Payne and Thullier independently have defined a canonical compactification of rational polyhedral cones, called extended rational polyhedral cones. We use "pointed" monoids, monoids with an absorbing element, to study extended rational polyhedral cones. The equivalence of toric monoids and rational polyhedral cones from previous results in toric geometry allow us to extend the equivalence to these more recently defined objects. This also allows us to explore affine toric varieties with new morphisms and constructions built from these extended cones and pointed monoids rather than their non-compact and unpointed analogs. Speaker(s): Alana Huszar (University of Michigan)