Abstract
This paper begins to develop a theory of non-commutative graded algebras and their Hilbert series which parallels the extensive already existing theory on commutative graded algebras. We begin with a discussion of “order ideals of monomials,” which give us vector space bases for these algebras. We develop next the concept of “weak summand,” which is a special relationship a subalgebra may have to an algebra, and the concept of “strongly free sets.” Strongly free sets have many properties which are analogous to the properties of regular sequences in commutative algebras. We discuss “combinatorially free” sets, which are useful for constructing examples of many of the preceding ideas. We conclude with an application of our results to algebraic topology by showing that there is a finite CW-complex with only nine positive-dimensional cells whose loop space has an irrational Poincaré series.
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