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
denotes the space of
orderings of the group
with the natural topology, then for fixed groups
and
our construction can be
considered a function
.
We show that this function is continuous and injective. Similar results hold for
left-ordered groups.
