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