Persistent properties and an application to algebras of logic (Q1272145)

Persistent properties of a class \(\mathcal K\) of algebraic structures are properties that are preserved under extensions (in \(\mathcal K\)) of structures in \(\mathcal K\). Persistent properties have been studied by \textit{L. Henkin} [1956] and \textit{A. Robinson} [1951, 1956]. Now the authors show that persistent cases of a first-order property are always definable by primitive existential formulas. For instance, every persistent atom in a class of simple relation algebras satisfies a primitive existential formula that defines atoms in all simple relation algebras. As a consequence of the definability of persistent atoms, a characterization of maximal relation algebras is obtained in the paper.