The number of locally invariant orderings of a group

We show that if a nontrivial group admits a locally invariant ordering, then it admits uncountably many locally invariant orderings. For the case of a left-orderable group, we provide an explicit construction of uncountable families of locally invariant orderings; for a general group we provide an existence theorem that applies compactness to yield uncountably many locally invariant orderings.Comment: 13 page