Incidence algebras and coalgebras of decomposition structures
The notion of incidence algebra was introduced by G.-C. Rota and others in order to supply a unified algebraic setting for a wide class of problems known from enumerative combinatorics. The procedure used was originally associated with locally finite posets and monoids with the finite factorization property, but then it was extended to the set of morphisms of a decomposition-finite category. The study of so-called incidence coalgebras has proved useful. In the present paper, the authors study the reciprocal relations between the set \({\mathcal S}\) of morphisms of a given category, its incidence algebra A(\({\mathcal S})\) and the incidence coalgebra C(\({\mathcal S})\). The main propositions generalize results relating to categories and posets. The authors prove that all automorphisms and derivations of an incidence algebra of a finitely generated decomposition structure are continuous with respect to finite topology.
scientific article; zbMATH DE number 4143980