On products of elementarily indivisible structures

Abstract

We say a structure $M$ in a first-order language is indivisible if for every coloring of its universe in two colors, there is a monochromatic substructure $M'$ of $M$ such that $M'$ is isomorphic to $M$. Additionally, we say that $M$ is symmetrically indivisible if $M'$ can be chosen to be symmetrically embedded in $M$ (that is, every automorphism of $M'$ can be extended to an automorphism of $M$). Similarly, we say that $M$ is elementarily indivisible if $M'$ can be chosen to be an elementary substructure. We define new products of structures in a relational language. We use these products to give recipes for construction of elementarily indivisible structures which are not transitive and elementarily indivisible structures which are not symmetrically indivisible, answering two questions presented by A. Hasson, M. Kojman and A. Onshuus.