A strongly minimal structure D is called unimodular if any two finite-to-one maps with the same domain and range have the same degree; that is if/4: (/-»• V is everywhere fc4-to-l, then kx = kc,. THEOREM. Unimodular strongly minimal structures are locally modular. This extends Zil'ber's theorem on locally finite strongly minimal sets, Urbanik's theorem on free algebras with the Steinitz propert...

We partially describe minimal, first-order structures which have a strong form of the strict order property. An infinite first-order structure is minimal if its each definable (possibly with parameters) subset is either finite or co-finite. It is strongly minimal if the minimality is preserved in elementarily equivalent structures. While strongly minimal structures were investigated more closel...

In this note I construct a proper o-minimal expansion of the ordered additive group of rationals.

In this note we point out that any strongly minimal pseudofinite structure (or set) is unimodular in the sense of [1], [5], [2], and hence measurable in the sense of Macpherson and Steinhorn [3], [2] as well as 1-based. The argument, involving nonstandard finite cardinalities, is straightforward. A few people asked about this issue in private conversations and communications, in particular Mart...

In this paper we study (strongly) locally o-minimal structures. We first give a characterization of the strong local o-minimality. We also investigate locally o-minimal expansions of (R,+, <).