Generalized rings of power series and finiteness spaces

Starting with an ordered monoid with an additional strictness condition, Ribenboim gave a construction of a ring, called ring of generalized power series. As examples, one can that way recover the rings of classical power series, the ring of Laurent power series, the ring of arithmetic functions, and so forth. Despite the unifying power of this construction, the strictness condition on the monoid seemed conceptually unclear. By further generalizing this construction using the notion of finiteness spaces from Ehrhard, we demonstrate that it is precisely the appropriate condition to represent such monoids as internal monoids in an appropriate category. Moreover, we obtain as additional examples the ring of Puiseux series, the ring of polynomials of degree at most n and the ring of formal power series on any alphabet. We then study the linear topology on these rings and give a characterization of the ones for which the topology is Cauchy complete. If time allows, we will discuss the particular example arising from an étale groupoid.