Following the classical approach of Birkhoff, we suggest an enriched version of universal algebra. Given a suitable base of enrichment \(\mathcal {V}\) , we define a language \(\mathbb L\) to be a collection of (X, Y)-ary function symbols whose arities are taken among the objects of \(\mathcal {V}\) . The class of \(\mathbb L\) -terms is constructed recursively from the symbols of \(\mathbb L\) , the morphisms in \(\mathcal {V}\) , and by incorporating the monoidal structure of \(\mathcal {V}\) . Then, \(\mathbb L\) -structures and interpretations of terms are defined, leading to enriched equational theories. In this framework we characterize algebras for finitary monads on \(\mathcal {V}\) as models of enriched equational theories.
