  Many important computational structures involve an intricate interplay
between algebraic features (given by operations on the underlying set) and
relational features (taking account of notions such as order or distance). This
paper investigates algebras over relational structures axiomatized by an
infinitary Horn theory, which subsume, for example, partial algebras, various
incarnations of ordered algebras, quantitative algebras introduced by Mardare,
Panangaden, and Plotkin, and their recent extension to generalized metric
spaces and lifted algebraic signatures by Mio, Sarkis, and Vignudelli. To this
end, we develop the notion of clustered equation, which is inspired by Mardare
et al.'s basic conditional equations in the theory of quantitative algebras, at
the level of generality of arbitrary relational structures, and we prove that
it is equivalent to an abstract categorical form of equation earlier introduced
by Milius and Urbat. Our main results are a family of Birkhoff-type variety
theorems (classifying the expressive power of clustered equations) and an
exactness theorem (classifying abstract equations by a congruence property).
