A Quasi - O - Minimal Groupwithout the Exchange

We construct an example of a quasi-o-minimal group without the Exchange Property. A structure hM;<;:::i is called quasi-o-minimal 1] if in any structure elementarily equivalent to it the deenable subsets are exactly the Boolean combinations of 0-deenable subsets and intervals. A complete theory T is said to have the Exchange Property if, for any model M of T and for any X M and b; c 2 M, if b 2 acl(X fcg) but b = 2 acl(X), then c 2 acl(X fbg). In general, in contrast to o-minimal theories, the theory of a quasi-o-minimal structure need not satisfy the Exchange Property; an easy counterexample can be found in 1]. However, that example does not carry a group structure; in fact it is also shown there that any quasi-o-minimal expansion of hZ;+;<i does have the Exchange Property. In this note we shall prove that the theory of the expansion A of the lexicographically ordered group ZZby the unary operation (x; y) 7 ! (0; x) and the constant (1; 0) is quasi-o-minimal and does not have the Exchange Property. It suuces to prove that the theory of a certain deenitional expansion M of A is quasi-o-minimal and does not have the Exchange Property. f is interpreted as the map (x; y) 7 ! (0; x), and D n (a) means`n divides a', for any positive integer n. Clearly, M is a deenitional expansion of A. We shall nd a set of rst order L-sentences T, each of which holds in M, and show that the theory T admits quantiier elimination. We shall prove that M is embeddable into any model of T. Therefore T is complete and hence is an axiom system for Th(M). Using quantiier elimination for T we shall deduce quasi-o-minimality of M. Even though in M itself the Exchange Property trivially holds (as every element of