Continuity in partially ordered sets
Time and place
4 PM on Thursday, November 1st, 2007; Mathematics Dept., Baruch Coll. Vertical Campus,
Venu Menon (Univ. of Connecticut)
Abstract
Continuous lattices and their generalizations, continuous domains, have been studied for more than three decades. Recall that given elements of a poset, p is way below q, if whenever q is less than or equal to each upper bound of a directed set D, then p is less than or equal to an element of D. Continuous lattices are complete lattices where each element is the supremum of elements way below it. A poset is a continuous domain, if it has sups of directed sets and satisfies the following two conditions: (i) each element is the sup of elements way below it, and (ii) for each element, the set of elements way below it is directed.
A continuous poset is any poset in which the conditions (i) and (ii) are satisfied. In a complete lattice, in fact in any sup-semilattice, condition (ii) above is automatically satisfied. The purpose of this talk is to look at posets which need not be dcpos or lattices but which satisfy the condition that each element is the sup of elements way below it. Since we don't require the set of elements way below any element to be directed, we will require a condition slightly stronger than condition (i) above. Several of the pleasing algebraic and topological properties of continuous domains extend to this setting.
