Abstract
AbstractWe compute the domination monoid in the theory of dense meet‐trees. In order to show that this monoid is well‐defined, we prove weak binarity of and, more generally, of certain expansions of it by binary relations on sets of open cones, a special case being the theory from [7]. We then describe the domination monoids of such expansions in terms of those of the expanding relations.
Highlights
We compute the domination monoid in the theory DMT of dense meettrees
Its complete first-order theory DMT is that of dense meet-trees: dense lower semilinear orders with everywhere infinite ramification
Such structures have received a certain amount of model-theoretic attention in the recent past. They appear in the classification of countable 2-homogeneous trees from [Dro85], and have since been important in the theory of permutation groups, see for instance [Cam87,DHM89,AN98]. They were shown to be dp-minimal in [Sim11], and the automorphism group of the unique countable one was studied in [KRS19], while the interest in similar structures goes back at the very least to [Per[73], Woo78], where they were used as a base to produce examples in the context of Ehrenfeucht theories
Summary
Lowercase Latin letters may denote tuples of variables or elements of a model. 3. We denote by Sxinv(U, A) the space of global A-invariant types in variables x, with A small, and by Sxinv(U) the union of all Sxinv(U, A) as A ranges among small subsets of U. When working in expansions of DMT, we will denote the closure of a set A under meets by dclLmt (A) This is justified by the previous remark. Types of kind (0), (Ia), or (Ib) correspond to cuts in a linearly ordered subset of the tree, where in kind (Ib), if the cut of p has a maximum a, we are specifying an existing open cone above a. Figure 2: some nonrealised B-invariant types, where points of B are denoted by triangles In this picture, the set of triangles below x has no maximum, solid lines lie in U, and dotted lines lie in a bigger U1 + U. The type of x is of kind (Ib), that of y of kind (II), and that of z of kind (IIIb)
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