A Note on Quasicontinuous Domains
Qingyu He
Department of Mathematics, Yangzhou University, Yangzhou 225002, P.R. China.
Luoshan Xu *
Department of Mathematics, Yangzhou University, Yangzhou 225002, P.R. China.
*Author to whom correspondence should be addressed.
Abstract
In this note, the quasicontinuity of a dcpo and its principal ideals/filters are considered. It is shown that every principal ideal, as a subposet, in a quasicontinuous domain is also a quasicontinuous domain. A dcpo is quasicontinuous if every principal ideal is quasicontinuous and it has finitely many maximal elements. For a bc-domain being quasicontinuous, the requirement of directedness of the approximation family of each element can be omitted.
Keywords: Quasicontinuous domain, principal ideal, Scott topology, semilattice.