Residually small variety

Summary

A variety is residually small when the sizes of its subdirectly irreducible members are bounded by a cardinal. By Birkhoff's subdirect representation theorem, every algebra of a variety is a subdirect product of subdirectly irreducible members, so these members control the whole variety. A residually small variety has, up to isomorphism, only a set of subdirectly irreducible members; otherwise it is called residually large. Residual smallness is related to the existence of enough injective algebras in a variety.