Algebras defined by patterns of zeros

If X is any set and T any subset of X×X, call V(T) the vector space of all functions with support T and values in a field of characteristic zero. The main theorem below shows that a necessary and sufficient condition that V(T) admit (and be closed under) a convolution f*g(x,y)=Σf(x, z) g(z, y), sum over all z∈X, is that T be a locally finite transitive relation. One special corollary is that, if V(T) consists of upper triangular (finite or infinite) matrices and contains the identity, then there is such a convolution if and only if V(T) is the incidence algebra, as defined by Rota, of the locally finite partial order T.