Makkai [10] produced an arithmetical structure of Scott rank ω1CK. In [9], Makkai's example is made computable. Here we show that there are computable trees of Scott rank ω1CK. We introduce a notion of “rank homogeneity”. In rank homogeneous trees, orbits of tuples can be understood relatively easily. By using these trees, we avoid the need to pass to the more complicated “group trees” of [10] and [9], Using the same kind of trees, we obtain one of rank ω1CK that is “strongly computably approximable”. We also develop some technology that may yield further results of this kind.

A generalization of the Seidel Entringer Arnold method for calculating the alternating permutation numbers (or secant tangent numbers) leads to a new operation on sequences, the boustrophedon transform.

For countable structure, "Scott rank" provides a measure of internal, model-theoretic complexity. For a computable structure, the Scott rank is at most [Formula: see text]. There are familiar examples of computable structures of various computable ranks, and there is an old example of rank [Formula: see text]. In the present paper, we show that there is a computable structure of Scott rank [Formula: see text]. We give two different constructions. The first starts with an arithmetical example due to Makkai, and codes it into a computable structure. The second re-works Makkai's construction, incorporating an idea of Sacks.

Abstract. Under P(4)− -amalgamation, we obtain the canonical hyperdefinable group from the group configuration.The group configuration theorem for stable theories given by Hrushovski [5], which extends Zilber's result for ω-categorical theories [17], plays a central role in producing deep results in geometric stability theory (For a complete exposition of it, see [14]). For example, it is pivotal in the proof of the dichotomy theorem for Zariski' structures (See [9]). It is fair to say the group configuration theorem is one of the foundational theorems in geometric stability theory and its applications to algebraic geometry.The theorem roughly says that one can get the canonical non-trivial type-definable group from the group configuration, a certain geometrical configuration, in stable theories. The complete generalization of the theorem into the context of simple theories seemed unreachable. In their topical paper [1], Ben-Yaacov, Tomasic and Wagner generalize the group configuration theorem by obtaining an invariant group from the group configuration in simple theories. However the group they produce does not completely fit into the first-order context.On the other hand, Kolesnikov in his important thesis [12], categorizes simple theories by strengthening the type-amalgamation property (the independence theorem [10]), along the lines of early suppositions by Shelah [15] and Hrushovski [6]. These works suggest to us the possibility of using higher amalgamation for the group configuration problem. This approach proves successful, and in this paper we succeed in getting the canonical hyperdefinable group from the group configuration under stronger type-amalgamation in simple theories. The element of the group is a hyperimaginary, an equivalence class of a type-definable equivalence relation, and the group operation is type-definable, hence the group belongs to the domain of the standard first-order logic.We assume that the reader is familiar with basics of simplicity theory [16]. Throughout the paper, T is a complete simple theory. We work in a saturated model M of T with hyperimaginaries, and a, b, ... are (possibly infinitary) hyperimaginaries, M, N are small elementary submodels. (Note that tuples from M eq are also hyperimaginaries). As usual,have the same type (Lascar strong type, resp.) over A. We point out that usually bdd(a) denotes the set of all countable hyperimaginaries definable over a [16, 3.1.7]. Here, depending on the context, it can be either a specific sequence which linearly orders the set bdd(a); or, since a sequence of hyperimaginaries is again a hyperimaginary (of a large arity), a fixed hyperimaginary interdefinable with the sequence.

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