Exact Borel subalgebras of quasi-hereditary monomial algebras

Green and Schroll give an easy criterion for a monomial algebra $A$ to be quasi-hereditary with respect to some partial order $\leq_A$. A natural follow-up question is under which conditions a monomial quasi-hereditary algebra $(A, \leq_A)$ admits an exact Borel subalgebra in the sense of K\"onig. In this article, we show that it always admits a Reedy decomposition consisting of an exact Borel subalgebra $B$, which has a basis given by paths, and a dual subalgebra. Moreover, we give an explicit description of $B$ and show that it is the unique exact Borel subalgebra of $A$ with a basis given by paths. Additionally, we give a criterion for when $B$ is regular, using a criterion by Conde.