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In this paper an equation means a homogeneous linear partial differential equation in $n$ unknown functions of $m$ variables which has real or complex polynomial coefficients. The solution set consists of all $n$-tuples of real or complex analytic functions that satisfy the equation. For a given system of equations we would like to characterize its Weyl closure, i.e. the set of all equations that vanish on the solution set of the given system. It is well-known that in many special cases the Weyl closure is equal to $B_m(F)N \cap A_m(F)^n$ where $F$ is either the field of real or complex numbers, $A_m(F)$ (respectively $B_m(F)$) consists of all linear partial differential operators with coefficients in $F[x_1,\ldots,x_m]$ (respectively $F(x_1,\ldots,x_m)$) and $N$ is the submodule of $A_m(F)^n$ generated by the given system. Our main result is that this formula holds in general. In particular, we do not assume that the module $A_m(F)^n/N$ has finite rank which used to be a standard assumption. Our approach works also for the real case which was not possible with previous methods. Moreover, our proof is constructive as it depends only on the Riquier-Janet theory., Comment: 10 pages, submitted

Every total ordering of a commutative domain can be extended uniquely to its field of fractions. This result is extended in two directions. Firstly, the notion of a total ordering is generalized so that a nonzero element can have more than two signs (in fact, these signs form a group). Secondly, commutative domains are replaced by noncommutative ones and we consider the following types of rings of fractions: Ore extensions, maximal (right or two-sided) rings of fractions, division hulls of free algebras and epic fields. Throughout the paper several examples are given to illustrate the theory.

We argue that Post hulls of spaces of signatures are a natural generalization of Boolean hulls of spaces of orderings.

We give an intrinsic characterization of semigroups that have a fixed cyclic group as their homomorphic image. Our techniques are inspired by the theory of orderings of higher level for noncommutative rings.