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The universal homogeneous binary tree. (English) Zbl 1444.03124

Summary: 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 \((\mathbb{S}_{2};\leq)\). We study the reducts of \((\mathbb{S}_{2};\leq)\), that is, the relational structures with domain \(\mathbb{S}_{2}\), all of whose relations are first-order definable in \((\mathbb{S}_{2};\leq)\). Our main result is a classification of the model-complete cores of the reducts of \(\mathbb{S}_{2}\). 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 \(\mathbb{S}_{2}\) that contain the automorphism group of \((\mathbb{S}_{2};\leq)\) and are closed with respect to the pointwise convergence topology.

MSC:

03C07 Basic properties of first-order languages and structures
03C50 Models with special properties (saturated, rigid, etc.)
20B35 Subgroups of symmetric groups
20B27 Infinite automorphism groups
06A06 Partial orders, general
03C60 Model-theoretic algebra
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