2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/141588This is the first of a series of papers in which we initiate and develop the theory of reflection monoids, motivated by the theory of reflection groups. The main results identify a number of important inverse semigroups as reflection monoids, introduce new examples, and determine their orders.22 pages. A number of minor revisions and a rewrite of the section on the Renner monoidsGroup TheoryCombinatoricstext