Vol. 8, No. 6, 2014

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Decompositions of commutative monoid congruences and binomial ideals

Thomas Kahle and Ezra Miller

Vol. 8 (2014), No. 6, 1297–1364

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.

commutative monoid, monoid congruence, primary decomposition, mesoprimary decomposition, binomial ideal, coprincipal ideal, associated prime
Mathematical Subject Classification 2010
Primary: 20M14, 05E40, 20M25
Secondary: 20M30, 20M13, 05E15, 13F99, 13C05, 13P99, 13A02, 68W30, 14M25, 20M14, 05E40
Received: 8 February 2012
Revised: 13 May 2014
Accepted: 18 June 2014
Published: 2 October 2014
Thomas Kahle
Fakultät für Mathematik
Otto-von-Guericke Universität Magdeburg
Institut Algebra und Geometrie
D-39106 Magdeburg
Ezra Miller
Mathematics Department
Duke University
Durham, NC 27708
United States