General Filters and Strict Extensions in Pointfree Topology

Abstract

Based on the fact that general filters(bounded meet-semilattice homomorphisms) in a frame are more suitable than classical filters to study convergence in frames, we here extend earlier work by constructing the strict extension L of a frame L for a set X of general rather than classical filters in L, using for this the adjointness between bounded meet-semilattices and frames. We characterize L by the fact that there exists a unique homomorphism L->L for which each filter in X is a trace filter and obtain various completions by this construction. Further we provide conditions on X for L to be regular or zero-dimensional