A cellular algebra with specific decomposition of the unity

Let $\mathbb{A}$ be a cellular algebra over a field $\mathbb{F}$ with a decomposition of the identity $1_{\mathbb{A}}$ into orthogonal idempotents $e_i$, $i \in I$ (for some finite set $I$) satisfying some properties. We describe the entire Loewy structure of cell modules of the algebra $\mathbb{A}$ by using the representation theory of the algebra $e_i \mathbb{A} e_i$ for each $i$. Moreover, we also study the block theory of $\mathbb{A}$ by using this decomposition.