Partially ordered sets

MacNeille, H. M.

doi:10.1090/s0002-9947-1937-1501929-x

Cited by 279 publications

(45 citation statements)

References 19 publications

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“…[7]) is a complete lattice containing P as a subposet and such that each element is a join and a meet of elements of P ; this lattice is unique, up to isomorphism, and denoted by N (P ). Let η be the order type of the chain made of the set Q of rational numbers with their natural order.…”

Section: Presentation Of the Main Resultsmentioning

confidence: 99%

Which posets have a scattered MacNeille completion?

2005

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A poset is order-scattered if it does not embed the chain η of the rational numbers. We prove that there are eleven posets such that N (P ), the MacNeille completion of P , is order-scattered if and only if P embeds none of these posets. Moreover these posets are pairwise non-embeddable in each other. This result completes a previous characterisation due to Duffus, Pouzet, Rival [4]. The proof is based on the "bracket relation": η → [η] 2 3 , a famous result of F. Galvin. Presentation of the main resultThe MacNeille completion of a poset P (cf. [7]) is a complete lattice containing P as a subposet and such that each element is a join and a meet of elements of P ; this lattice is unique, up to isomorphism, and denoted by N (P ). Let η be the order type of the chain made of the set Q of rational numbers with their natural order. A poset P is order-scattered, or in brief scattered, if it contains no chain of type η. The following characterization of posets P for which N (P ) is scattered was given in [4] (Theorem 4, p. 49).Theorem 1.1. Let P be an ordered set, then N (P ) contains a subset isomorphic to η if and only if P contains a subset isomorphic to η, D(ℵ 0 ), or some member of E(B(η)).In this statement, D(ℵ 0 ) denotes the cartesian product N × 2, where 2 := {0, 1}, ordered so that (x, i) < (y, j) if x = y and i < j. Also B(η) denotes the set Q × 2 ordered so that (x, i) < (y, j) if x < y and i < j; whereas E(B(η))denote the collection of posets made of Q × 2 and any order ε extending the order of B(η) with no additional comparabilities between Q × {0} and Q × {1}.The proof of Theorem 1.1 is based on the "arrow relation" η → (η, ℵ 0 ) 2 2 , obtained by Erdös and Rado [5], asserting that if the pairs of rational numbers are divided into two classes A and B then there is a subset X of the rationals such that either Presented by R. W. Quackenbush.

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Section: Presentation Of the Main Resultsmentioning

confidence: 99%

Which posets have a scattered MacNeille completion?

2005

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“…Pour les algébres qui ont une structure complètement déterminée par une relation d'ordre partiel, M. H. M. Mac Neille a donné une théorie des extensions canoniques [7] ( < ). Mais il y a lieu de considérer aussi les algébres partiellement ordonnées où existent une ou plusieurs opérations définies indépendamment de l'ordre, mais vérifiant une relation d'homogénéité l'exemple le plus typique étant celui des groupes réticulés.…”

Section: Les Algèbres Partiellement Ordonnées Et Leurs Extensions;unclassified

“…est le produit ^ défini par Fordre. L^extension basique de K s^identifîe alors avec Pextension canonique de ce système multiplicatif K pour les sommes distributives, telle quelle est définie par M. H. M. Mac Nielle [7]. Parmi les extensions de K, celles dans lesquelles la multiplication, est la même que le produits-défini par Pordre, sont toutes normales.…”

Section: Donc+(a)=^((3)^(y)unclassified

Les algèbres partiellement ordonnées et leurs extensions

Krishnan¹

1950

Bul. Soc. Math. France

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“…Generalizing Dedekind's pioneering construction of the real line by cuts of rational numbers (see [3]), MacNeille [17] has introduced the normal completion for arbitrary posets. It is well known that modularity and distributivity are not Xiaoquan Xu xiqxu2002@163.com 1 preserved under the formation of normal completions (see [2]).…”

Section: Introductionmentioning

confidence: 99%

Frink quasicontinuous posets

Zhang

2015

Semigroup Forum

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In this paper, the concept of Frink quasicontinuous posets is introduced. The main results are: (1) a poset is a Frink quasicontinuous poset if and only if its normal completion is a quasicontinuous lattice; (2) a poset is precontinuous if and only if it is Frink quasicontinuous and meet precontinuous; (3) when a Frink quasicontinuous poset satisfies certain conditions, the way below relation has the interpolation property; (4) the category of quasicontinuous lattices with complete homomorphisms is a full reflective subcategory of the category of Frink quasicontinuous posets with cut-stable maps.

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Partially ordered sets

Cited by 279 publications

References 19 publications

Which posets have a scattered MacNeille completion?

Which posets have a scattered MacNeille completion?

Les algèbres partiellement ordonnées et leurs extensions

Frink quasicontinuous posets

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