The relations between the order convergence of nets and filters are studied. It is shown that Birkhoff-Frink′s order convergence is equivalent to Erné-Gatzke′s strong order convergence, and Wolk′s order convergence is equivalent to Erné-Gatzke′s order convergence. These two kinds of order convergence induce the same order topology if the poset is a lattice. The concept of order convergence lattices is introduced. It is shown that the order topology induced by an order convergence lattice is Hausdorff and regular, finite products of order convergence lattices are still order convergence lattices. As a subcategory of the category of posets, the category consisting of all order convergence complete lattices is Cartesian closed.

filter; net; o1-convergence; o2-convergence; order topology; order convergence lattice; Cartesian closed