  The paper presents algebraic and logical developments. From the algebraic
viewpoint, we introduce Monadic Equational Systems as an abstract enriched
notion of equational presentation. From the logical viewpoint, we provide
Equational Metalogic as a general formal deductive system for the derivability
of equational consequences. Relating the two, a canonical model theory for
Monadic Equational Systems is given and for it the soundness of Equational
Metalogic is established. This development involves a study of clone and
double-dualization structures. We also show that in the presence of free
algebras the model theory of Monadic Equational Systems satisfies an internal
strong-completeness property.
