Vol. 294, No. 1, 2018

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Ordered groups as a tensor category

Dale Rolfsen

Vol. 294 (2018), No. 1, 181–194

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) × O(G) O(F G). We show that this function is continuous and injective. Similar results hold for left-ordered groups.

ordered group, free product, coproduct, tensor category, monoidal category
Mathematical Subject Classification 2010
Primary: 18D10, 20F60
Received: 9 April 2017
Revised: 26 September 2017
Accepted: 27 September 2017
Published: 5 January 2018
Dale Rolfsen
Mathematics Department
University of British Columbia
Vancouver, BC