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Mathematica Slovaca

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Volume 64, Issue 3

Issues

Hölder categories

Anthony Hager / Jorge Martínez
Published Online: 2014-07-05 | DOI: https://doi.org/10.2478/s12175-014-0230-x

Abstract

Hölder categories are invented to provide an axiomatic foundation for the study of categories of archimedean lattice-ordered algebraic structures. The basis of such a study is Hölder’s Theorem (1908), stating that the archimedean totally ordered groups are precisely the subgroups of the additive real numbers ℝ with the usual addition and ordering, which remains the single most consequential result in the studies of lattice-ordered algebraic systems since Birkhoff and Fuchs to the present.

This study originated with interest in W*, the category of all archimedean lattice-ordered groups with a designated strong order unit, and the ℓ-homomorphisms which preserve those units, and, more precisely, with interest in the epireflections on W*. In the course of this study, certain abstract notions jumped to the forefront. Two of these, in particular, seem to have been mostly overlooked; some notion of simplicity appears to be essential to any kind of categorical study of W*, as are the quasi-initial objects in a category. Once these two notions have been brought into the conversation, a Hölder category may then be defined as one which is complete, well powered, and in which

(a) the initial object I is simple, and

(b) there is a simple quasi-initial coseparator R.

In this framework it is shown that the epireflective hull of R is the least monoreflective class. And, when I = R — that is, the initial element is simple and a coseparator — a theorem of Bezhanishvili, Morandi, and Olberding, for bounded archimedean f-algebras with identity, can be be generalized, as follows: for any Hölder category subject to the stipulation that the initial object is a simple coseparator, every uniformly nontrivial reflection — meaning that the reflection of each non-terminal object is non-terminal — is a monoreflection.

Also shown here is the fact that the atoms in the class of epireflective classes are the epireflective hulls of the simple quasi-initial objects. From this observation one easily deduces a converse to the result of Bezhanishvili, Morandi, and Olberding: if in a Hölder category every epireflection is a monoreflection, then the initial object is a coseparator.

MSC: Primary 18A20, 18A40, 06F20; Secondary 08C05, 46A40, 54C40

Keywords: Holder category; (E,M)-category; epireflective subcategory; quasi-initial object; simple object; coseparator; archimedean l-group; strong unit

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About the article

Published Online: 2014-07-05

Published in Print: 2014-06-01


Citation Information: Mathematica Slovaca, Volume 64, Issue 3, Pages 607–642, ISSN (Online) 1337-2211, DOI: https://doi.org/10.2478/s12175-014-0230-x.

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© 2014 Mathematical Institute, Slovak Academy of Sciences. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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