  Hecke-Kiselman monoids $\textrm{HK}_{\Theta}$ and their algebras
$K[\textrm{HK}_{\Theta}]$, over a field $K$, associated to finite oriented
graphs $\Theta$ are studied. In the case $\Theta $ is a cycle of length
$n\geqslant 3$, a hierarchy of certain unexpected structures of matrix type is
discovered within the monoid $C_n=\textrm{HK}_{\Theta}$ and it is used to
describe the structure and the properties of the algebra $K[C_n]$. In
particular, it is shown that $K[C_n]$ is a right and left Noetherian algebra,
while it has been known that it is a PI-algebra of Gelfand-Kirillov dimension
one. This is used to characterize all Noetherian algebras
$K[\textrm{HK}_{\Theta}]$ in terms of the graphs $\Theta$. The strategy of our
approach is based on the crucial role played by submonoids of the form $C_n$ in
combinatorics and structure of arbitrary Hecke-Kiselman monoids
$\textrm{HK}_{\Theta}$.
