What do we call collections of subsets of a monoid that satisfy these axioms?

Consider a monoid $M$ and a semiring $S$. Then there’s an $S$-algebra freely generated by the monoid $M$, which can be described explicitly as the set of all finitely supported functions $M \rightarrow S$, with multiplication given by convolution. For example, let $x$ denote an indeterminate. Then the monoid $\{x^n \mid n \in \mathbb{N}\}$ yields the $S$-algebra of univariate polynomials (with coefficients in $S$) in the above manner.

Okay, but perhaps we want more than just the finitely supported functions. Maybe we want power series, or Laurent series, or something like that. Then we’ll need to choose a collection of “small” subsets of $M$, and it needs to have the following properties:

  • Cofilter axioms.
    • Subsets of small sets are small.
    • $\emptyset_M$ is a small subset of $M$.
    • If $A$ and $B$ are small , then $A \cup B$ is small.
  • Monoid axioms.
    • $\{1_M\}$ is a small subset of $M$.
    • If $A$ and $B$ are small, then $AB$ is small.
  • Finiteness axiom.
    • For all small subsets $A$ and $B$, the function $f_{A,B}:A \times B \rightarrow M$ given by $f(a,b)=ab$ has the property that for all $m \in M$, the subset $f^{-1}(m)$ is finite.

Question. What do we call collections of subsets of a monoid that satisfy the above axioms?

I’m also interested in terminology for a monoid equipped with a collection of subsets satisfying the above axioms, and/or for the category of all such entities.

Observe that, having chosen such a collection of small subsets of $M$, the set of all functions $f : M \rightarrow S$ whose support is small forms an $S$-algebra, where multiplication is given by convolution.

Example. Let $x$ denote an indeterminate. Then we can declare that the small subsets of the monoid $\{x^n \mid n \in \mathbb{N}\}$ are the finite ones, in which case the above construction gives us the $S$-algebra of univariate polynomials. On the other hand, if we declare that every subset of $\{x^n \mid n \in \mathbb{N}\}$ is small, then we get the $S$-algebra of univariate power series. Now consider the monoid $M$ whose underlying set is $\{x^n \mid n \in \mathbb{Z}\}$ with the obvious notion of product. We’re not allowed to declare that every subset of $M$ is small, because this would violate the finiteness axiom. So instead, declare that a subset of $M$ is small iff it is bounded below. Then the above construction gives us all Laurent series.

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