PhD Theses.Valuated matroids are a generalisation of matroids; matroids themselves being an abstraction
of the notion of independence. Valuated matroids have many equivalent de nitions
including via independent sets and circuits, and in this thesis we show that a valuated
matroid has an equivalent de nition in terms of a rank function which we construct by
analogy with the matroid rank function by looking at matroid and valuated matroid
polytopes. We separately construct a hyperoperation which is an extension of a previously
studied operation of composing valuated matroids, this being the composition
of valuated linking systems. The composition of valuated linking systems can be seen
as a generalisation of matrix multiplication to tropical linear spaces. In particular, the
hyperoperation we introduce has been in
uenced by viewing matrices as representing
linear spaces, which we can do by looking at their row space, and consequently by how
these relate to Pl ucker coordinates. Working tropically, since tropical linear spaces are
equivalent to valuated matroids, which are also known as tropical Pl ucker vectors, we
create the hyperoperation by using the parallels with matrices representing linear spaces
over a eld. We describe the hyperproduct completely for small rank, where this operation
forms a hypergroup. In higher rank we investigate what known matroid subdivisions
it contains, as well as also showing that it does not form a fan, and nor is it convex in
general. We also conjecture this hyperoperation forms a hypergroup for higher rank, and
present some investigation towards this
