Reducts of the Random Partial Order

We determine, up to the equivalence of first-order interdefinability, all structures which are first-order definable in the random partial order. It turns out that these structures fall into precisely five equivalence classes. We achieve this result by showing that there exist exactly five closed permutation groups which contain the automorphism group of the random partial order, and thus expose all symmetries of this structure. Our classification lines up with previous similar classifications, such as the structures definable in the random graph or the order of the rationals; it also provides further evidence for a conjecture due to Simon Thomas which states that the number of structures definable in a homogeneous structure in a finite relational language is, up to first-order interdefinability, always finite. The method we employ is based on a Ramsey-theoretic analysis of functions acting on the random partial order, which allows us to find patterns in such functions and make them accessible to finite combinatorial arguments. 1. Reducts of homogeneous structures The random partial order P := (P ;≤) is the unique countable partial order which is universal in the sense that it contains all countable partial orders as induced suborders and which is homogeneous, i.e., any isomorphism between two finite induced suborders of P extends to an automorphism of P. Equivalently, P is the Fräıssé limit of the class of finite partial orders – confer the textbook [Hod97]. As the “generic order” representing all countable partial orders, the random partial order is of both theoretical and practical interest. The latter becomes in particular evident with the recent applications of homogeneous structures in theoretical computer science; see for example [BP11a, BP11b, BK09, Mac11]. It is therefore tempting to classify all structures which are first-order definable in P, i.e., all relational structures on domain P all of whose relations can be defined from the relation ≤ by a first-order formula. Such structures have been called reducts of P in the literature [Tho91, Tho96]. It is the goal of the present paper to obtain such a classification up to first-order interdefinability, that is, we consider two reducts Γ,Γ′ equivalent iff they are reducts of one another. We will show that up to this equivalence, there are precisely five reducts of P. Our result lines up with a number of previous classifications of reducts of similar generic structures up to first-order interdefinability. The first non-trivial classification of this kind was obtained by Cameron [Cam76] for the order of the rationals, i.e., the Fräıssé limit of the class of finite linear orders; he showed that this order has five reducts up to first-order Date: July 10, 2014.