Decompositions of commutative monoid congruences and binomial ideals

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.

62 pages, 7 figures, v2: small improvements over v1, v3: added Problem 17.7, Corollary 4.15 and other small refinements, v4: major revision: definition of witness adjusted (Section 4), and incorrect claims concerning binomial irreducible decomposition excised (Section 15), v5: various corrections and improvements. Final version, accepted by Algebra and Number Theory

References in corpus (2)

• Decompositions of Binomial Ideals in Macaulay 2
• Positive margins and primary decomposition

Cited by in corpus (11)

• Detecting Binomiality
• Parity binomial edge ideals
• Irreducible decomposition of binomial ideals
• Multistationarity in the space of total concentrations for systems that admit a monomial parametrization
• The Koszul property for spaces of quadrics of codimension three
• Minimal Generating Sets of Lattice Ideals
• Eigenschemes and the Jordan canonical form
• Decompositions of Cellular Binomial Ideals
• Some algebraic aspects of mesoprimary decomposition
• Binomial ideals and congruences on ${\mathbb{N}}^n$
• Mesoprimary decomposition of binomial submodules