A Semiring Structure for Generalised P\'olya Urns

We define the notions of disjoint unions and products for generalised P\'olya urns, proving that this turns the set of isomorphism classes of urns into a commutative semiring. The set of square matrices up to similarity by a permutation matrix is also a commutative semiring under the operations of direct sum and Kronecker sum, and we prove that assigning to an urn its intensity matrix leads to a morphism of semirings. Moreover, we show that a second semiring morphism exists, sending intensity matrices to their spectra. This, together with the existence of the first morphism has implications for the asymptotic behaviour of product urns, which are discussed.