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) = ~f(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,

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... n 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.