a note on the problem when fs-domains coincide with rb-domains

In this paper, we introduce the notion of super finitely separating functions which gives a characterization of rb-domains. then we prove that fs-domains and rb-domains are equivalent in some special cases by the following three claims: a dcpo is an rb-domain if and only if there exists an approximate identity for it consisting of super finitely separating functions; a consistent join-semilattice is an fs-domain if and only if it is an rb-domain; an l-domain is an fs-domain if and only if it is an rb-domain. these results are expected to provide useful hints to the open problem of whether fs-domains are identical with rb-domains.