Reducts of the random partial order

Abstract

Abstract. 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. Reducts of homogeneous structuresThe 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 Fraïssé 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 Fraïssé limit of the class of finite linear orders; he showed that this order has five reducts up to first-order interdefinability. Thomas [Tho91] proved that the random graph has five reducts up to firstorder interdefinability as well, and later generalized this result by showing that for all k ≥ 2, Date: February 26, 2018. The second author is grateful for support through an APART-fellowship of the Austrian Academy of Sciences.

THE REDUCTS OF THE HOMOGENEOUS BINARY BRANCHING C-RELATION

“…It turns out that several fundamental homogeneous structures with finite relational signatures have only finitely many reducts up to first-order interdefinability. This was shown for (Q; <) by Cameron [24] (and, independently and in somewhat different language, by Frasnay [29]), by Thomas for the the random graph [45], by Junker and Ziegler for the expansion of (Q; <) by a constant [34], by Pach, Pinsker, Pluhár, Pongrácz, and Szabó for the homogeneous universal poset [42], and by Bodirsky, Pinsker and Pongrácz for the random ordered graph [19]. Thomas has conjectured that all homogeneous structures with a finite relational signature have finitely many reducts [45].…”

Infinitely many reducts of homogeneous structures

“…Although the strategy works only under very special conditions, the first few reducts were possible to determine in several cases. Applying these techniques several structures have been analyzed: the random poset [14], [13], equality [1] and the random graph revisited [2]. They all have finitely many reducts.…”

Permutation groups containing infinite symplectic linear groups and reducts of linear spaces over the two element field

“…Later in [4] and [5] a general technique was introduced to investigate first order definable reducts of homogeneous structures on a finite language. Then several structures were analyzed from this aspect: the pointed Henson-graphs [15], the random poset [12], [13], equality [2] and the random graph revisited [3].…”