Bodirsky, Manuel; Bradley-Williams, David; Pinsker, Michael; Pongrácz, András The universal homogeneous binary tree. (English) Zbl 1444.03124 J. Log. Comput. 28, No. 1, 133-163 (2018). 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. Cited in 9 Documents 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 Keywords:semilinear order; reduct; model companion; model-complete core; permutation group; endomorphism monoid; constraint satisfaction problem PDFBibTeX XMLCite \textit{M. Bodirsky} et al., J. Log. Comput. 28, No. 1, 133--163 (2018; Zbl 1444.03124) Full Text: DOI arXiv Link