Abstract
It is shown that, in a category with a specified class ℳ of monics and under some mild hypothesis,there is a monoreflection maximum among those whose reflection maps lie in ℳ. Thus, for example, any variety, and “most” SP-classes in a variety, have both amaximum monoreflection and amaximum essential reflection (which might be the same, but frequently aren't, and which might be the identity functor, but frequently aren't). And, for example, under some mild hypotheses, beneath each “completion” lies a maximum monoreflection, so that, for example, any “category of rings” has amaximum functorial ring of quotients.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
F. W. Anderson: Lattice-ordered rings of quotients,Canad. J. Math. 17 (1965), 434–448.
M. Anderson and P. Contrad: Epicomplete l-groups,Alg. Univ. 12 (1981), 224–241.
M. Anderson and T. Feil:Lattice Ordered Groups, Reidel TMS, Kluwer (1988), Dordrecht.
R. N. Ball and A. W. Hager: Characterization of epimorphisms in archimedean l-groups and vector lattices,Lattice-Ordered Groups, Advances and Techniques; A. M. W. Glass and W. C. Holland, eds. Kluwer (1989) Dordrecht, Chapter 8.
R. N. Ball and A. W. Hager: Epicomplete archimedean l-groups,Trans. Amer. Math. Soc. 322 (1990), 459–478.
R. N. Ball and A. W. Hager: Epicompletion of archimedean l-groups and vector lattices with weak unit,J. Austral. Math (Series A) 48 (1990), 25–56.
R. N. Ball and A. W. Hager: Algebraic extensions of an archimedean l-group I,J. Pure and Appl. Algebra 85 (1993), 1–20.
R. N. Ball and A. W. Hager: Algebraic extensions of an archimedean l-group II, in preparation.
A. Bigard, K. Keimel, and S. Wolfenstein:Groups et Anneaux Reticules, Springer Lecture Notes608 (1977), Berlin, Heidelberg, New York.
G. Birkhoff: The meaning of completeness,Ann. of Math. 38 (1937), 57–60.
P. F. Conrad: The essential closure of an archimedean lattice-ordered group,Proc. London Math. Soc. 38 (1971), 151–160.
P. Freyd:Abelian Categories, Harper and Row (1965), New York.
A. W. Hager and J. Martinez: Functorial Rings of Quotients I,Proc. Conf. Ordered Alg. Struc., Gainesville, 1991, W. C. Holland and J. Martinez, Eds., Kluwer (1993), Dordrecht, 133–157.
A. W. Hager and J. Martinez: Functorial rings of Quotients II,Forum Math., to appear.
A. W. Hager and J. Martinez: Pushout Invariant Extensions, to appear.
A. W. Hager and L. C. Robertson: Representing and ringifying a Riesz space,Sympos. Math. XXI (1977), 411–431.
D. Hajek and G. Strecker: Direct Limit of Hausdorff Spaces,Proceedings of the Third Prague Topological Symposium, 1971 (1972), 165–169.
H. Herrlich and G. Strecker: H-closed spaces and reflective subcategories,Math. Ann. 177 (1968), 302–309.
H. Herrlich and G. Strecker:Category Theory, Sigma Ser. PM 1, Heldermann Verlag (1979), Berlin.
J. Isbell: Epimorphisms and dominions, pp. 232–246 ofProceedings of the Conference on Categorical Algebra, La Jolla, 1965, Springer 1966, Berlin.
N. Jacobson:Basic Algebra II (Second Edition), Freeman (1989), New York.
J. Lambek:Lectures on Rings and Modules; Chelsea Publ. Co. (1976), New York.
J. Madden: κ-frames,J. Pure Appl. Algebra 70 (1991), 107–127.
J.-P. Olivier: Anneaux absolument plats universels et epimorphismes a buts reduits; Sem. d'Alg. Comm. 1967–68, Dir. P. Samuel; (les Epimorphismes d'Anneaux;) ENSdJF (1968), Paris.
J.-P. Olivier: Le foncteurT −∞; globalisation du foncteur T. Sem. d'Alg. Comm. 1967–68, Dir. P. Samuel; (les Epimorphismes d'Anneaux;) ENSdJF (1968), Paris.
N. Schwartz:Epimorphisms of f-Rings; Proc. Conf. Ordered Alg. Struc., Curacao 1988, J. Martinez, Ed., Kluwer (1989), Dordrecht.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hager, A.W., Martinez, J. Maximum monoreflections. Appl Categor Struct 2, 315–329 (1994). https://doi.org/10.1007/BF00873037
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00873037