On the congruence lattice of a frame

Banaschewski, Bernhard; Frith, John; Gilmour, Christopher

doi:10.2140/pjm.1987.130.209

Cited by 8 publications

(5 citation statements)

References 6 publications

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“…Compactifications of CL are also of independent interest. It is easy to see that the frame of lattice congruences C lat L is compact and it is shown in [3] that CL is compact if and only if it coincides with C lat L. This prompts us to consider the relationship between these two frames more generally.…”

Section: Resultsmentioning

confidence: 99%

The congruence biframe as a quasi-uniform bicompletion

Manuell

2020

Topology and its Applications

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Künzi and Ferrario have shown that a T 0 space is sober if and only if it is bicomplete in the well-monotone quasi-uniformity. We prove a pointfree version of this result: a strictly zero-dimensional biframe is a congruence biframe if and only if it is bicomplete in the same quasi-uniformity. As a corollary we obtain a new proof of a result of Plewe that a congruence frame is ultraparacompact. The main result makes use of a new construction of the bicompletion of a quasi-uniform biframe as a quotient of the Samuel compactification.2010 Mathematics Subject Classification. 06D22, 06B10, 54E15, 54E55.

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Section: Resultsmentioning

confidence: 99%

The congruence biframe as a quasi-uniform bicompletion

Manuell

2020

Topology and its Applications

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“…This is the analogue, in pointfree context, of the Skula modification of a topological space (indeed, the spectrum of CL is homeomorphic to the Skula modification of the spectrum of L and, consequently, the Skula topology of the spectrum of L is the spatial reflection of CL [4]). A good presentation of the congruence frame is given by Frith [14].…”

Section: The Skula Biframe Of a Frame Lmentioning

confidence: 97%

Functorial Quasi-Uniformities on Frames

Ferreira

Picado

2005

Appl Categor Struct

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“…A result of [5] states that every compact congruence frame is the congruence frame of a Noetherian frame (and conversely). In fact, every compact strictly zerodimensional biframe is of this form.…”

Section: First Parts and The Congruence Functormentioning

confidence: 99%

“…Using similar techniques one can show that Str0DBiFrm is the largest full subcategory of RegBiFrm for which F restricts to a coreflector. A result of [5] states that every compact congruence frame is the congruence frame of a Noetherian frame (and conversely). In fact, every compact strictly zerodimensional biframe is of this form.…”

Section: First Parts and The Congruence Functormentioning

confidence: 99%

Strictly Zero-Dimensional Biframes and a Characterisation of Congruence Frames

Manuell¹

2017

Appl Categor Struct

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Abstract. Strictly zero-dimensional biframes were introduced by Banaschewski and Brümmer as a class of strongly zero-dimensional biframes including the congruence biframes. We consider the category of strictly zero-dimensional biframes and show it is both complete and cocomplete. We characterise the extremal monomorphisms in this category and explore the special position that congruence biframes hold in it. Finally, we provide an internal characterisation of congruence biframes, and hence, of congruence frames.

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On the congruence lattice of a frame

Cited by 8 publications

References 6 publications

The congruence biframe as a quasi-uniform bicompletion

The congruence biframe as a quasi-uniform bicompletion

Functorial Quasi-Uniformities on Frames

Strictly Zero-Dimensional Biframes and a Characterisation of Congruence Frames

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