The congruence frame and the Madden quotient for partial frames

Frith, John; Schauerte, Anneliese

doi:10.1007/s00012-018-0554-4

Cited by 7 publications

(5 citation statements)

References 22 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: 77%

“…Our papers with a more topological flavour are [6], [8], [10], [13] and [14]. Our papers with a more algebraic flavour are [9], [11] and [12]. Crucial for this paper is [15].…”

Section: Introductionmentioning

confidence: 94%

“…(iv) The quotient map induced by the Madden congruence, p : L → L/π L is dense, onto and the universal such. We refer to this as the Madden quotient of L. (See [11].) Definition 2.4 For any S-frame L, define e L : H S L → C S L to be the unique frame map such that e L (↓ a) = ∇ a for all a ∈ L; that is, making the following diagram commute:…”

Section: -4mentioning

confidence: 99%

“…We know that (see [11]) that C S is a functor from S-frames to frames which is natural in the sense that for any S-frame h : L → M we have a frame map C S h : C S L → C S M making the following diagram commute: (a) The quotient map q : L → L/θ is closed if and only if θ is a closed S-congruence; i.e. θ = ∇ a for some a ∈ L. (b) The quotient map q : L → L/θ is open if and only if θ is an open S-congruence; that is, θ = ∆ a for some a ∈ L. Definition 4.4 Let h : L → M be an S-frame map.…”

Section: Right and Left Adjointsmentioning

confidence: 99%

“…A frame is then Boolean if and only if it is isomorphic to its Booleanization. Following Madden's lead in [19], in [11] we constructed a least dense quotient for partial frames. The codomain need not be Boolean, however, as Madden already noted in the case of κ-frames.…”

Section: Boolean Properties For Partial Framesmentioning

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: 77%

“…Our papers with a more topological flavour are [6], [8], [10], [13] and [14]. Our papers with a more algebraic flavour are [9], [11] and [12]. Crucial for this paper is [15].…”

Section: Introductionmentioning

confidence: 94%

Section: -4mentioning

confidence: 99%

Section: Right and Left Adjointsmentioning

confidence: 99%

Section: Boolean Properties For Partial Framesmentioning

confidence: 99%

See 3 more Smart Citations

Partial frames, their free frames and their congruence frames

Schauerte¹,

Frith²

2023

Electronic Notes in Theoretical Informatics and Computer Science

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Nuclear ranges in implicative semilattices

Erné

2022

Algebra Univers.

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A nucleus on a meet-semilattice A is a closure operation that preserves binary meets. The nuclei form a semilattice $$\mathrm{N }A$$ N A that is isomorphic to the system $${\mathcal {N}}A$$ N A of all nuclear ranges, ordered by dual inclusion. The nuclear ranges are those closure ranges which are total subalgebras (l-ideals). Nuclei have been studied intensively in the case of complete Heyting algebras. We extend, as far as possible, results on nuclei and their ranges to the non-complete setting of implicative semilattices (whose unary meet translations have adjoints). A central tool are so-called r-morphisms, that is, residuated semilattice homomorphisms, and their adjoints, the l-morphisms. Such morphisms transport nuclear ranges and preserve implicativity. Certain completeness properties are necessary and sufficient for the existence of a least nucleus above a prenucleus or of a greatest nucleus below a weak nucleus. As in pointfree topology, of great importance for structural investigations are three specific kinds of l-ideals, called basic open, boolean and basic closed.

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Closed and Open Maps for Partial Frames

Frith

Schauerte

2023

Appl Categor Struct

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This paper concerns the notions of closed and open maps in the setting of partial frames, which, in contrast to full frames, do not necessarily have all joins. Examples of these include bounded distributive lattices, $$\sigma $$ σ - and $$\kappa $$ κ -frames and full frames. We define closed and open maps using geometrically intuitively appealing conditions involving preservation of closed, respectively open, congruences under certain maps. We then characterize them in terms of algebraic identities involving adjoints. We note that partial frame maps need have neither right nor left adjoints whereas frame maps of course always have right adjoints. The embedding of a partial frame in either its free frame or its congruence frame has proved illuminating and useful. We consider the conditions under which these embeddings are closed, open or skeletal. We then look at preservation and reflection of closed or open maps under the functors providing the free frame or the congruence frame. Points arise naturally in the construction of the spectrum functor for partial frames to partial spaces. They may be viewed as maps from the given partial frame to the 2-chain or as certain kinds of filters; using the former description we consider closed and open points. Any point of a partial frame extends naturally to a point on its free frame and a point on its congruence frame; we consider the closedness or openness of these.

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The congruence frame and the Madden quotient for partial frames

Cited by 7 publications

References 22 publications

Partial frames, their free frames and their congruence frames

Partial frames, their free frames and their congruence frames

Nuclear ranges in implicative semilattices

Closed and Open Maps for Partial Frames

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