We provide a unified approach, via a deformation theory for incidence algebras that
we introduce, to several types of representations with finiteness conditions, as well
as to the combinatorial algebras which produce them. We show that for
finite-dimensional algebras over infinite fields, modules with finitely many orbits, or
with finitely many invariant subspaces, or that are distributive, coincide (and further
coincide with thin modules in the acyclic case). Incidence algebras produce examples
of such modules via their principal projective modules, and we show that algebras
which are locally hereditary, and whose indecomposable projectives are distributive,
or equivalently, which have finitely many ideals, are precisely the deformations of
incidence algebras. New characterizations of incidence algebras are obtained, such as
that they are exactly the algebras which have a faithful thin module. As a main
consequence, we show that every thin module comes from an incidence algebra, i.e., if
is thin (and, in
particular, if
is
distributive and is
acyclic), then
is an
incidence algebra and
can be presented as its defining representation. As applications, other results in the
literature are rederived and a positive answer to the accessibility question of Ringel
and Bongartz, in the distributive case, is given.
M. C. Iovanov dedicates this work to
the memory of his late friend and former student, G. D.
Koffi.
