The thesis of this paper is that considering the general setting of
monoids admitting such a triangularity, namely *J*-trivial monoids,
sheds further light on the topic. This is a step in an ongoing effort
to use representation theory to automatically extract combinatorial
structures from (monoid) algebras, often in the form of posets and
lattices, both from a theoretical and computational point of view, and
with an implementation in Sage.

Motivated by ongoing work on related monoids associated to Coxeter
systems, and building on well-known results in the semi-group
community (such as the description of the simple modules or the
radical), we describe how most of the data associated to the
representation theory (Cartan matrix, quiver) of the algebra of any
*J*-trivial monoid *M* can be expressed combinatorially by counting
appropriate elements in *M* itself. As a consequence, this data does
not depend on the ground field and can be calculated in
*O*(*n*^{2}), if
not *O*(*nm*), where *n*=|*M*| and *m* is the number of generators. Along
the way, we construct a triangular decomposition of the identity into
orthogonal idempotents, using the usual M\"obius inversion formula in
the semi-simple quotient (a lattice), followed by an algorithmic
lifting step.

Applying our results to the 0-Hecke algebra (in all finite types),
we recover previously known results and additionally provide an
explicit labeling of the edges of the quiver. We further explore
special classes of *J*-trivial monoids, and in particular monoids of
order preserving regressive functions on a poset, generalizing known
results on the monoids of nondecreasing parking functions.

Received: October 23, 2010. Revised: February 7, 2011. Accepted: February 17, 2011.

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