Notes on the products of the lower topology and Lawson topology on posets

In this paper, we investigate the relation between the lower topology respectively the Lawson topology on a product of posets and their corresponding topological product. We show that (1) if S and T are nonsingleton posets, then Ω ( S × T ) = Ω ( S ) × Ω ( T ) iff both S and T are finitely generated upper sets; (2) if S and T are nontrivial posets with σ ( S ) or σ ( T ) being continuous, then Λ ( S × T ) = Λ ( S ) × Λ ( T ) iff S and T satisfy property K, where for a poset L, Ω ( L ) means the lower topological space, Λ ( L ) means the Lawson topological space, and L is said to satisfy property K if for any x ∈ L , there exist a Scott open U and a finite F ⊆ L with x ∈ U ⊆ ↑ F .