The universal homogeneous binary tree

Abstract

A partial order is called semilinear if the upper bounds of each element are linearly ordered and any two elements have a common upper bound. There exists, up to isomorphism, a unique countable existentially closed semilinear order, which we denote by (S2; ≤). We study the reducts of (S2; ≤), that is, the relational structures with domain S2, all of whose relations are first-order definable in (S2; ≤). Our main result is a classification of the model-complete cores of the reducts of S2. From this, we also obtain a classification of reducts up to first-order interdefinability, which is equivalent to a classification of all subgroups of the full symmetric group on S2 that contain the automorphism group of (S2; ≤) and are closed with respect to the pointwise convergence topology.

On the Automorphism Group of the Universal Homogeneous Meet-Tree

“…It is an ℵ 0 -categorical binary tree, but in contrast to T 2 , which is a meet-tree, no two incomparable elements have a meet. (See [1] for more details.) It might be interesting to ask whether it has a generic automorphism, but the methods based on Fact 2.9 used in this paper do not seem to apply directly.…”

“…We use a criterion for having a generic automorphism that was established by Truss [17,Theorem 2.1] and then improved to a characterisation independently by Ivanov [5,Theorem 1.2] and Kechris and Rosendal [9,Theorem 6.2]. Namely, to show that an ultrahomogeneous structure M with age K has a generic n-tuple of automorphisms, one needs to prove that the class K n of pairs (A, p) such that A ∈ K and p is an n-tuple of partial automorphisms of A, has the joint embedding property (JEP) and a version of the amalgamation property which we call the existential amalgamation property (EAP) 1 (see Fact 2.9).…”

“…To show EAP in the case of finite meet-trees, we find a cofinal subclass of K 1 consisting of amalgamation bases (which is the aforementioned criterion from [17]). Starting with some member (A, p A ), we extend p A to a partial automorphism p B on a bigger domain B in such a way that (B, p B ) is an amalgamation base in K 1 . The idea in finding p B is model-theoretic: instead of giving a precise description of p B , we define it as being "pseudo existentially closed" in the sense that, roughly, any behaviour that happens in some extension of p B is already witnessed in p B itself (see Definition 6.3).…”

Homogenizable structures and model completeness

“…A structure over a finite relational language is homogenizable if we can add new relational symbols to the structure's signature representing a finite number of formulas, such that the new expanded structure is homogeneous (see Definition 2.2 for details). The homogenizable structures are found in a variety of areas of mathematics, especially when studying random structures or structures with some excluded subgraphs, also called H−free structures [2,3,4,7,14,16]. In 1953 Fraïssé [10] studied homogeneous structures and found that for each set of finite structures K satisfying the properties HP, JEP and AP there is a unique infinite countable homogeneous structure M such that K is exactly the set of finite substructures of M (up to isomorphism).…”

Pairwise Nonisomorphic Maximal-Closed Subgroups of Sym(ℕ) via the Classification of the Reducts of the Henson Digraphs

“…With further developments [4,7], canonical functions have become powerful tools in studying reducts. The robustness and relative ease of the methodology is becoming more evident as several classifications have been achieved by their use, e.g., [1,2,6,10,19,23].…”