The signature of a monomial ideal

The irreducible decomposition of a monomial ideal has played an important role in combinatorial commutative algebra, with applications beyond pure mathematics, such as biology. Given a monomial ideal $I$ of a polynomial ring $S=k[\mathbf{x}]$ over a field $k$ and variables $\mathbf{x}=\{x_1,\dots ,x_n\}$, its incidence matrix, is the matrix whose rows are indexed by the variables $\mathbf{x}$ and whose columns are indexed by its minimal generators. The main contribution of this paper is the introduction of a novel invariant of a monomial ideal $I$, termed its signature, which could be thought of as a type of canonical form of its incidence matrix, and the proof that two monomial ideals with the same signature have essentially the same irreducible decomposition.