Computability Theory and Ordered Groups

Ordered abelian groups are studied from the viewpoint of computability theory. In particular, we examine the possible complexity of orders on a computable abelian group. The space of orders on such a group may be represented in a natural way as the set of infinite paths through a computable tree, but not all such sets can occur in this way. We describe the connection between the complexity of a basis for a group and an order for the group, and completely characterize the degree spectra of the set of bases for a group. We describe some restrictions on the possible degree spectra of the space of orders, including a connection to algorithmic randomness