The Duality between Incidence Algebras and Coalgebras. A Few Remarks

Luigi Azzena ; Francesco Piras

Séminaire lotharingien de combinatoire, Tome 16 (1987)

Résumé

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.