Matrix relation algebras. (Q1771950)

The collection of all \(n\times n\) matrices with entries from a given relation algebra \(\mathcal R\) forms a relation algebra \(M_n(\mathcal R)\) with naturally defined operations. The paper contains lists of properties of \(\mathcal R\) (e.g.\ atomicity, simplicity, representability, non-embeddability, density, etc.) which are preserved also in \(M_n(\mathcal R)\). The authors introduce so-called \(M_n\)-algebras as relation algebras enriched with \(n-1\) constants and subject to suitable equational axioms. The main result is that for fixed \(n>0\), the class \(M_n(\mathcal R)\) for some \(\mathcal R\) coincides with the class of reducts of \(M_n\)-algebras and hence has a first-order characterization. As a consequence, homomorphic images and simple proper extensions of matrix relation algebras are isomorphic to matrix relation algebras.