Representation of HH-closed monoreflections in archimedean ellell-groups with weak unit

The category of the title is called $mathcal{W}$. This has all free objects $F(I)$ ($I$ a set). For an object class $mathcal{A}$, $Hmathcal{A}$ consists of all homomorphic images of $mathcal{A}$-objects. This note continues the study of the $H$-closed monoreflections $(mathcal{R}, r)$ (meaning $Hmathcal{R} = mathcal{R}$), about which we show ({em inter alia}): $A in mathcal{A}$ if and  only if $A$ is a countably up-directed union from $H{rF(omega)}$. The meaning of this is then analyzed for two important cases: the maximum essential monoreflection $r = c^{3}$, where $c^{3}F(omega) = C(RR^{omega})$, and $C in H{c(RR^{omega})}$ means $C = C(T)$, for $T$ a closed subspace of $RR^{omega}$; the epicomplete, and maximum, monoreflection, $r = beta$, where $beta F(omega) = B(RR^{omega})$, the Baire functions, and $E in H{B(RR^{omega})}$ means $E$ is {em an} epicompletion (not ``the'') of such a $C(T)$

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 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 ℝω, B(ℝω) ; label the inclusion eS: F(ω) → S. Then (a) inj eS (the class of objects injective for eS) is such an R, with eS a reflection map iff S is closed under countable composition with itself (called ccc), (b) each such 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 βF(ω) = B(ℝω). The paper continues our earlier work along similar lines