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

(2000). Complete precones on noncommutative integral domains. Communications in Algebra: Vol. 28, No. 1, pp. 103-119.

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.

Let $A$ be the algebra of all $n \times n$ matrices with entries from $\RR[x_1,\ldots,x_d]$ and let $G_1,\ldots,G_m,F \in A$. We will show that $F(a)v=0$ for every $a \in \RR^d$ and $v \in \RR^n$ such that $G_i(a)v=0$ for all $i$ if and only if $F$ belongs to the smallest real left ideal of $A$ which contains $G_1,\ldots,G_m$. Here a left ideal $J$ of $A$ is real if for every $H_1,\ldots,H_k \in A$ such that $H_1^T H_1+\ldots+H_k^T H_k \in J+J^T$ we have that $H_1,\ldots,H_k \in J$. We call this result the one-sided Real Nullstellensatz for matrix polynomials. We first prove by induction on $n$ that it holds when $G_1,\ldots,G_m,F$ have zeros everywhere except in the first row. This auxiliary result can be formulated as a Real Nullstellensatz for the free module $\RR[x_1,\ldots,x_d]^n$., v1 7 pages. v2 9 pages: revised abstract, extended introduction and references. To appear in J. Algebra