Term algebras of elementarily equivalent atom structures
H. Andréka, I. Németi
TL;DR
The paper addresses whether elementary equivalence of relation-algebra atom structures transfers to elementary equivalence of their term algebras in $L_{\omega\omega}$. It constructs a completely representable, simple atom structure $\mathcal{S}$ and a nontrivial ultrapower $\mathcal{S}'$ that are elementarily equivalent at the atom-structure level, yet yield term algebras that are not elementarily equivalent. A plant-and-leaf configuration is used to distinguish finite versus infinite plants via a first-order sentence $\varphi$, which holds in the term algebra of $\mathcal{S}$ but fails in that of $\mathcal{S}'$ due to the presence of infinitely many leaves in the ultrapower. This provides a negative answer to Problem $14.19$ and highlights a separation between elementary equivalence of atom structures and that of their term algebras; the paper also notes an additional example by Hirsch and Hodkinson.
Abstract
We exhibit two relation algebra atom structures such that they are elementarily equivalent but their term algebras are not. This answers Problem 14.19 in the book Hirsch, R. and Hodkinson, I., "Relation Algebras by Games", North-Holland, 2002.