Non-integer multigraded algebra

The polynomial ring in n variables is a direct sum of one-dimensional vector spaces spanned by monomials. Modules that are similarly multigraded as direct sums of finite dimensional vector spaces over the integer lattice enjoy concrete, effective algebra. For example, monomial ideals have unique minimal primary decompositions as intersections of primary monomial ideals. What is it about the integer lattice that drives this good fortune? Would the same work if, say, the underlying multigrading were a direct sum over a real vector space instead? The ambient ring would then consist of real-exponent polynomials, and the answer has important implications for applications to topological data analysis. This talk explores how far the hypotheses for multigradings can be relaxed while maintaining fundamental results from commutative and homological algebra, with a particular focus on primary decomposition.