Algebraic and analytic finite spaces of orderings

In this paper, the finite subspaces of orderings of the ring of regular functions on an algebraic set $V$ are compared with those of the ring of analytic function germs at a point of $V$. Necessary and sufficient conditions for subspaces to be isomorphic are given, both from a purely algebraic and from a more geometric point of view. As a result, a criterion for analytic separation of semialgebraic sets is proved. Extendability of such subspaces is also proved to be stable under suitable approximations.