Decompositions of commutative monoid congruences and binomial ideals

Abstract

Abstract. Primary decomposition of commutative monoid congruences is insensitive to certain features of primary decomposition in commutative rings. These features are captured by the more refined theory of mesoprimary decomposition of congruences, introduced here complete with witnesses and associated prime objects. The combinatorial theory of mesoprimary decomposition lifts to arbitrary binomial ideals in monoid algebras. The resulting binomial mesoprimary decomposition is a new type of intersection decomposition for binomial ideals that enjoys computational efficiency and independence from ground field hypotheses. Binomial primary decompositions are easily recovered from mesoprimary decomposition.

Decompositions of cellular binomial ideals

“…The first half of this statement is [4, Theorem 8.1]. Theorem 15.11 in [10] generalizes this result, and also provides a converse.…”

“…The above definition is an extension of a construction of Kahle [9], that has its roots in the work of Kahle and Miller [10]. We note that in [9], the definition of embedded associated lattice requires only proper containment of the lattices involved; however, in the case of interest in [9], when I is ∆-cellular with a unique minimal prime (over the algebraic closure of ) and char( ) = 0, proper containment of those lattices implies that their ranks are not the same.…”

“…The notion of associated mesoprime from [10] is an evolved version of the concept of associated lattices that is used in this article (see [10,Remark 12.8]). In the case of cellular binomial ideals, every associated mesoprime is supported on an associated lattice.…”

“…Combinatorial descriptions of the primary components of a binomial ideal were provided by Dickenstein, Matusevich and Miller [2], in terms of connected components of graphs with infinitely many vertices, under the assumption that the base field is algebraically closed of characteristic zero. Later Kahle and Miller [10] removed all base field assumptions, and introduced new decompositions of binomial ideals, called mesoprimary decompositions, that are built to capture the combinatorial structure of a binomial ideal, and from which binomial primary decompositions can be recovered. The results in [2] and [10] are important and useful for theoretical purposes, but are currently computationally infeasible.…”

“…Later Kahle and Miller [10] removed all base field assumptions, and introduced new decompositions of binomial ideals, called mesoprimary decompositions, that are built to capture the combinatorial structure of a binomial ideal, and from which binomial primary decompositions can be recovered. The results in [2] and [10] are important and useful for theoretical purposes, but are currently computationally infeasible.…”