A Partial Order Where All Monotone Maps Are Definable

It is consistent that there is a partial order (P,≤) of size א1 such that every monotone function f : P → P is first order definable in (P,≤) It is an open problem whether there can be an infinite lattice L such that every monotone function from L to L is a polynomial. Kaiser and Sauer [KS] showed that such a lattice would have to be bounded, and cannot be countable. Sauer then asked the weaker question if there can be an infinite partial order (P,≤) such that all monotone maps from P to P are at least definable. (Throughout the paper, “definable” means “definable with parameters by a first order formula in the structure (P,≤).) Since every infinite partial order P admits at least c = 20 many monotone maps from P to P , our partial order must have size (at least) continuum. We show: 0.1. Theorem. The statement There is a partial order (P,≤) of size c = ω1 such that all monotone functions f : P → P are definable in P is consistent relative to ZFC. Moreover, the statement holds in any model obtained by adding (iteratively) ω1 Cohen reals to a model of CH. We do not know if Sauer’s question can be answered outright (i.e., in ZFC), or even from CH. Structure of the paper. In section 1 we give four conditions on a partial order on (P,v) of size κ and we show that they are sufficient to ensure the conclusion of the theorem. This section is very elementary. The two main conditions of section 1 are (1) a requirement on small sets, namely that they should be definable (2) two requirements on large sets (among them: “there are no large antichains”) Here, “small” means of size < |P |, and “large” means of size = |P |. In section 2 we show how to take care of requirement 1 in an inductive construction of our partial order in c many steps. Each definability requirement will be satisfied at some stage α < c. Finally, in section 4 we deal with the problem of avoiding large antichains. Here the inductive construction is not so straightforward, as we have to “anticipate” Date: February 1997. This is number 554 in the second author’s publication list. Supported by the Israeli Academy of Sciences.