  All irreducible representations of the Chinese monoid $C_n$, of any rank $n$,
over a nondenumerable algebraically closed field $K$, are constructed. It turns
out that they have a remarkably simple form and they can be built inductively
from irreducible representations of the monoid $C_2$. The proof shows also that
every such representation is monomial. Since $C_n$ embeds into the algebra
$K[C_n]/J(K[C_n])$, where $J(K[C_n])$ denotes the Jacobson radical of the
mooned algebra $K[C_n]$, a new representation of $C_n$ as a subdirect product
of the images of $C_n$ in the endomorphism algebras of the constructed simple
modules follows.
