The Slice Algorithm for irreducible decomposition of monomial ideals

monomialideal.lib

“…The first one only uses the characterization of irreducible monomial ideals, it is very easy but sometimes is inefficient since it generates many superfluous components that have to be removed. The other methods that we have implemented are based on works by Gao and Zhu [8], Miller [9], Milowski [10], and Roune [11], [12]. One can choose a particular method using an optional argument when calling the function.…”

Siphons in Chemical Reaction Networks

“…Displayed below is a portion the output of the last command above; the list tells the number of minimal siphons of each possible size. The current version of dual in Macaulay 2 uses Roune's implementation of his slice algorithm [19]. For background on the relation of Alexander duality and primary decomposition of monomial ideals, see the textbook [18].…”

“…Rather than implementing any such algorithm from scratch, it is convenient to harness existing tools for monomial and binomial primary decomposition [12,13]. We recommend the widely-used computer algebra system Macaulay 2 [15], and the implementations developed by Kahle [16] and Roune [19].…”

“…The current version of dual in Macaulay 2 uses Roune's implementation of his slice algorithm [19]. For background on the relation of Alexander duality and primary decomposition of monomial ideals, see the textbook [18].…”

(n-1)-st Koszul homology and the structure of monomial ideals

“…The components of the irredundant irreducible decomposition of a monomial ideal I also correspond to the minimal generators of its Alexander dual [8], to the facets of its Scarf complex [9], and to the maximal standard monomials associated to I [11]. These correspondences have originated different algorithms for the computation of irredundant irreducible decompositions.…”