An axiomatic formulation is presented for point processes which may be interpreted as ordered sequences of points randomly located on the real line. Such concepts as forward recurrence times and number of points in intervals are defined and related in set-theoretic Note that for α∈A, G^α may not cover G_α as a convex subgroup and so we cannot use Theorem 1.1 to prove this result. Moreover, all that we know about the G^α/G_α is that each is an extension of a trivially ordered subgroup by a subgroup of R. It B is a plenary subset of A, then there exists a v-isomorphism μ of G into V(B, G^β/G_β), but whether or not μ is an o-isomorphism is not known.