The σ1-topology and λ1-topology on s1-quasicontinuous posets

In this paper, we consider a common generalization of both s1s1-continuous posets and quasicontinuous domains, and we introduce new concepts of way below relations and s1s1-quasicontinuous posets. The main results are: (1) A poset is an s1s1-quasicontinuous poset iff the σ1σ1-topology is a hypercontinuous lattice iff the S⁎S⁎-convergence is topological with respect to the σ1σ1-topology; (2) A poset is s1s1-continuous iff it is meet s1s1-continuous and s1s1-quasicontinuous; (3) The λ1λ1-topology on an s1s1-quasicontinuous poset is Tychonoff; (4) A poset P   is s1s1-quasicontinuous and the σ1σ1-topology is sober iff P   is a quasicontinuous domain and the σ1σ1-topology coincides with the Scott topology.