Meet Precontinuous Posets
Wenfeng Zhang, Xiaoquan Xu · Electronic Notes in Theoretical Computer Science · 2014
In this paper, we introduce the concept of meet precontinuous posets, a generalization of meet continuous lattices to posets. The main results are: (1) A poset P is meet precontinuous iff its normal completion is a meet continuous lattice iff a certain system γ ( P ) which is, in the case of complete lattices, the lattice of all Scott-closed sets is a complete Heyting algebra; (2) A poset P is precontinuous iff the way below relation is the smallest approximating auxiliary relation iff P is meet precontinuous and there is a smallest approximating auxiliary relation on P . Finally, given a poset P and an auxiliary relation on P , we characterize those join-dense subsets of P whose way-below relation agrees with the given auxiliary relation.