2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/210460We 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.Final version, minor changes, to appear in Moscow Mathematical JournalAlgebraic Geometry14L10, 14L30, 14M17, 20M20text