What general set-theoretical definitions of the notion of "structure" are there?
By general definition of "structure" I mean a formula $\Phi(x)$ in the first-order language of set theory such that
a set $x$ is a structure iff $\Phi(x)$
The definitions of structures in standard model theory would qualify, but they are restricted to first-order structures (or second-order, at most).
Suppes' set-theoretical predicates à la
a set $x$ is an algebraic structure iff $\phi(x)$
would qualify, but they are obviously too specific. (A rigorous meta-definition of "set-theoretical predicate" seems to be missing.)
Are there other approaches I'm just not aware of? Or is such a general definition considered useless?