Π10 classes and orderable groups Original Research Article

It is known that the spaces of orders on orderable computable fields can represent all View the MathML source classes up to Turing degree. We show that the spaces of orders on orderable computable abelian and nilpotent groups cannot represent View the MathML source classes in even a weak manner. Next, we consider presentations of ordered abelian groups, and we show that there is a computable ordered abelian group for which no computable presentation admits a computable set of representatives for its Archimedean classes.