Meet continuity properties of posets (Q732011)

A meet continuous lattice is a complete lattice in which binary meets distribute over directed suprema. The authors generalize the concept of meet continuity to the setting of posets which are not directed complete. Using a newly-defined intrinsic topology, called the local Scott topology, the concept of locally meet continuous posets is introduced. Characterizations and properties of (locally) meet continuous posets are obtained in terms of the lattice properties of the Scott topology and of the local Scott topology. Posets for which the Scott topology coincides with the local Scott topology are meet continuous iff they are locally meet continuous iff they are PI-meet continuous, while the three kinds of meet continuities do not imply each other in general.