Vol. 294, No. 1, 2018

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Ordered groups as a tensor category

Dale Rolfsen

Vol. 294 (2018), No. 1, 181–194
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 $O\left(G\right)$ 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 $O\left(F\right)×O\left(G\right)\to O\left(F\ast G\right)$. We show that this function is continuous and injective. Similar results hold for left-ordered groups.

Keywords
ordered group, free product, coproduct, tensor category, monoidal category
Mathematical Subject Classification 2010
Primary: 18D10, 20F60