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Sep 20, 1981 - sup (j)ÇX). On the other hand, we have sup ... Further, let V ^ Lx be open. We have to show that U = (]rl(V) is open. In view of Proposition 1, we ...

ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 14, Number 3, Summer 1984

A NOTE ON ORDER CONVERGENCE IN COMPLETE LATTICES

H. DOBBERTIN, M. ERNÉ AND D. C. KENT

ABSTRACT. Order convergence is studied in a complete lattice Lx which is the image of another complete lattice L under a complete homomorphism. The goal is to relate order convergence in L to that in Lx. For instance, we show that order convergence in Lx is pretopological if it is pretopological in L, while topological order convergence is in general not preserved under complete images. We conclude with some applications and examples.

1. Complete homomorphisms and order convergence. Throughout this note, let L and Lx be complete lattices and suppose that Lx is a surjective, complete homomorphism (i.e., cp(L) = Lx and

L and the upper adjoint cp* : Lx -> L of cp are given by (p*(y) = inf{x e L: