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We develop “local NIP group theory” in the context of pseudofinite groups. In particular, given a sufficiently saturated pseudofinite structure G expanding a group, and left invariant NIP formula δ... |
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We observe simple links between equivalence relations, groups, fields and groupoids (and between preorders, semi-groups, rings and categories), which are type-definable in an arbitrary structure, and apply these observations to the particular context of small and simple structures. Recall that a structure is small if it has countably many n-types with no parameters for each natural number n. We show that a θ-type-definable group in a small structure is the conjunction of definable groups, and extend the result to semi-groups, fields, (...) |
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According to Belegradek, a first order structure is weakly small if there are countably many $1$-types over any of its finite subset. We show the following results. A field extension of finite degree of an infinite weakly small field has no Artin-Schreier extension. A weakly small field of characteristic $2$ is finite or algebraically closed. A weakly small division ring of positive characteristic is locally finite dimensional over its centre. A weakly small division ring of characteristic $2$ is a field. |
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For a group G definable in a first order structure M we develop basic topological dynamics in the category of definable G-flows. In particular, we give a description of the universal definable G-ambit and of the semigroup operation on it. We find a natural epimorphism from the Ellis group of this flow to the definable Bohr compactification of G, that is to the quotient ${G^{\rm{*}}}/G_M^{{\rm{*}}00}$. More generally, we obtain these results locally, i.e., in the category of Δ-definable G-flows for any (...) |
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The "space" of Lascar strong types, on some sort and relative to a given complete theory T, is in general not a compact Hausdorff topological space. We have at least three aims in this paper. The first is to show that spaces of Lascar strong types, as well as other related spaces and objects such as the Lascar group Gal L of T, have well-defined Borel cardinalities. The second is to compute the Borel cardinalities of the known examples as well (...) |
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Lascar described E KP as a composition of E L and the topological closure of E L (Casanovas et al. in J Math Log 1(2):305–319). We generalize this result to some other pairs of equivalence relations. Motivated by an attempt to construct a new example of a non-G-compact theory, we consider the following example. Assume G is a group definable in a structure M. We define a structure M′ consisting of M and X as two sorts, where X is an (...) |
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