Title: Closure operators on dcpos,
Abstract: We examine collective properties of closure operators on posets that are at
least dcpos. The first theorem sets the tone of the paper: it tells how a set
of preclosure maps on a dcpo determines the least closure operator above them,
and pronounces the related induction principle, and its companion, the obverse
induction principle. Using this theorem we prove that the poset of closure
operators on a dcpo is a complete lattice, and then provide a constructive
proof of the Tarski's theorem for dcpos. We go on to construct the joins in the
complete lattice of Scott-continuous closure operators on a dcpo, and to prove
that the complete lattice of nuclei on a preframe is a frame, giving some
constructions in the special case of the frame of all nuclei on a frame. In the
rather drawn-out proof if the Hofmann-Mislove-Johnstone theorem we show off the
utility of the obverse induction, applying it in the proof of the crucial
lemma. After that we shift a viewpoint and prove some results, analogous to
results about dcpos, for posets in which certain special subsets have enough
maximal elements; these results actually specialize to dcpos, but at the price
of using the axiom of choice. We conclude by pointing out two convex geometries
associated with closure operators on a dcpo.