Topologies on products of partially ordered sets III

Order convergence and order topology

verfasst von

Marcel Erné

Abstract

This paper deals with the question under which circumstances filter-theoretical order convergence in a product of posets may be computed componentwise, and the same problem is treated for convergence in the order topology (which may differ from order convergence). The main results are: (1) Order convergence in a product of posets is obtained componentwise if and only if the number of non-bounded posets occurring in this product is finite (1.5). (2) For any product of posets, the projections are open and continuous with respect to the order topologies (2.1). (3) A product L of chains L _i has topological order convergence iff all but a finite number of the chains are bounded. In this case, the order topology on L agrees with the product topology (2.7). (4) If (L _i :j ∈J) is a countable family of lattices with topological order convergence and first countable order topologies then order topology of the product lattice and product topology coincide (2.8). (5) Let P ₁ be a poset with topological order convergence and locally compact order topology. Then for any poset P ₂, the order topology of P ₁⊗P ₂ coincides with the product topology (2.10). (6) A lattice L which is a topological lattice in its order topology is join- and meet-continuous. The converse holds whenever the order topology of L⊗L is the product topology (2.15). Many examples are presented in order to illustrate how far the obtained results are as sharp as possible.

Organisationseinheit(en)

Institut für Algebra, Zahlentheorie und Diskrete Mathematik

Typ

Artikel

Journal

Algebra universalis

Band

13

Seiten

1-23

Anzahl der Seiten

23

ISSN

0002-5240

Publikationsdatum

12.1981

Publikationsstatus

Veröffentlicht

Peer-reviewed

Ja

ASJC Scopus Sachgebiete

Algebra und Zahlentheorie, Logik

Elektronische Version(en)

https://doi.org/10.1007/BF02483819 (Zugang: Geschlossen)