Ideals in ordered sets, a unifying approach

Niederle, Josef

doi:10.1007/bf02874708

Cited by 5 publications

(3 citation statements)

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“…Also note that if I is a lower set then (2) is immediate. In fact, if P is a lattice then the Z-complete lower sets are precisely the P Z (P)-ideals defined in [Nie06], where P Z (P) denotes the collection of all Z-disjoint subsets of P. While if P is an effect algebra with centre Z then the Z-complete lower sets are precisely the strongly type determining (STD) subsets defined in [FP10b] §4. And in this case P itself will be Z-complete precisely when P is centrally orthocomplete, as defined in [FP10a].…”

Section: Type Decompositionmentioning

confidence: 99%

Type decomposition in posets

Bice¹

2016

Mathematica Slovaca

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Section: Type Decompositionmentioning

confidence: 99%

Type decomposition in posets

Bice¹

2016

Mathematica Slovaca

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“…That is, a completion of a poset P whose elements are downsets of P with certain closure properties. Unfortunately, the word 'ideal' appears in the order theory literature with a variety of similar but different meanings (see [24] for a unifying treatment). Here we will be interested in downsets closed under certain families of existing joins.…”

Section: Introductionmentioning

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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“…We also refer the reader to [2] for standard definitions and notations for lattice structures. For facts concerning the notion of distributivity in posets we refer to papers [4], [5] and [6] by Erné and to [3,8,13,14], for ideals in posets to [11,15]. Our terminology and notation agree with the book [12] and the paper [14].…”

Section: Introductionmentioning

confidence: 99%