A two-sided cell is called [*strongly regular*]{}, see [@MM1 Subsection 4.8], provided that

-   different left (resp. right) cells inside $\mathcal{J}$ are not comparable with respect to $\leq_L$ (resp. $\leq_R$);

-   the intersection of any left and any right cell inside $\mathcal{J}$ consists of exactly one element.

By [@KM2 Corollary 19], the first condition is automatically satisfied for all fiat $2$-categories.

By [@CM Subsection 3.2], each simple transitive $2$-representation $\mathbf{M}$ of ${\sc\mbox{C}\hspace{1.0pt}}$ has an [*apex*]{}, which is defined as a unique two-sided cell that is maximal in the set of all two-sided cells whose elements are not annihilated by $\mathbf{M}$.

Cell $2$-representations {#s2.5}
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For every $\mathtt{i}\in {\sc\mbox{C}\hspace{1.0pt}}$, we will denote by $\mathbf{P}_{\mathtt{i}}:={\sc\mbox{C}\hspace{1.0pt}}_A(\mathtt{i},{}_-)$ the corresponding [*principal*]{} $2$-representation. For a left cell $\mathcal{L}$ in ${\sc\mbox{C}\hspace{1.0pt}}$, there is $\mathtt{i}\in {\sc\mbox{C}\hspace{1.0pt}}$ such that all $1$-morphisms in $\mathcal{L}$ start at $\mathtt{i}$. Let $\mathbf{N}_{\mathcal{L}}$ denote the $2$-subrepresentation of $\mathbf{P}_{\mathtt{i}}$ given by the additive closure of all $1$-morphisms $\mathbf{F}$ satisfying $\mathbf{F}\geq_L \mathcal{L}$. Then $\mathbf{N}_{\mathcal{L}}$ has a unique maximal ${\sc\mbox{C}\hspace{1.0pt}}$-invariant ideal and the corresponding quotient $\mathbf{C}_{\mathcal{L}}$ is called the [*cell*]{} $2$-representation associated to $\mathcal{L}$. We refer the reader to [@MM1 Section 4] and [@MM2 Subsection 6.5] for more details.

Matrices in the Grothendieck group {#s2.6}
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Let 