Term algebras of elementarily equivalent atom structures

TL;DR

This paper demonstrates that two relation algebra atom structures can be elementarily equivalent while their term algebras are not, addressing a specific open problem in the theory of relation algebras.

Contribution

It provides a counterexample showing the non-equivalence of elementary equivalence and term algebra isomorphism in relation algebras.

Findings

Two relation algebra atom structures are elementarily equivalent but have non-isomorphic term algebras

Answers an open problem from Hirsch and Hodkinson's book on relation algebras

Highlights limitations of elementary equivalence in predicting algebraic structure

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.