In this paper we establish that the causal order determined by an Ol'shanski semigroup on the corresponding homogeneous space is globally hyperbolic. Using this fact, we present sufficient conditions for a special class of Lie semigroups to admit a canonical ``triple decomposition,'' namely those for which the Lie algebra is of Cayley type. This theory applies in particular to semigroups which are naturally associated to euclidean Jordan algebras as the semigroup of compressions of the symmetric cone of the algebra.