Robinson and Zakon gave necessary and sufficient conditions for an abelian ordered group to satisfy the same first‐order sentences as an archimedean abelian ordered group (i.e., which embeds in the group of real numbers). The present paper generalizes their work to obtain similar results for infinite subgroups of the group of unimodular complex numbers. Furthermore, the groups which satisfy the same first‐order sentences as ultraproducts of finite cyclic groups are characterized.

We show that any countable model of a model complete theory has an elementary extension with a “pseudofinite-like” quasi-dimension that detects dividing.

We introduce the concept of o-asymptotic classes of finite structures, melding ideas coming from 1-dimensional asymptotic classes and o-minimality. Along with several examples and non-examples of these classes, we present some classification theory results of their infinite ultraproducts: Every infinite ultraproduct of structures in an o-asymptotic class is superrosy of U^þ-rank 1, and NTP2 (in fact, inp-minimal).

In connection with the work of Anscombe, Macpherson, Steinhorn and the present author in [1] we investigate the notion of a multidimensional exact class (R-mec), a special kind of multidimensional asymptotic class (R-mac) with measuring functions that yield the exact sizes of definable sets, not just approximations. We use results about smooth approximation [24] and Lie coordinatization [13] to prove the following result (Theorem 4.6.4), as conjectured by Macpherson: For any countable language$\mathcal {L}$and any positive integerdthe class$\mathcal {C}(\mathcal {L},d)$of all (...) finite$\mathcal {L}$-structures with at mostd4-types is a polynomial exact class in$\mathcal {L}$, where a polynomial exact class is a multidimensional exact class with polynomial measuring functions. (shrink)

We show that every definable subset of an uncountably categorical pseudofinite structure has pseudofinite cardinality which is polynomial (over the rationals) in the size of any strongly minimal subset, with the degree of the polynomial equal to the Morley rank of the subset. From this fact, we show that classes of finite structures whose ultraproducts all satisfy the same uncountably categorical theory are polynomial R-mecs as well as N-dimensional asymptotic classes, where N is the Morley rank of the theory.

We study a family of ultraproducts of finite fields with the Frobenius automorphism in this paper. Their theories have the strict order property and TP2. But the coarse pseudofinite dimension of the definable sets is definable and integer-valued. Moreover, we establish a partial connection between coarse dimension and transformal transcendence degree in these difference fields.

We study a family of ultraproducts of finite fields with the Frobenius automorphism in this paper. Their theories have the strict order property and TP2. But the coarse pseudofinite dimension of the definable sets is definable and integer-valued. Moreover, we also discuss the possible connection between coarse dimension and transformal transcendence degree in these difference fields.