  The goal of this paper is to use topological methods to compute
$\mathrm{Ext}$ between irreducible representations of von Neumann regular
monoids in which Green's $\mathscr L$- and $\mathscr J$-relations coincide
(e.g., left regular bands). Our results subsume those of S.~Margolis,
F.~Saliola, and B.~Steinberg, \emph{Combinatorial topology and the global
dimension of algebras arising in combinatorics}, J. Eur. Math. Soc. (JEMS),
\textbf{17}, 3037--3080 (2015).
  Applications include computing $\mathrm{Ext}$ between arbitrary simple
modules and computing a quiver presentation for the algebra of Hsiao's monoid
of ordered $G$-partitions (connected to the Mantaci-Reutenauer descent algebra
for the wreath product $G\wr S_n$). We show that this algebra is Koszul,
compute its Koszul dual and compute minimal projective resolutions of all the
simple modules using topology. More generally, these results work for CW left
regular bands of abelian groups. These results generalize the results of
S.~Margolis, F.~V. Saliola, and B.~Steinberg. \emph{Cell complexes, poset
topology and the representation theory of algebras arising in algebraic
combinatorics and discrete geometry}, Mem. Amer. Math. Soc., \textbf{274},
1--135, (2021).
