JAFFARD ([3], Chapt. t , Sect. 3) has shown that a directed partially ordered group may be imbedded, by means of an ideal extension, in a complete lattice semigroup. In this paper we show how this procedure may be generalised to a wide class of partially ordered semigroups, with, in addition, preservation of existing least upper bounds. In particular, we g~ve a construction which is applicable to a class of semigroups which includeS that of all residuated semigroups. Further, we obtain necessary and sufficient conditions under which a partially ordered semigroup m a y be imbedded in a conditionally complete group. We introduce in Section l the terminology used in the paper, and in the next Section the notion, of an ideal extension,-in terms of which we find necessary and sufficient conditions that a partially ordered semigroup may be imbedded, with preservation of least upper bounds, in a (conditionally) complete lattice semigroup. I t is shown that not every partially ordered semigroup may be so imbedded. We describe the result mentioned above for residuated semigroups, and complete the Section by proving some general results on ideal extensions. In Section 3 relations between different ideal systems are considered, and in the fiflal Section we establish necessary and sufficient conditions under which a partially ordered semigroup may be imbedded in a conditionally complete group.