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Vector spaces have also have two natural operations on them that act like addition and multiplication, the direct sum and kronecker product. These operations do form a semiring, although again not on the nose.

If we’re working with vector spaces (without a choice of a basis) we usually talk about the direct sum and tensor product (instead of the Kronecker product). Then the set of vector spaces over a given field [1] forms a semiring once we quotient by isomorphisms, but it’s not a very exciting semiring: As each vector space is determined by its dimension, we just get something isomorphic to the semiring of non-negative integers.

However, there is a way to get something exciting out of this: We fix a space and allow our vector spaces to vary continuously on that space. An example would be the unit ring in the plane, where each vector space is the line tangent to a point on the ring. This again gives a semiring like before, but once we quotient by isomorphisms we can get a bunch of interesting invariants of the space. This is basically what K-theory is and it’s an active field of research.

[1] Things get a lot more complicated if we consider modules, that is, the equivalent of vector spaces but over a ring instead of a field.