  The matrix representation of the set $\Delta({\bf d}^3)$, ${\bf
d}^3=(d_1,d_2, d_3)$, of the integers which are unrepresentable by
$d_1,d_2,d_3$ is found. The diagrammatic procedure of calculation of the
generating function $\Phi({\bf d}^3;z)$ for the set $\Delta({\bf d}^3)$ is
developed. The Frobenius number $F({\bf d}^3)$, genus $G({\bf d}^3)$ and
Hilbert series $H({\bf d}^3;z)$ of a graded subring for non--symmetric and
symmetric semigroups ${\sf S}({\bf d}^3)$ are found. The upper bound for the
number of non--zero coefficients in the polynomial numerators of Hilbert series
$H({\bf d}^m;z)$ of graded subrings for non--symmetric semigroups ${\sf S}
({\bf d}^m)$ of dimension, $m\geq 4$, is established.
