When is the ideal completion of abstract basis algebraic

The concepts of abstract basis and its ideal completion play an important role in domain theory because the category of bases (and approximate relations) is equivalent to the category of continuous domains (and continuous mappings) (cf. [1, Theorem 2.2.28]). We know that if the relation equipped with an abstract basis is reflexive then its ideal completion is an algebraic domain (see [1, Proposition 2.2.22.4]). In this short note, we give a sufficient and necessary condition under which the ideal completion of an abstract basis is algebraic