Suppose $A$ is a finite dimensional algebra over a field $\Bbbk$ such that there is a filtration by two-sided ideals $$A=I_0\supset I_1\supset \cdots \supset I_n=0$$ such that for each $k$, either:

- I_k/I_{k+1} is a heredity ideal in A/I_{k+1} (in the sense of Cline, Parshall and Scott);
- or, $(I_k/I_{k+1})^2$ is $0$ in $A/I_{k+1}$, i.e., $I_k^2\subseteq I_{k+1}$.

Is there a name for algebras with such a filtration? If only 1. occurs, then we have the definition of a quasi-hereditary algebra.

My reason for asking is that in characteristic $0$ (or more generally "good" characterstic) algebras of finite semigroups have such a filtration (with even better properties in the heredity case) and I would like to know if there is a more general theory of such algebras and more general examples beyond the quasi-hereditary case.