Order-Topological lattices

authored by: 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].
Organisation(s): Institute of Algebra, Number Theory and Discrete Mathematics
Type: Article
Journal: Glasgow mathematical journal
Volume: 21
Pages: 57-68
No. of pages: 12
ISSN: 0017-0895
Publication date: 01.01.1980
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: General Mathematics
Electronic version(s): https://doi.org/10.1017/S0017089500003980 (Access: Open)

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",
}