Order-Topological lattices

verfasst von: Marcel Erné
Abstract: The observation that convergence of real sequences may be defined in terms of limits inferior and limits superior as by means of neighbourhoods in the Euclidean topology leads to the question: for which lattices does order convergence coincide with convergence in the order topology? This problem has been attacked by D. C. Kent [10], A. Gingras [7] and others. We hope to present a satisfactory solution in this paper. Although there are known several characterizations of lattices, with topological order convergence (cf. Propositions 1, 2), an evaluation of these criteria already requires some knowledge of the order topology of the given lattice. In the present paper, we establish a purely lattice-theoretical description of those lattices for which order convergence is not only topological, but moreover, the lattice operations are continuous. Henceforth, such lattices will be referred to as order-topological lattices. All convergence statements will be formulated in terms of filters rather than nets. For an introduction to convergence functions, the reader may consult D. C. Kents's paper [9].
Organisationseinheit(en): Institut für Algebra, Zahlentheorie und Diskrete Mathematik
Typ: Artikel
Journal: Glasgow mathematical journal
Band: 21
Seiten: 57-68
Anzahl der Seiten: 12
ISSN: 0017-0895
Publikationsdatum: 01.01.1980
Publikationsstatus: Veröffentlicht
Peer-reviewed: Ja
ASJC Scopus Sachgebiete: Allgemeine Mathematik
Elektronische Version(en): https://doi.org/10.1017/S0017089500003980 (Zugang: Offen)

BibTeX

@article{6dcf0626886b40908a7fb1331064a62f,
title = "Order-Topological lattices",
abstract = "The observation that convergence of real sequences may be defined in terms of limits inferior and limits superior as by means of neighbourhoods in the Euclidean topology leads to the question: for which lattices does order convergence coincide with convergence in the order topology? This problem has been attacked by D. C. Kent [10], A. Gingras [7] and others. We hope to present a satisfactory solution in this paper. Although there are known several characterizations of lattices, with topological order convergence (cf. Propositions 1, 2), an evaluation of these criteria already requires some knowledge of the order topology of the given lattice. In the present paper, we establish a purely lattice-theoretical description of those lattices for which order convergence is not only topological, but moreover, the lattice operations are continuous. Henceforth, such lattices will be referred to as order-topological lattices. All convergence statements will be formulated in terms of filters rather than nets. For an introduction to convergence functions, the reader may consult D. C. Kents's paper [9].",
author = "Marcel Ern{\'e}",
year = "1980",
month = jan,
day = "1",
doi = "10.1017/S0017089500003980",
language = "English",
volume = "21",
pages = "57--68",
journal = "Glasgow mathematical journal",
issn = "0017-0895",
publisher = "Cambridge University Press",
number = "1",
}