A countably infinite first-order structure M is ω-categorical if every countable N which has the same first-order theory as M is isomorphic to M. By a theorem due independently to Engeler[8], Ryll–Nardzewski[14] and Svenonius[19], this is equivalent to the condition that Aut M has finitely many orbits on Mn for all positive integers n. It is well known that ω-categorical algebraic structures are often intimately related to Boolean algebras. As examples, we mention Theorem 1·3, which is used later, and also the following theorem of J. S. Wilson (proved in [3]), which partly motivates our work.