Furstenberg (1) and the author (4), among others, have exhibited postulate systems for groups in terms of the operation x — y = x + ( — y).Furstenberg also investigated a system obtained by removing one of his postulates which defines what he called a half-group. A structure theorem for half-groups was given in (1). In the present paper, we prove another structure theorem for half-groups. A more restricted entity, called a pseudo-group, is introduced, and its structure, together with that of a half-group satisfying the left and right cancellation laws, is studied. Finally, some topological questions concerning their structure are also considered.