Geometry of commutative semigroups

Redei's theory of finitely generated commutative semigroups [1] describes a geometric embedding of the finitely generated free $\mathbb{N}$-semimodule $F(S)$. The first realisation is that the semimodule $\mathbb{N}$ can be completed to form the $\mathbb{Z}$ module $F^{\circ}(S)$. Then the points of a finitely generated $\mathbb{Z}$ module can be embedded in a $\mathbb{R}$ vector space, which is also the manifold $\mathbb{R}^n$. \[ F(S) \hookrightarrow F^{\circ}(S) \hookrightarrow \mathbb{R}^n \] Redei's embedding [1] will be briefly discussed. A consequence of this is that we can use geometric intuition in commutative semigroup theory. As embedded in $\mathbb{R}^n$ the free $\mathbb{N}$-semimodule is a set of "lattice points" in the sense of discrete geometry. In $\mathbb{R}^2$ this can visualized as a grid in the plane. In higher dimensions it is of course harder to perform a visualization. As topological semimodules $F(S)$ are both $F^{\circ}(S)$ are both trivial, because they are discrete. On the other hand, as a topological vector space $\mathbb{R}^n$ is far more interesting because among other things it is also a manifold. The important thing for semigroup theory, is any finitely generated commutative semigroup is a quotient of $F(S)$.

If we consider $F(S)$ to consist of multisets, then the multiplicities in $F(S)$ are natural numbers, because $F(S)$ is a $\mathbb{N}$ semimodule. The extension from $F(S)$ to $\mathbb{R}^n$ can be seen as extending $\mathbb{N}$ to form a full continuum of multiplicities. The elements of $\mathbb{R}^n$ are real-valued sets.

Finally, it is worth asking why this approach works for commutative semigroups and if it can be generalized. In fact, it cannot be further generalized because both associativity and commutativity are fundamental to this construction. In a commutative magma all you would get are trees, and the words of the free non-commutative semigroup cannot be separated from one another to form points in a geometric space.

References:
[1] Finitely generated commutative semigroups