In this article, we develop tame topology over dp-minimal structures equipped with definable uniformities satisfying certain assumptions. Our assumptions are enough to ensure that definable sets are tame: there is a good notion of dimension on definable sets, definable functions are almost everywhere continuous, and definable sets are finite unions of graphs of definable continuous “multivalued functions.” This generalizes known statements about weakly o-minimal, C-minimal, and P-minimal theories.

We show that if a first-order structure${\cal M}$, with universe ℤ, is an expansion of (ℤ,+,0) and a reduct of (ℤ,+,<,0), then${\cal M}$must be interdefinable with (ℤ,+,0) or (ℤ,+,<,0).

We study structures equipped with generic predicates and/or automorphisms, and show that in many cases we obtain simple theories. We also show that a bounded PAC field is simple. 1998 Published by Elsevier Science B.V. All rights reserved.

Generalizing and unifying the known theorems for difference and differential fields, it is shown that for every finite free algebra scheme.