(Strongly) Zero-Dimensional Partially Ordered Spaces

Nailana, Koena Rufus

doi:10.1007/978-94-017-2529-3_27

Cited by 3 publications

(3 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The notion of an order-zero-dimensional space coincides with that of a zero-dimensional ordered topological space of Nailana [15,Section 1]. Let X = (X, τ, ≤) be an ordered topological space, CpUp(X) be the set of clopen upsets of X, and CpDn(X) be the set of clopen downsets of X.…”

Section: Remark 39mentioning

confidence: 99%

Priestley Rings and Priestley Order-Compactifications

Bezhanishvili

Morandi

2010

Order

View full text Add to dashboard Cite

We introduce Priestley rings of upsets (of a poset) and prove that inequivalent Priestley ring representations of a bounded distributive lattice L are in 1-1 correspondence with dense subspaces of the Priestley space of L. This generalizes a 1955 result of Bauer that inequivalent reduced field representations of a Boolean algebra B are in 1-1 correspondence with dense subspaces of the Stone space of B. We also introduce Priestley order-compactifications and Priestley bases of an ordered topological space, and show that they are in 1-1 correspondence. This generalizes a 1961 result of Dwinger that zero-dimensional compactifications of a topological space are in 1-1 correspondence with its Boolean bases.

show abstract

Section: Remark 39mentioning

confidence: 99%

Priestley Rings and Priestley Order-Compactifications

Bezhanishvili

Morandi

2010

Order

View full text Add to dashboard Cite

show abstract

“…In [7], Bezhanishvili and Morandi study what they call Priestley order-compactifications for a suitable class of ordered spaces, which includes those that are discretely topologised. Crucially for our purposes they demonstrate that β (Y) is a Priestley space for any Y ∈ P [7, Corollary 4.7]; this result was also proved, by a different method, by Nailana [42]. As a consequence, Proposition 4.1 now provides the following noteworthy result.…”

Section: Natural Extensions and Bohr Compactifications: Making Use Ofmentioning

confidence: 52%

Bohr Compactifications of Algebras and Structures

Davey

Haviar

Priestley

2016

Appl Categor Struct

View full text Add to dashboard Cite

Abstract. This paper provides a unifying framework for a range of categorical constructions characterised by universal mapping properties, within the realm of compactifications of discrete structures. Some classic examples fit within this broad picture: the Bohr compactification of an abelian group via Pontryagin duality, the zero-dimensional Bohr compactification of a semilattice, and the Nachbin order-compactification of an ordered set.The notion of a natural extension functor is extended to suitable categories of structures and such a functor is shown to yield a reflection into an associated category of topological structures. Our principal results address reconciliation of the natural extension with the Bohr compactification or its zero-dimensional variant. In certain cases the natural extension functor and a Bohr compactification functor are the same; in others the functors have different codomains but may agree on all objects. Coincidence in the stronger sense occurs in the zero-dimensional setting precisely when the domain is a category of structures whose associated topological prevariety is standard. It occurs, in the weaker sense only, for the class of ordered sets and, as we show, also for infinitely many classes of ordered structures.Coincidence results aid understanding of Bohr-type compactifications, which are defined abstractly. Ideas from natural duality theory lead to an explicit description of the natural extension which is particularly amenable for any prevariety of algebras with a finite, dualisable, generator. Examples of such classes-often varieties-are plentiful and varied, and in many cases the associated topological prevariety is standard.

show abstract

“…Moreover, by [2,Prop. 4.4] (see also [19,Sec. 2]), X is strongly order-zero-dimensional iff the Nachbin ordercompactification n(X) of X is a Priestley space.…”

Section: Priestley Order-compactifications Of Totally Ordered Spacesmentioning

confidence: 99%

Order-Compactifications of Totally Ordered Spaces: Revisited

Bezhanishvili

Morandi

2011

Order

View full text Add to dashboard Cite

Order-compactifications of totally ordered spaces were described by Blatter (J Approx Theory 13:56-65, 1975) and by Kent and Richmond (J Math Math Sci 11(4):683-694, 1988). Their results generalize a similar characterization of order-compactifications of linearly ordered spaces, obtained independently by Fedorčuk (Soviet Math Dokl 7:1011-1014, 1966 Sib Math J 10:124-132, 1969) and Kaufman (Colloq Math 17:35-39, 1967). In this note we give a simple characterization of the topology of a totally ordered space, as well as give a new simplified proof of the main results of Blatter (J Approx Theory 13: [56][57][58][59][60][61][62][63][64][65] 1975) and Kent and Richmond (J Math Math Sci 11(4):683-694, 1988). Our main tool will be an order-topological modification of the Dedekind-MacNeille completion. In addition, for a zero-dimensional totally ordered space X, we determine which order-compactifications of X are Priestley order-compactifications.

show abstract

(Strongly) Zero-Dimensional Partially Ordered Spaces

Cited by 3 publications

References 20 publications

Priestley Rings and Priestley Order-Compactifications

Priestley Rings and Priestley Order-Compactifications

Bohr Compactifications of Algebras and Structures

Order-Compactifications of Totally Ordered Spaces: Revisited

Contact Info

Product

Resources

About