The generalised regularity classification for matrix semigroups over semirings

Let $\mathfrak{L}$ be the semiring under consideration, and let $M_n(\mathfrak{L})$, $UT_n(\mathfrak{L})$, and $U_n(\mathfrak{L})$ denote the corresponding matrix semigroups. The table records their generalised regularity properties according to $n$:

$$\begin{array}{c|c|c|c|c} & n=1 & n=2 & n=3 & n\geq 4\\ \hline M_n(\mathfrak{L}) & \text{Regular} & \text{Regular} & \text{Fountain (?) } & \text{Not Fountain}\\ UT_n(\mathfrak{L}) & \text{Regular} & \text{Abundant} & \text{Fountain (?) } & \text{Not Fountain}\\ U_n(\mathfrak{L}) & \text{Regular (group)} & \text{Regular (band)} & \text{Fountain} & \text{Fountain} \end{array}$$

Generalised regularity classification. The displayed classification summarises the generalised regularity properties of $M_n(\mathfrak{L})$, $UT_n(\mathfrak{L})$, and $U_n(\mathfrak{L})$ in the indicated cases.

The question marks in the $n=3$ entries indicate that those classifications are tentative rather than established assertions. Since the source presents this as a summary with unresolved entries, the precise status of the tentative cases should be checked against the paper's surrounding discussion.