  Stanley decompositions are used in invariant theory and the theory of normal
forms for dynamical systems to provide a unique way of writing each invariant
as a polynomial in the Hilbert basis elements. Since the required Stanley
decompositions can be very long, we introduce a more concise notation called a
block decomposition, along with three notions of shortness (incompressibility,
minimality of Stanley spaces, and minimality of blocks) for block
decompositions. We give two algorithms that generate different block
decompositions, which we call elementary and gnomon decompositions, and give
examples. Soleyman-Jahan's criterion for a Stanley decomposition to come from a
prime filtration is reformulated to apply to block decompositions. We simplify
his proof, and apply the theorem to show that elementary and gnomon
decompositions come from "subprime" filtrations. In a sequel to this paper we
will introduce two additional algorithms that generate block decompositions
that may not always be subprime, but are always incompressible.
