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arXiv · 1704.02666

Ordered groups as a tensor category

Dale Rolfsen
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Abstract

It is a classical theorem that the free product of ordered groups is orderable. In this note we show that, using a method of G. Bergman, an ordering of the free product can be constructed in a functorial manner, in the category of ordered groups and order-preserving homomorphisms. With this functor interpreted as a tensor product this category becomes a tensor (or monoidal) category. Moreover, if $O(G)$ denotes the space of orderings of the group $G$ with the natural topology, then for fixed groups $F$ and $G$ our construction can be considered a function $O(F) \times O(G) \to O(F * G)$. We show that this function is continuous and injective. Similar results hold for left-ordered groups.

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Dale Rolfsen. 2017-04-09. Ordered groups as a tensor category. https://doi.org/10.2140/pjm.2018.294.181

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