Classification of weakly Noetherian monomial algebras

We describe weakly Noetherian (i.e. satisfying the ascending chain condition on two-sided ideals) monomial algebras as follows. Let $A$ be a weakly Noetherian monomial algebra. Then there exists a Noetherian set of (super-)words $\mathcal U$ such that every non-zero word in $A$ is a subword of a word belonging to $\mathcal U$. A finite set of words or superwords $\mathcal U$ is said to be Noetherian, if every element of $\mathcal U$ is either a finite word or a product of a finite word and one or two uniformly-recurring superwords (in the last case one of these superwords is infinite to the left and the other one to the right).