Generalization of convolution and incidence algebras

The paper extends semigroup rings and incidence algebras into a single algebra called general convolution algebra. It is based on three ingredients: (1) a partial hypergroupoid on a set BB, (2) an algebra on a set AA whose addition is a commutative monoid and whose neutral element 0 is a both-sided absorbing element of the multiplication and (3) an ideal of the lattice of subsets of BB. The elements of the structure are the maps from BB into AA whose supports (with respect to 0) belong to the ideal and satisfy certain finiteness restriction with respect to the partial hypergroupoid. The sum is pointwise and the product, called a general convolution, extends the product in semigroup-rings and incidence algebras. We characterize general convolution algebras with some common properties, like commutativity, associativity and idempotence.