  First order formulas in a relational signature can be considered as
operations on the relations of an underlying set, giving rise to multisorted
algebras we call first order algebras. We present universal axioms so that an
algebra satisfies the axioms iff it embeds into a first order algebra.
Importantly, our argument is modular and also works for, e.g., the positive
existential algebras (where we restrict attention to the positive existential
formulas) and the quantifier-free algebras. We also explain the relationship to
theories, and indicate how to add in function symbols.
