Extension Monads: Some Structure Theorems

We investigate the behavior of extension monads, introduced in the 1990s by the second author, in terms of structure results for infinitely many finitary operations and common constructions in varieties or categories of algebras. Specifically, we see that for finite collections of algebras of the same signature there the extension monad operation commutes with the direct product operation, and that the only retractions from an enlargement or extension monad are trivial.