Large Finite Structures with Few Lk-Types

Large Finite Structures with Few Lk-Types

For each k 3, we show that there is no recursive bound for the size of the smallest finite model of an Lk-theory in terms of its k-size. Here Lk denotes the k-variable fragment of first-order logic. An Lk-theory is a maximal consistent set of Lk-sentences, and the k-size of an Lk-theory is the number of Lk-types realized in its models. Our result answers a question of Dawar [3, 2]. As a corolla...

Finite Structures with Few Types

Finite structures with few types / by Gregory Cherlin and Ehud Hrushovski. p. cm.– (Annals of mathematics studies ; no. 152) Includes bibliographical references and index. The publisher acknowledges the authors for providing the camera-ready copy from which this book was printed.

Hrushovski : Finite Structures with Few Types

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Modules with Few Types over Some Finite-Dimensional Algebras

Fields with few types

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 ...