**Papers Accepted**

- Kahle, T; Miller, E; O’Neill, C,
*Irreducible decomposition of binomial ideals*, Compositio Mathematica, vol. 152 no. 06 (June, 2016), pp. 1319-1332

(last updated on 2018/08/15)**Abstract:**

Building on coprincipal mesoprimary decomposition [Kahle and Miller, 2014], we combinatorially construct an irreducible decomposition of any given binomial ideal. In a parallel manner, for congruences in commutative monoids we construct decompositions that are direct combinatorial analogues of binomial irreducible decompositions, and for binomial ideals we construct decompositions into ideals that are as irreducible as possible while remaining binomial. We provide an example of a binomial ideal that is not an intersection of irreducible binomial ideals, thus answering a question of Eisenbud and Sturmfels [1996].