Decomposition of tournament limits

TL;DR

This paper extends the theory of tournament limits and kernels, demonstrating their decomposition into irreducible components with a focus on transitivity and irreducibility, and establishing the uniqueness of this decomposition.

Contribution

It introduces a decomposition framework for tournament limits and kernels into irreducible components, highlighting their structure and uniqueness.

Findings

Decomposition of tournament kernels into irreducible components

Decomposition of tournament limits into irreducible components

Uniqueness of the decomposition process

Abstract

The theory of tournament limits and tournament kernels (often called graphons) is developed by extending common notions for finite tournaments to this setting; in particular we study transitivity and irreducibility of limits and kernels. We prove that each tournament kernel and each tournament limit can be decomposed into a direct sum of irreducible components, with transitive components interlaced. We also show that this decomposition is essentially unique.