Towards enriched universal algebra

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