Séminaire Lotharingien de Combinatoire, B16g (1988), 14 pp.
[Formerly: Publ. I.R.M.A. Strasbourg, 1987/S-16, p.
109-122.]
Luigi Azzena and Francesco Piras
The Duality between Incidence Algebras and Coalgebras. A Few Remarks
Abstract.
A set S with a suitable "decomposition law" gives rise to a coalgebra
C(S), the so-called Incidence Coalgebra of S. Its
dual algebra is the
better known Incidence Algebra of S, A(S). In the
last few years, a
few particular cases of these structures have been studied in great
detail. These concepts have shown themselves to be
powerful tools in enumerative combinatorial problems with which they
are concerned. We are interested in them from a general point of
view. In fact, in our opinion they are likely to be the source of a
correct algebraic counterpart of combinatorial structures.
In the quoted cases, S is the set of the intervals of a locally
finite ordered set or the set of the morphisms of a Möbius
Category. In both cases, the use of the so-called
"standard topology" plays a central role. Nevertheless, this use is
not essential. As we shall see, it may be substituted by the duality
between A(S) and C(S). Making use of this
duality, as well as of new
results about Incidence Coalgebras, in the present work, we also
generalize a few properties about Incidence Algebras due to Leroux.
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