Generalizations of the Burns-Hale Theorem

The Burns-Hale theorem states that a group G is left-orderable if and only if G is locally projectable onto the class of left-orderable groups. Similar results have appeared in the literature in the case of UPP groups and Conradian left-orderable groups, with proofs using varied techniques in each case. This note presents a streamlined approach to showing that if C is the class of either Conradian left-orderable, left-orderable, or UPP groups, then C contains all groups that are locally projectable onto C; and shows that this streamlined approach works for the class of diffuse groups as well. It also includes an investigation of the extent to which a similar theorem can hold for the classes of bi-orderable, circularly orderable or recurrent orderable groups.