Robinson and Zakon gave necessary and sufficient conditions for an abelian ordered group to satisfy the same first‐order sentences as an archimedean abelian ordered group (i.e., which embeds in the group of real numbers). The present paper generalizes their work to obtain similar results for infinite subgroups of the group of unimodular complex numbers. Furthermore, the groups which satisfy the same first‐order sentences as ultraproducts of finite cyclic groups are characterized.

Let l be a commutative field; Bauval [1] showed that the theory of the ring l[X1,...,Xm] is the same as the weak second-order theory of the field l. Now, l[X1,...,Xm] is the ring of the monoid m, so it may be asked what properties of m we can deduce from the theory of l[;m], that is, if l[m] is elementarily equivalent to the ring of monoid k[G], with k, a field and G, a monoid, what do we know not only (...) about the first-order theory of G but also about more properties of G. We prove that in this case G is isomorphic to m.Bauval [1, Theorem V.2.1] proved that if a factorial ring A is elementarily equivalent to l[;X1,...,Xm], then A is isomorphic to F[X1,...,Xm], with F being the field of invertible elements of A.Our result is different from this property because, a priori, the field k is not the field of invertible elements . Furthermore, k[G]; is not always factorial or Neotherian: if G is isomorphic to the cartesian product ∏i<ω, then the number of irreducible elements that divide elements that divide X is infinite, and the ideal generated by the monomials X is not finitely generated.We recall that to be factorial of Neotherian are not elementary properties. (shrink)