, i.e., if its cozero part coz L is a Boolean σ-frame. (See [3] for several characterizations of P -frames, together with information on their role in the general theory.) Perhaps the handiest of the several well-known characterizations of P -spaces is that a countable intersection of open sets remains open.…”

Section: Exact and Strongly Exact Meets In Framesmentioning

confidence: 99%

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Notes on Exact Meets and Joins

Ball

Picado

Pultr

2013

Appl Categor Struct

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An exact meet in a lattice is a special type of infimum characterized by, inter alia, distributing over finite joins. In frames, the requirement that a meet is preserved by all frame homomorphisms makes for a slightly stronger property. In this paper these concepts are studied systematically, starting with general lattices and proceeding through general frames to spatial ones, and finally to an important phenomenon in Scott topologies.

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Section: Proposition (Proposition 110 [2]mentioning

confidence: 99%

Section: Proposition (Proposition 110 [2]mentioning

confidence: 99%

Section: Exact and Strongly Exact Meets In Framesmentioning

confidence: 99%

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Notes on Exact Meets and Joins

Ball

Picado

Pultr

2013

Appl Categor Struct

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“…This shows that L is boolean, and completes the proof. 2 A quotient of a P -frame need not be a P -frame [7]. However, a C-quotient of a P -frame is clearly a P -frame, for a C-quotient f : L → M is coz-onto, meaning every cozero element of M is the image under f of a cozero element of L. Since the cozero elements of L are complemented, so are their images.…”

Section: P -Frames and Boolean Framesmentioning

confidence: 99%

Pointfree pointwise suprema in unital archimedean ℓ-groups

Ball

Hager

Walters-Wayland

2015

Journal of Pure and Applied Algebra

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We generalize the concept of the pointwise supremum of real-valued functions to the pointfree setting. The concept itself admits a direct and intuitive formulation which makes no mention of points. But our aim here is to investigate pointwise suprema of subsets of RL, the family of continuous real valued functions on a locale, or pointfree space. Thus our setting is the category W of archimedean lattice-ordered groups ( -groups) with designated weak order unit, with morphisms which preserve the group and lattice operations and take units to units. This is an appropriate context for this investigation because every W-object can be canonically represented as a subobject of some RL. We show that the suprema which are pointwise in the Madden representation can be characterized purely algebraically. They are precisely the suprema which are context-free, in the sense of being preserved by every W homomorphism out of G. We show that closure under such suprema characterizes the W-kernels among the convex -subgroups. Finally, we prove that all existing joins in a W-object G are pointwise iff its Madden frame L is boolean, and that all existing countable joins in G are pointwise if L is a P -frame, but not conversely. This leads up to the appropriate analog of the Nakano-Stone Theorem: a (completely regular) locale L has the feature that RL is conditionally pointwise complete (σ-complete), i.e., every bounded (countable) family from RL has a pointwise supremum in RL, iff L is boolean (a P -locale). We adopt a maximally broad definition of unconditional pointwise completeness (σ-completeness): a divisible W-object G is pointwise complete (σ-complete) if it contains a pointwise supremum for every subset which has a supremum in any extension. We show that the pointwise complete (σ-complete) W-objects are those of the form RL for L a boolean locale (P -locale). Finally, we show that a W-object G is pointwise σ-complete iff it is epicomplete.

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“…The latter kind we will call quasi P -frames, and the definition we use is motivated by the definition of quasi P -spaces [22]. Although P -frames generalise P -spaces in the sense that a Tychonoff space is a P -space precisely when the frame of its open sets is a P -frame, it has recently been shown by Ball, Walters-Wayland and Zenk [4] that, in stark contrast with P -spaces, there are P -frames with quotients which are not P -frames. In the last part of this article we examine how far the theory of quasi P -frames parallels that of quasi P -spaces as defined by Henriksen, Martínez and Woods [22].…”

Section: Introductionmentioning

confidence: 99%

When certain prime ideals in rings of continuous functions are minimal or maximal

Dube

Nsayi

2015

Topology and its Applications

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A Tychonoff space X is called a quasi m-space if every prime d-ideal of the ring C(X) is either a maximal ideal or a minimal prime ideal. These spaces were characterised by Azarpanah and Karavan. In this paper we look at some properties of these spaces from a ring-theoretic perspective. We observe, for instance, that among subspaces which inherit this property are (i) cozero subspaces, (ii) dense z-embedded subspaces, and (iii) regular-closed subspaces among the normal quasi m-spaces. The ring-theoretic approach actually yields the above results within the broader context of frames. The latter part of the paper discusses completely regular frames L for which every prime z-ideal in the ring RL is a maximal ideal or a minimal prime ideal.

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The P-frame reflection of a completely regular frame

Cited by 16 publications

References 12 publications

Notes on Exact Meets and Joins

Notes on Exact Meets and Joins

Pointfree pointwise suprema in unital archimedean ℓ-groups

When certain prime ideals in rings of continuous functions are minimal or maximal

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