Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58539120180701Representation of $H$-closed monoreflections in archimedean $\ell$-groups with weak unit1136147510.29252/cgasa.9.1.1ENBernhardBanaschewskiDepartment of Mathematics and Statistics, McMaster University, Hamilton, Ontario L85 4K1, Canada.Anthony W.HagerDepartment of Mathematics and CS, Wesleyan University, Middletown, CT 06459.Journal Article20170722 The category of the title is called $\mathcal{W}$. This has all free objects $F(I)$ ($I$ a set). For an object class $\mathcal{A}$, $H\mathcal{A}$ consists of all homomorphic images of $\mathcal{A}$-objects. This note continues the study of the $H$-closed monoreflections $(\mathcal{R}, r)$ (meaning $H\mathcal{R} = \mathcal{R}$), 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 $C(T)$.https://cgasa.sbu.ac.ir/article_61475_f777dd362fb1959c3a9aa5115a63f9a9.pdf