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]. The authors acknowledge the support by the Swiss National Science Foundation, 21-30585.91, 20-50579.97 and 2000-041475.94/2 respectively, the last of which has funded a research stay of the second author at the University of Berne. 1 The ideal completion We recall the definition of the ideal completion (e.g. [DP90]) and of the sequential version of the ideal completion known as the chain completion. If (P,v) is a partial order and A is a nonempty subset of P , then A is an ideal iff ∀y ∈ A. x v y ⇒ x ∈ A and A is directed; that is ∀x, y ∈ A∃z ∈ A. x v z and y v z. The ideal completion of a partial order (P,v,⊥) with a least element ⊥, is the partial order (Q,⊆, {⊥}) where Q is the set of all ideals. Let (P,≤) be a partial order. A sequence (xn)n in P is eventually increasing iff ∃n0 ∀m,n ≥ n0.m ≤ n ⇒ xm ≤ xn. We let S(P ) denote the set of eventually increasing sequences for this partial order. We remark that in the following we will use the standard terminology “chain completion” as used in theoretical computer science (e.g. [BvBR98]), where the notion of a chain refers to a countable linear order. This replaces the standard mathematical definition of a chain as a linear order.