Limit-Preserving Embeddings of Partially Ordered Sets in Directed Sets
J. R. Isbell, Herman Rubin · Proceedings of the American Mathematical Society · 1956
Let (A, >) be a partially ordered set.Then we say [3] that a function / on A decides for S (on A) if {a|/(a)£5} contains a cofinal residual subset of A, i.e., if yEA, there is a ß^y such that for all a^ß, f(a)ES.Clearly the intersection of any two cofinal residual sets is cofinal residual.Ginsburg [2] showed that any partially ordered set (A, >) can be embedded in an everywhere branching set (S, >■) in such a manner that the "natural" extension of / to S decides for the same sets as / does.Day [l ] has an embedding in a directed set preserving decision, but possibly creating new decisions, even turning a nonconvergent function into a convergent one.We show here that the embedding can be made precise.First, let us notice that Lemma.Let (A, >) and (B, >) be partially ordered sets and let map B into A. If (i) for every T cofinal residual in A, ~1 T is cofinal residual in B, and (ii) for every U cofinal residual in B, decides for S on B.We may assume (by a trivial preliminary embedding) that the given set A has no least cofinal residual set, i.e., the set of maximal elements is not cofinal.Let S be the set of all pairs (T, a) such that aET and T is cofinal residual in A. We define (7\, cti)>-(72, a2) if TiET2 or Ti = T2 and ai>a2.We define ) is directed and therefore cofinal residual reduces to residual.Now if we replace (A, a) by a, we see that Theorem.Every partially ordered set (A, >) can be embedded in a directed set (B, >) in such a manner that for any function f on A, the "natural" extension of f to B decides for precisely those sets for which f decides.