Cyclic 2-structures and spaces of orderings of power series fields in two variables

We consider the space of orderings of the field R((x,y)) and the space of orderings of the field R((x))(y), where R is a real closed field. We examine the structure of these objects and their relationship to each other. We define a cyclic 2-structure to be a pair (S,Φ) where S is a cyclically ordered set and Φ is an equivalence relation on S such that each equivalence class has exactly two elements. We show that each of these spaces of orderings is described by a cyclic 2-structure, in a natural way. We also show that if the real closed field R is archimedean then the space of R-places of these fields is describable in terms of the cyclic 2-structure.