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 B , (2) an algebra on a set A 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 B . The elements of the structure are the maps from B into A 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.