Irreducible decomposition of monomial ideals

In this paper we present two algorithms for irreducible decomposition of monomial ideals. We first use staircase structures to study the monomial ideals. We generalize the shifting degrees rule from two variables to three variables and then arbitrary case. With the aid of monomial tree representation, a new algorithm for irreducible decomposition of monomial ideals is provided. For the second method, we associate a monomial ideal with a Scarf complex, which is introduced by Herbert Scarf. Every facet of the Scarf complex corresponds to an irreducible component, and vice versa. Milowski(2004) developed a method to enumerate all the facets based on reverse search by exchanging one monomial at a time. We define a facet graph where nodes are facets and there is an edge from a facet FA to another facet FB if FB can be obtained from FA by exchanging one vertex. We give a new generic deformation called ordinal deformation and prove that the facet graph of an ordinally generic monomial ideal is strongly connected. This yields a simpler and more efficient enumeration algorithm.