Hölder categories

Rok, strany: 2014, 607 - 642

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 $\mathbb{R}$ 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 $\mathbf{W}^*$, the category of all archimedean lattice-ordered groups with a designated strong order unit, and the $\ell$-homomorphisms which preserve those units, and, more precisely, with interest in the epireflections on $\mathbf{W}^*$. 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 $\mathbf{W}^*$, 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.

ISO 690:

Hager, A., Martínez, J. 2014. Hölder categories. In Mathematica Slovaca, vol. 64, no.3, pp. 607-642. 0139-9918. DOI: https://doi.org/10.2478/s12175-014-0230-x

APA:

Hager, A., Martínez, J. (2014). Hölder categories. Mathematica Slovaca, 64(3), 607-642. 0139-9918. DOI: https://doi.org/10.2478/s12175-014-0230-x