Towards enriched universal algebra

Following the classical approach of Birkhoff, we suggest an enriched version of enriched 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 an equational theories.