On the groups with homogeneous theory
If \(2\leq m < n\) then a structure \(M\) is \((m,n)\)-homogeneous if for any \(\bar{a},\bar{b} \in M^n\), \(\bar{a}\) and \(\bar{b}\) have the same type if and only if corresponding \(m\)-subtuples have the same type. A complete theory is \((m,n)\)-homogeneous if all its subtuples are. The author discusses the conjecture that if \((M,\cdot)\) is a group (possibly with extra structure), then its theory is not \((m,n)\)-homogeneous for any \(m,n\) with \(2\leq m<n\). Handa has shown that no infinite abelian \(p\)-group can be interpreted in a theory which (for some \(m\)) is \((m,n)\)-homogeneous for all \(n>m\), and the above conjecture would imply Handa's result without the `abelian \(p\)-group' assumption. The author proves the conjecture for \(\omega\)-categorical theories, stable theories, o-minimal theories, and also shows that if \((M,\cdot)\) has simple theory then its theory is not \((2,3)\)-homogeneous.
scientific article; zbMATH DE number 1247897