Representation of $H$-closed monoreflections in archimedean $\ell$-groups with weak unit

Document Type : Research Paper


1 Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario L85 4K1, Canada.

2 Department of Mathematics and CS, Wesleyan University, Middletown, CT 06459.



 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 $C(T)$.


[BH89] Ball, R.N. and Hager, A.W., Characterization of epimorphisms in Archimedean
lattice-ordered groups and vector lattices, In: “Lattice-Ordered Groups”, Math.
Appl. 48, Kluwer Academic Publisher, 1989, 175-205.
[BH90] Ball, R.N. and Hager, A.W., Epicompletion of Archimedean `-groups and vector
lattices with weak unit, J. Austral. Math. Soc. Ser. A 48(1) (1990), 25-56.
[BHW-W15] Ball, R.N., Hager, A.W., and Walters-Wayland, J., Pointfree pointwise
suprema in unital archimedean `-groups, J. Pure Appl. Algebra 219(11) (2015),
[B05] Banaschewski, B., On the function ring functor in pointfree topology, Appl. Categ.
Structures 13 (2005), 305-328.
[B08] Banaschewski, B., On the function rings of pointfree topology, Kyungpook Math.
J. 48(2) (2008), 195-206.
[B14] Banaschewski, B. On the characterization of the function rings in pointfree topology,
Lecture in the conference: Aspects of Contemporary Topology V, Vrije Universiteit
Brussels, September 2014.
[E89] Engelking, R. “General Topology”, Revised and completed edition, Sigma Series
in Pure Mathematics 6, Heldermann Verlag, 1989.
[GJ76] Gillman, L. and Jerison, M., “Rings of Continuous Functions”, Graduate Texts
in Mathematics 43, Springer-Verlag, 1976.
[H85] Hager, A.W. Algebraic closures of `-groups of continuous functions, In: “Rings of
Continuous Functions” (C.E. Aull, eds.). Dekker Notes 95, 1985, 165-194.
[HM85] Hager, A.W. and Madden, J., Essential reflections versus minimal embeddings,
J. Pure Appl. Algebra 37(1) (1985), 27-32.
[HM16] Hager, A.W. and Madden, J., The H-closed monoreflections, implicit operations,
and countable composition, in archimedean lattice-ordered groups with weak unit,
Appl. Categ. Structures 24(5) (2016), 605-617.
[HR77] Hager, A.W. and Robertson, L.C., Representing and ringifying a Riesz space,
Symposia Mathematica XXI, Academic Press (1977), 411-431.
[HIJ61] Henriksen, M., Isbell, J., and Johnson, D., Residue class fields of lattice-ordered
algebras, Fund. Math. 50 (1961), 107-117.
[I64] Isbell, J., “Uniform Spaces”, Math. Surveys 12, American Math. Society, 1964.
[P65] Pasynkov, B.A., On the spectral decomposition of topological spaces, Mat. Sb.
66(108) (1965), 35-79, and Amer. Math. Soc. Translations Series 2 (74) (1968),
[S69] Sikorski, R. “Boolean Algebras”, Third Edition, Springer, 1969.