Characterizations of Various Continuities of Posets Via Approximated Elements

In this paper, various continuities of posets which may not be dcpos are considered. The concepts of approximated elements and hyper-approximated elements on posets are introduced. New characterizations of continuous posets and hypercontinuous posets are given. Meanwhile, as a generalization of approximated elements, the concept of quasi-approximated elements on dcpos is introduced and some characterizations of quasicontinuous domains are also obtained. It is proved that under some reasonable conditions, the set B(L) (resp., QB(L)) of approximated elements (resp., quasi-approximated elements) in the induced order of a dcpo L is a continuous domain (resp., a quasicontinuous domain).