Colon Structure of Associated Primes of Monomial Ideals

We find an explicit expression of the associated primes of monomial ideals as a colon by an element v, using the unique irredundant irreducible decomposition whose irreducible components are monomial ideals. An algorithm to compute v is given using Macaulay2. For squarefree monomial ideals the problem is related to the combinatorics of the underlying clutter or graph. For ideals of Borel type, the monomial v takes a simpler form, and we classify when v is unique.