The Equivalence of Two Modes of Order Convergence

Abstract

It is well known that if a poset satisfies Property A and its dual form, then the o-convergence and o2-convergence in the poset are equivalent. In this paper, we supply an example to illustrate that a poset in which the o-convergence and o2-convergence are equivalent may not satisfy Property A or its dual form, and carry out some further investigations on the equivalence of the o-convergence and o2-convergence. By introducing the concept of the local Frink ideals (the dually local Frink ideals) and establishing the correspondence between ID-pairs and nets in a poset, we prove that the o-convergence and o2-convergence of nets in a poset are equivalent if and only if the poset is ID-doubly continuous. This result gives a complete solution to the problem of E.S. Wolk in two modes of order convergence, which states under what conditions for a poset the o-convergence and o2-convergence in the poset are equivalent.