On the consistency of score sheets of a round-robin football tournament

TL;DR

This paper explores algebraic structures called submonoids related to score sheets in round-robin football tournaments, analyzing their invariants and properties using theoretical and computational methods.

Contribution

It introduces and studies the submonoids  and  of score sheets, providing a description of their Hilbert basis and showing  is Gorenstein for n>2.

Findings

The Hilbert basis of  is explicitly described.

 is Gorenstein for n>2.

The invariants of these monoids are characterized.

Abstract

In this paper we introduce the submonoids $\mathscr{R}_n$, resp. $\mathscr{C}_n$, of the monoid $\mathscr{M}_n$ of ordered score sheets of a robin-round tournament played by $n$ teams for which the order is preserved after the leader team is disqualified, resp. all principal submatrices preserve the given ordering. We study (using both theoretical and computational methods) the most important invariants of these monoids, namely the Hilbert basis, the multiplicity, the Hilbert series and the Hilbert function. In particular we give a general description of the Hilbert basis of $\mathscr{R}_n$ and we show that $\mathscr{C}_n$ is Gorenstein for $n>2$.