The Stone-Čech compactification of a partial frame via ideals and cozero elements

Frith, John; Schauerte, Anneliese

doi:10.2989/16073606.2015.1023866

Cited by 13 publications

(5 citation statements)

References 14 publications

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“…The remainder of this section gives a lot of information about H S L, the free frame over the S-frame L, as well as C S L, the frame of S-congruences of L, and the relationship between the two. These results come from [7] on H S L, [9] and [11] on C S L.…”

Section: Introductionmentioning

confidence: 76%

“…See [7] for details about this notion. In particular, Lemma 4.3 of that paper characterizes the S-Lindelöf elements of H S L as being the principal S-ideals.…”

Section: Note Thatmentioning

confidence: 99%

“…The basics of our approach to partial frames can be found in [4], [5] and [7]. Our papers with a more topological flavour are [6], [8], [10], [13] and [14].…”

Section: Introductionmentioning

confidence: 99%

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Partial frames, their free frames and their congruence frames

Schauerte¹,

Frith²

2023

Electronic Notes in Theoretical Informatics and Computer Science

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The context of this work is that of partial frames; these are meet-semilattices where not all subsets need have joins. A selection function, S, specifies, for all meet-semilattices, certain subsets under consideration, which we call the ``designated'' ones; an S-frame then must have joins of (at least) all such subsets and binary meet must distribute over these. A small collection of axioms suffices to specify our selection functions; these axioms are sufficiently general to include as examples of partial frames, bounded distributive lattices, sigma-frames, kappa-frames and frames. We consider right and left adjoints of S-frame maps, as a prelude to the introduction of closed and open maps. Then we look at what might be an appropriate notion of Booleanness for partial frames. The obvious candidate is the condition that every element be complemented; this concept is indeed of interest, but we pose three further conditions which, in the frame setting, are all equivalent to it. However, in the context of partial frames, the four conditions are distinct. In investigating these, we make essential use of the free frame over a partial frame and the congruence frame of a partial frame. We compare congruences of a partial frame, technically called S-congruences, with the frame congruences of its free frame. We provide a natural transformation for the situation and also consider right adjoints of the frame maps in question. We characterize the case where the two congruence frames are isomorphic and provide examples which illuminate the possible different behaviour of the two. We conclude with a characterization of closedness and openness for the embedding of a partial frame into its free fame, and into its congruence frame.

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Section: Introductionmentioning

confidence: 76%

“…See [7] for details about this notion. In particular, Lemma 4.3 of that paper characterizes the S-Lindelöf elements of H S L as being the principal S-ideals.…”

Section: Note Thatmentioning

confidence: 99%

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Partial frames, their free frames and their congruence frames

Schauerte¹,

Frith²

2023

Electronic Notes in Theoretical Informatics and Computer Science

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“…( 6) If (at most) countable joins are specified, we have the notion of a σ-quantale. (7) If joins of subsets with cardinality less than some (regular) cardinal κ are specified, we have the notion of a κ-quantale. (8) In the case that directed joins are specified, we have the notion of a prequantale.…”

Section: Explicitly Posgr S (A) Is Isomorphic To Posmentioning

confidence: 99%

Reflectors to quantales

Zhang

Paseka²,

Feng³

et al. 2023

Fuzzy Sets and Systems

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“…A scheme associating each poset P with a join-specification U P is a special case of what Erné calls a subset selection, which is just a rule Z associating with each poset P a set of its subsets ZP . [15] uses the term selection rule for a similar purpose to us, with a similar use of ideals closed under specified joins, though in the context of partial frames.…”

mentioning

confidence: 99%

Categories of frame-completions and join-specifications

Egrot

2018

Preprint

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Given a poset P , a join-specification U for P is a set of subsets of P whose joins are all defined. The set IU of downsets closed under joins of sets in U forms a complete lattice, and is, in a sense, the free U-join preserving join-completion of P . The main aim of this paper is to address two questions. First, given a join-specification U, when is IU a frame? And second, given a poset P , what is the structure of its set of framegenerating join-specifications?To answer the first question we provide a number of equivalent conditions, and we use these to investigate the second. In particular, we show that the set of frame-generating join-specifications for P forms a complete lattice ordered by inclusion, and we describe its meet and join operations. We do the same for the set of 'maximal' such join-specifications, for a natural definition of 'maximal'. We also define functors from these lattices, considered as categories, into a suitably defined category of framecompletions of P , and construct right adjoints for them.

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The Stone-Čech compactification of a partial frame via ideals and cozero elements

Cited by 13 publications

References 14 publications

Partial frames, their free frames and their congruence frames

Partial frames, their free frames and their congruence frames

Reflectors to quantales

Categories of frame-completions and join-specifications

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