  We describe the local structure of an irreducible algebraic monoid $M$ at an
idempotent element $e$. When $e$ is minimal, we show that $M$ is an induced
variety over the kernel $MeM$ (a homogeneous space) with fibre the two-sided
stabilizer $M_e$ (a connected affine monoid having a zero element and a dense
unit group). This yields the irreducibility of stabilizers and centralizers of
idempotents when $M$ is normal, and criteria for normality and smoothness of an
arbitrary $M$. Also, we show that $M$ is an induced variety over an abelian
variety, with fiber a connected affine monoid having a dense unit group.
