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Functorial polar functions. (English) Zbl 1265.06050

Summary: \(\mathbf {W}_{\infty }\) denotes the category of Archimedean \(\ell \)-groups with designated weak unit and complete \(\ell \)-homomorphisms that preserve the weak unit. \(\mathbf {CmpT}_{2,\infty }\) denotes the category of compact Hausdorff spaces with continuous skeletal maps. This work introduces the concept of a functorial polar function on \(\mathbf {W}_{\infty }\) and as its dual a functorial covering function on \(\mathbf {CmpT}_{2,\infty }\). We demonstrate that functorial polar functions give rise to reflective hull classes in \(\mathbf {W}_{\infty }\) and that functorial covering functions give rise to coreflective covering classes in \(\mathbf {CmpT}_{2,\infty }\). We generate a variety of reflective and coreflecitve subcategories and prove that for any regular uncountable cardinal \(\alpha \), the class of \(\alpha \)-projectable \(\ell \)-groups is reflective in \(\mathbf {W}_{\infty }\) and the class of \(\alpha \)-disconnected compact Hausdorff spaces is coreflective in \(\mathbf {CmpT}_{2,\infty }\). Lastly, the notion of a functorial polar function (respectively functorial covering function) is generalized to sublattices of polars (respectively sublattices of regular closed sets).

MSC:

06F15 Ordered groups
18A40 Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
54B30 Categorical methods in general topology
54G05 Extremally disconnected spaces, \(F\)-spaces, etc.
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[1] BALL, R. N.– HAGER, A. W.– MACULA, A. J.: An {\(\alpha\)}-disconnected space has no proper monic preimages, Topology Appl. 37 (1990), 141–151. · Zbl 0734.54005 · doi:10.1016/0166-8641(90)90059-B
[2] CONRAD, P. F.: The essential closure of an archimedean lattice-ordered group, Duke Math. J. 38 (1971), 151–160. · Zbl 0216.03104 · doi:10.1215/S0012-7094-71-03819-1
[3] CARRERA, R. E.: Operators on the Category of Archimedean Lattice-ordered Groups with Designated Weak Unit. Doctoral Dissertation, University of Florida, Gainesville, 2004.
[4] HAGER, A. W.: Minimal covers of topological spaces, Ann. New York Acad. Sci. 552 (1989), 44–59. · Zbl 0881.54025 · doi:10.1111/j.1749-6632.1989.tb22385.x
[5] HAGER, A. W.– MARTÍNEZ, J.: The laterally {\(\sigma\)}-complete reflection of an archimedean lattice-ordered group. In: Ordered Algebraic Structures. Cura cao 1995 (W. C. Holland, J. Martínez, eds.), Kluwer Acad. Publ., Dordrecht, 1997, pp. 217–236.
[6] HAGER, A. W.– MARTÍNEZ, J.: Hulls for various kinds of {\(\alpha\)}-completeness in archimedean lattice-ordered groups, Order 16 (1999), 89–103. · Zbl 0953.06017 · doi:10.1023/A:1006323031986
[7] HAGER A. W.– ROBERTSON, L. C.: Representing and ringifying a Riesz space. In: Sympos. Math. 21, Academic Press, London, 1977, pp. 411–431. · Zbl 0382.06018
[8] HERRLICH, H.– STRECKER, G.: Category Theory. Sigma Series in Pure Math No. 1, Heldermann Verlag, Berlin, 1979. · Zbl 0437.18001
[9] MARTÍNEZ, J.: Polar functions, I: The summand-inducing hull of an archimedean -group with unit. In: Ordered Algebraic Structures. Proc. Gainesville Conf., 2001 (J. Martínez, ed.), Kluwer Acad. Pub., Dordrecht, 2002, pp. 275–299.
[10] MARTÍNEZ, J.: Polar functions, III: On irreducilbe maps vs. essential extensions of archimedean -groups with unit, Tatra Mt. Math. Publ. 27 (2003), 189–211.
[11] MARTÍNEZ, J.: Hull classes of archimedean lattice-ordered groups with unit: A Survey. In: Ordered Algebraic Structures. Proc. Gainesville Conf., 2001 (J. Martínez, ed.), Kluwer Acad. Pub., Dordrecht, 2002, pp. 89–121.
[12] MACULA, A. J.: {\(\alpha\)}-Dedekind complete archimedean vector lattices versus {\(\alpha\)}-quasi-F, Topology Appl. 44 (1992), 217–234. · Zbl 0774.06010 · doi:10.1016/0166-8641(92)90097-J
[13] PORTER, J. R.– WOODS, R. G. Extensions and Absolutes of Hausdorff Spaces, Springer-Verlag, Berlin-Heidelberg-New York, 1989. · Zbl 0652.54016
[14] VERMEER, J.: On perfect irreducible preimages, Topology Proc. 9 (1984), 173–189. · Zbl 0561.54029
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