Propositional logic is intimately related to Boolean algebra, in the sense that both its syntax and semantics can be given an algebraic interpretation.
Are there any algebras that stand to second or higher order logics (or type theory) in the same way that boolean algebras stand to propositional logic?
Is there any well-known way of relating an algebraic structure to a fragment of a higher order logic?
On a related note, are there ways of relating the completeness of FOL to corresponding results in universal algebra, and ways of relating the theorem that second-order validity is not recursively enumerable to corresponding results in universal algebra?