  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 $n>2$.
