omega-saturated quasi-minimal models of Th(Qomega, +, sigma, 0)

B. Zil’ber introduced the notion of quasi-minimality; Definition 1.1 An uncountable first-order structure M is called quasi-minimal, if for any definable (allowing parameters) subset X of M either X or M −X is finite or countable. Zil’ber conjectures that Cexp = (C,+, · , exp, 1, 0) is quasi-minimal. This conjecture is still open. Inspired by this conjecture the authors together with A. Tsuboi investigated basic properties of quasi-minimal structures and considered a way of constructing ω-saturated quasi-minimal structures, see [3]. When we were writing this paper we were unable to present a good concrete example shedding light to the argument. Here in this short note we present a quasi-minimal structure explaining the whole situation. In this sense this is a supplement to [3]. It is well known that the theory of torsion-free divisible abelian groups, denoted by DAG, admits elimination of quantifiers and is strongly minimal, hence ω-stable, (cf. [4, Theorem 3.1.9 and Corollary 3.1.11]). Consider the structure (Q,+) where the addition is defined component by component. It is easy to see that (Q,+) DAG, therefore this structure is also strongly minimal. We now define the shift function σ : Q −→ Q as follows: For x = (x0, x1, x2, . . .) ∈ Q let σ(x) = (x1, x2, . . .) ∈ Q. If you like a more formal definition, the function σ can be defined as follows: If x ∈ Q, then σ(x)(n) = x(n + 1) for each n ∈ ω. In the expanded structure (Q,+, σ, 0) consider for example the definable set X = {x ∈ Q : σ(x) = x}. Clearly X = {(r, r, . . .) : r ∈ Q} is countable. In this note we show first that Th(Q,+, σ, 0) admits elimination of quantifiers and that (Q,+, σ, 0) is quasi-minimal. Then we present an ω-saturated quasi-minimal model of Th(Q,+, σ, 0). In this note for a set A, A denotes its complement. The word countable always means countably infinite.