\(Z\)-semicontinuous posets
A basic result in the theory of continuous lattices is that in a distributive continuous lattice, the way-below relation is multiplicative if and only if every weak prime element is prime if and only if the set of weak primes is closed in the Lawson topology. The authors generalize these results to the case of \(Z\)-semicontinuous posets, which include semicontinuous lattices and \(Z\)-continuous posets as special cases. For \(Z\)-prime and \(Z\)-semicontinuous posets, \(\ll_Z\) is multiplicative if and only if every weak prime is prime, and in the case that the poset \(P\) is further assumed to be a complete lattice, the set of weak primes is closed in a modified version of the Lawson topology \(\lambda_Z(P)\).
scientific article; zbMATH DE number 2153980