The Ideal Completion is Not Sequentially Adequate

It is well known that for the case of a countable partial order, the ideal completion and the chain completion coincide. We investigate the boundary at which the chain and ideal completion do not coincide. We show in particular that the ideal completion is not sequentially adequate; that is it is not possible in general to simply replace the ideal completion with a completion based on sequences as for instance the chain completion. The implications of this result for the Yoneda completion ([BvBR98]) and for the Smyth completion ([Smy89],[Smy91],[Smy94],[Sün93] and [Sün95]) which are based on the ideal completion, are discussed in an extended version of this paper, reported in [KS98]