Covers, q-binomial series, and q-lattices

An ordered cover of a finite set S is a finite sequence of nonempty subsets of S with union S. If the subsets are pairwise disjoint, then the cover is an ordered partition. An ordered cover of a finite vector space V is a finite sequence of nontrivial subspaces of V with sum V. If the sum is, in fact, a direct sum, then the cover is an ordered partition of V. Certain power series are generating functions for ordered partitions (in both senses—i.e., for finite sets and finite vector spaces). This dissertation develops a new class of infinite series that are generating functions for ordered covers (as distinct from partitions) in both senses. These series are elements of the algebra of q-binomial series that results from equipping the set of arithmetic functions (sequences of complex numbers) with the usual addition and scalar multiplication and with a new multiplication, the q-Newton product. In a particular sort of binomial poset, the q-lattice, the union of sets and the sum of vector spaces are both special cases of the join of lattice elements. Thus q-lattices unify and generalize several pairs of identities involving, on the one hand, binomial coefficients, and on the other hand, q-binomial coefficients. Formal q-binomial series are defined in terms of the join of q-lattice elements. As a consequence, particular q-binomial series turn out to be generating functions both for ordered covers of sets and ordered covers of vector spaces. When equipped with suitable operations, the collection of complex-valued functions on the intervals of a q-lattice becomes an algebra—the covering algebra of the lattice. A subalgebra—the reduced covering algebra—is isomorphic to the algebra of g-binomial series. These results provide an algebraic underpinning for q-binomial series much like that provided by incidence algebras and reduced incidence algebras for classical generating functions.