On products of elementarily indivisible structures

TL;DR

This paper introduces new product constructions of structures in a relational language to generate examples of elementarily indivisible structures with specific properties, addressing open questions in the field.

Contribution

It provides novel methods for constructing elementarily indivisible structures that are neither transitive nor symmetrically indivisible, answering previously open questions.

Findings

Constructed elementarily indivisible structures that are not transitive.

Constructed elementarily indivisible structures that are not symmetrically indivisible.

Developed new product techniques for relational 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.