  We consider Ribenboim's construction of rings of generalized power series.
Ribenboim's construction makes use of a special class of partially ordered
monoids and a special class of their subsets. While the restrictions he imposes
might seem conceptually unclear, we demonstrate that they are precisely the
appropriate conditions to represent such monoids as internal monoids in an
appropriate category of Ehrhard's finiteness spaces. Ehrhard introduced
finiteness spaces as the objects of a categorical model of classical linear
logic, where a set is equipped with a class of subsets to be thought of as
finitary. Morphisms are relations preserving the finitary structure. The notion
of finitary subset allows for a sharper analysis of computational structure
than is available in the relational model. For example, fixed point operators
fail to be finitary.
  In the present work, we take morphisms to be partial functions preserving the
finitary structure rather than relations. The resulting category is symmetric
monoidal closed, complete and cocomplete. Any pair of an internal monoid in
this category and a ring induces a ring of generalized power series by an
extension of the Ribenboim construction based on Ehrhard's notion of
linearization of a finiteness space. We thus further generalize Ribenboim's
constructions. We give several examples of rings which arise from this
construction, including the ring of Puiseux series and the ring of formal power
series generated by a free monoid.

Category Theory;Logic;Rings and Algebras