Decompositions of commutative monoid congruences and binomial ideals

Kahle, Thomas; Miller, Ezra

doi:10.2140/ant.2014.8.1297

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Decompositions of commutative monoid congruences and binomial ideals

Thomas Kahle and Ezra Miller

Vol. 8 (2014), No. 6, 1297–1364

DOI: 10.2140/ant.2014.8.1297

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.

Keywords

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

Milestones

Received: 8 February 2012

Revised: 13 May 2014

Accepted: 18 June 2014

Published: 2 October 2014

Authors

Thomas Kahle
Fakultät für Mathematik Otto-von-Guericke Universität Magdeburg Institut Algebra und Geometrie D-39106 Magdeburg Germany

Ezra Miller
Mathematics Department Duke University Durham, NC 27708 United States

Vol. 8, No. 6, 2014