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.

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
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