Compactification and local connectedness of frames

Baboolal, Dharmanand; Banaschewski, Bernhard

doi:10.1016/0022-4049(91)90003-k

Cited by 35 publications

(10 citation statements)

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“…For a completely regular L, the frame of its completely regular ideals is denoted by βL. The join map βL → L is dense onto and referred to as the Stone-Čech compactification of L. We denote its right adjoint by r. A straightforward calculation shows that r(a) = {x ∈ L | x ≺≺ a} for each a ∈ L. If L is normal then r preserves finite joins as was shown in [3,Lemma 3.1]. We shall frequently use the fact that if I, J ∈ βL and I ≺≺ J, then I ∈ J.…”

Section: Preliminariesmentioning

confidence: 99%

Concerning P-frames, essential P-frames, and strongly zero-dimensional frames

Dube

2009

Algebra Univers.

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We give characterizations of P -frames, essential P -frames and strongly zero-dimensional frames in terms of ring-theoretic properties of the ring of continuous real-valued functions on a frame. We define the m-topology on the ring RL and show that if L belongs to a certain class of frames properly containing the spatial ones, then L is a P -frame iff every ideal of RL is m-closed. We define essential P -frames (analogously to their spatial antecedents) and show that L is a proper essential Pframe iff all the nonmaximal prime ideals of RL are contained in one maximal ideal. Further, we show that L is strongly zero-dimensional iff RL is a clean ring, iff certain types of ideals of RL are generated by idempotents.

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Section: Preliminariesmentioning

confidence: 99%

Concerning P-frames, essential P-frames, and strongly zero-dimensional frames

Dube

2009

Algebra Univers.

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“…Following Baboolal and Banaschewski [2], an element x ( = 0) of a frame L is said to be connected if whenever x = a ∨ b and a ∧ b = 0 then either a = 0 or b = 0. The frame L is connected if its top element e is connected; and L is locally connected if each of its elements is a join of connected elements.…”

Section: Introductionmentioning

confidence: 99%

A spread extension associated with Madden-type generators

Siweya

2006

Acta Math Hung

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A frame homomorphism f : L → M between locally connected frames is called a localic spread if u∈L S u is a basis for M , whereis a component of h(u)". Maddentype generators and relations are applied on L to form a freely generated frame CM induced by j : M → CM leading to a spread extension j • f : L → CM of f .In this article, we discuss properties of a local spread extension (which is not complete) between locally connected frames.

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“…In [3] and [17] these statements are claimed to be equivalent for frames, but no demonstrations are given. We give a proof that indeed they are equivalent for completely regular frames.…”

Section: Introductionmentioning

confidence: 99%