Minimal monomial reductions and the reduced fiber ring of an extremal ideal

Let I be a monomial ideal in a polynomial ring A= K[x₁,…,xₙ]. We call a monomial ideal J a minimal monomial reduction ideal of I if there exists no proper monomial ideal L ⊂ J such that L is a reduction ideal of~I. We prove that there exists a unique minimal monomial reduction ideal J of I and we show that the maximum degree of a monomial generator of J determines the slope p of the linear function (Iᵗ)=pt+c for t 0. We determine the structure of the reduced fiber ring F(J)_ of J and show that F(J)_ is isomorphic to the inverse limit of an inverse system of semigroup rings determined by convex geometric properties of J.