ABSTRACT
In this paper, the authors mainly concerned with some special choices of t which are of particular interest for the theory of topological lattices and semilattices, namely, the upper topology , the Scott topology , the Lawson topology , the lim inf topology ; further, “self-dual” topologies such as the interval topology , the Bi-Scott topology , the order topology , and the convex order topology. They obtain a new necessary and sufficient condition for a complete lattice to be continuous, namely, that its square is a topological meet semilattice with respect to the lim inf topology. Several of general results on continuity and convergence in partially ordered sets have been established earlier for complete lattices. The authors investigate continuity properties of isotone maps with respect to the convergence relations and topologies.
