Meet continuity properties of posets

In this paper we consider a number of properties of posets that are not directed complete: in particular, meet continuous posets, locally meet continuous posets and PI-meet continuous posets are introduced. Characterizations of (locally) meet continuous posets are presented. The main results are: (1) A poset is meet continuous iff its lattice of Scott closed subsets is a complete Heyting algebra; (2) A poset is a meet continuous poset with a lower hereditary Scott topology iff its upper topology is contained in its local Scott topology and the lattice of all local Scott closed sets is a complete Heyting algebra; and (3) A poset with a lower hereditary Scott topology is meet continuous iff it is locally meet continuous, iff it is PI-meet continuous.