2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/219154We described in [M1] a monoid acting on the integrable highest weight modules of a symmetrizable Kac-Moody algebra. It has similar structural properties as a reductive algebraic monoid with unit group a Kac-Moody group. Now we find natural extensions of the action of the Kac-Moody group on its building to actions of this monoid. These extensions are partly motivated by representation theory and the combinatorics of the faces of the Tits cone.37 pages, a new introduction, new results added, some typos correctedGroup TheoryRepresentation Theory20E42text