Finitary Incidence Algebras

It is well known that incidence algebras can be defined only for locally finite partially ordered sets (Doubilet et al., 1972 Doubilet , P. , Rota , G.-C. , Stanley , R. ( 1972 ). On the foundations of combinatorial theory (VI): The idea of generating function . In: Proc. of the Sixth Berkely Symp. on Math. Stat. and Probab . v. II , Univ. of Calif. Press , pp. 267 – 318 . [Google Scholar]; Stanley 1986 Stanley , R. P. ( 1986 ). Enumerative Combinatorics . v. 1 . Monterey, CA : Wadsworth &; Brooks/Cole .[Crossref] , [Google Scholar]). At the same time, for example, the poset of cells of a noncompact cell partition of a topological space is not locally finite. On the other hand, some operations, such as the order sum and the order product (Stanley, 1986 Stanley , R. P. ( 1986 ). Enumerative Combinatorics . v. 1 . Monterey, CA : Wadsworth &; Brooks/Cole .[Crossref] , [Google Scholar]), do not save the locally finiteness. So it is natural to try to generalize the concept of incidence algebra.

In this article, we consider the functions in two variables on an arbitrary poset (finitary series), for which the convolution operation is defined. We obtain the generalization of incidence algebra—finitary incidence algebra and describe its properties: invertibility, the Jackobson radical, idempotents, regular elements. As a consequence a positive solution of the isomorphism problem for such algebras is obtained.