Regularity in terms of reductions in local Noether lattices

An $(n,d)$-sequence in a local Noether lattice $(L,M)$ is a sequence of words which satisfy certain factorization properties. If $(L,M)$ satisfies the union condition, there exist $(n,d)$-sequences which can be extended to minimal bases for the powers of $M$. Consequently, if $(L,M)$ satisfies the union condition, $(L,M)$ is regular if and only if $M$ is a minimal reduction.