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 %constructions the model theory of Monadic Equational Systems satisfies an internal strong-completeness property.
Keywords: Monoidal action; strong monad; clones; double dualization; equational presentation; free algebra; equational logic; soundness; (strong) completeness
2010 MSC: 18A15; 18C10; 18C15; 18C20; 18C50; 18D20; 18D25; 68Q55; 03B22
Theory and Applications of Categories, Vol. 27, 2013, No. 18, pp 464-492.