On the archimedean kernels of function rings in pointfree topology

Banaschewski, Bernhard; Bhattacharjee, Papiya; Walters-Wayland, Joanne

doi:10.1016/j.topol.2015.12.011

Cited by 4 publications

(8 citation statements)

References 6 publications

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“…The representation result for the completion described above, in terms of normal semicontinuous real functions (studied in Section 4), is presented in Section 5. As an immediate consequence of it, we get that for a completely regular frame L, C(L) is Dedekind complete if and only if L is extremally disconnected, a result originally due to Banaschewski and Hong [6].…”

Section: Introductionmentioning

confidence: 61%

“…As an immediate corollary we get the following result from Banaschewski-Hong [6]: (1) L is extremally disconnected.…”

Section: The Normal Completion Of C(l) and C * (L)mentioning

confidence: 70%

“…In order to describe D(C(L)) there is no loss of generality if we restrict ourselves to completely regular frames (see the discussion in [6,Section 2]). (2) Ψ is well defined: First note that since f ∈ F cb (L), there exists a g ∈ C(L) such that g f .…”

Section: The Normal Completion Of C(l) and C * (L)mentioning

confidence: 99%

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Normal semicontinuity and the Dedekind completion of pointfree function rings

2016

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This paper supplements an earlier one by the authors which constructed the Dedekind completion of the ring of continuous real functions on an arbitrary frame L in terms of partial continuous real functions on L. In the present paper we provide three alternative views of it, in terms of (i) normal semicontinuous real functions on L, (ii) the Booleanization of L (in the case of bounded real functions) and the Gleason cover of L (in the general case) and (iii) Hausdorff continuous partial real functions on L. The first is the normal completion and extends Dilworth's classical construction to the pointfree setting. The second shows that in the bounded case the Dedekind completion is isomorphic to the lattice of bounded continuous real functions on the Booleanization of L and that in the non-bounded case it is isomorphic to the lattice of continuous real functions on the Gleason cover of L. Finally, the third is the pointfree version of Anguelov's approach in terms of interval-valued functions. Two new classes of frames, cb-frames and weak cb-frames, emerge naturally in the first two representations. We show that they are conservative generalizations of their classical counterparts.

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Section: Introductionmentioning

confidence: 61%

“…As an immediate corollary we get the following result from Banaschewski-Hong [6]: (1) L is extremally disconnected.…”

Section: The Normal Completion Of C(l) and C * (L)mentioning

confidence: 70%

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Normal semicontinuity and the Dedekind completion of pointfree function rings

2016

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“…Section 6 is devoted to conditional pointwise completeness; the main result here is the pointfree NakanoStone Theorem for conditional pointwise completeness (σ-completeness), Theorem 6.2.1. This result makes heavy use of the pointfree generalization of the classical theorem, a beautiful result of Banaschewski and Hong [10] which appears here in embellished form as Theorem 6.1.2.…”

Section: Introductionmentioning

confidence: 86%

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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“…Next, (zMax) implies (dMax) because every d-ideal is a z-ideal; (dMax) implies (vNR) because every minimal prime ideal is a d-ideal; and, finally, (vNR) implies (zMax) because every prime ideal in a von Neumann regular ring is a maximal ideal. Thus, RL satisfies any (and hence all) of these three if and only if L is a P -frame because L is a P -frame if and only if RL is von Neumann regular [7].…”

Section: Introductionmentioning

confidence: 93%

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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On the archimedean kernels of function rings in pointfree topology

Cited by 4 publications

References 6 publications

Normal semicontinuity and the Dedekind completion of pointfree function rings

Normal semicontinuity and the Dedekind completion of pointfree function rings

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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