Lattices of Equivalence Relations Closed Under the Operations of Relation Algebras

One of the longstanding problems in universal algebra is the question of which finite lattices are isomorphic to the congruence lattices of finite algebras. This question can be phrased as which finite lattices can be represented as lattices of equivalence relations on finite sets closed under certain first order formulas. We generalize this question to a different collection of first-order formulas, giving examples to demonstrate that our new question is distinct. We then prove that every lattice $\m M_n$ can be represented in this new way. [This is an extended version of a paper submitted to \emph{Algebra Universalis}.]