The pp conjecture for the space of orderings of the field R(x,y)R(x,y)

The paper considers the space of orderings (XR(x,y),GR(x,y))(XR(x,y),GR(x,y)) of the field of rational functions over RR in two variables. It is shown that the pp conjecture fails to hold for such a space; an example of a positive primitive formula which is not product-free and one-related is investigated and it is proven, that although the formula holds true for every finite subspace of (XR(x,y),GR(x,y))(XR(x,y),GR(x,y)), it is false in general. This provides a negative answer to one of the questions raised in [M. Marshall, Open questions in the theory of spaces of orderings, J. Symbolic Logic 67 (2002) 341–352]. This work is a sequel to the previous results presented in [P. Gładki, M. Marshall, The pp conjecture for spaces of orderings of rational conics, J. Algebra Appl. 6 (2) (2007) 245–257]. Both spaces of orderings of conic sections and the space (XR(x,y),GR(x,y))(XR(x,y),GR(x,y)) are important examples of spaces of stability index 2 that are within the scope of our research.