韩国裸舞

Encoding modules with posets

Bar codes arise from exploration of data using one multiscale parameter. When the context allows or demands more than one parameter, partially ordered sets step in to encode topological summaries in finite fashion. The theory governing the tame behavior of topological summaries of data depends on the algebra of graded modules over rings of polynomials where the exponents are allowed to be real numbers instead of integers. The finiteness condition that gives rise to effective data structures for statistical analyses replaces the noetherian hypothesis, which routinely fails in this context. Out of this tameness surprisingly falls much of ordinary commutative algebra, including primary decomposition and syzygy theorems.