The H-Closed Monoreflections, Implicit Operations, and Countable Composition, in Archimedean Lattice-Ordered Groups with Weak Unit

In the category of the title, called W, we completely describe the monoreflections \(\mathcal {R}\) which are H-closed (closed under homomorphic image) by means of epimorphic extensions S of the free object on ω generators, F(ω), within the Baire functions on \(\mathbb {R}^{\omega }\), \(B(\mathbb {R}^{\omega })\); label the inclusion \(e_{S} : F(\omega ) \rightarrow S\). Then (a) inj eS (the class of objects injective for eS) is such an \(\mathcal {R}\), with eS a reflection map iff S is closed under countable composition with itself (called ccc), (b) each such \(\mathcal {R}\) is inj eS for a unique S with ccc, and (c) if S has ccc, then A∈inj eS iff A is closed under countable composition with S. We think of (c) as expressing: A is closed under the implicit operations of W represented by S (and these are of at most countable arity). In particular, the family of H-closed monoreflections is a set, whereas the family of all monoreflections is consistently a proper class. There is a categorical framework to the proofs, valid in any sufficiently complete category with free objects and epicomplete monoreflection β which is H-closed and of bounded arity; in W the β is of countable arity, and \(\beta F(\omega ) = B(\mathbb {R}^{\omega })\). The paper continues our earlier work along similar lines.