As cylindric algebra is the algebraization of FO with equality: How can be this algebra extended s.t. ESO is captured by it, e.g. how to algebraize quantifications of relation variables?
How can one define algebras for extensions of FO like inclusion logic FO$(\subseteq)$ or exclusion logic FO$(|)$. As FO$(\subseteq, |)$ captures ESO, it would be interesting to compare their algebras. Is there any research about that?
ESO: only existential quantifications of relation variables are allowed.