A Semiring Structure for Generalised P\'olya Urns

TL;DR

This paper introduces a semiring framework for generalized Pólya urns, linking their algebraic structure to matrix operations and spectral properties, with implications for understanding their asymptotic behavior.

Contribution

It defines disjoint unions and products for urns, establishing a semiring structure and connecting it to matrix operations and spectra, advancing the algebraic understanding of urn models.

Findings

Urns form a commutative semiring under defined operations.

Intensity matrices induce semiring morphisms.

Spectral mapping provides insights into asymptotic behavior.

Abstract

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.