  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 (Theorem 3.1).
An algorithm to compute $v$ is given using Macaulay2 (Section 7). For
squarefree monomial ideals the problem is related to the combinatorics of the
underlying clutter or graph (Proposition 4.3). For ideals of Borel type the
monomial $f$ takes a simpler form (Proposition 5.2). The authors classify when
$f$ is unique (Proposition 6.2).
