This paper develops properties of l-negatively ordered (l.n.o.), r‑negatively ordered (r.n.o.), and negatively ordered (n.o.) semigroups. Each of those three classes of semigroups is closed under forming homomorphic images, subsemigroups, direct products, and on adjoining a zero. Certain subsets of such p.o. semigroups are shown to be one or two-sided ideals, and the consequences of this for simple and 0-simple semigroups are shown. For example, the set of all nilpotent elements in an l.n.o. semigroup with zero is an ideal. Various types of regularity conditions on p.o. semigroups are considered. Exemplary of this is the following: if S is l.n.o. and right weakly regular, then S is a band. The effects of the l.n.o., r.n.o., and n.o. conditions on the Green’s relations are investigated. Numerous examples are given to motivate the study of l.n.o., r.n.o., and n.o. semigroups and these also illustrate and delimit the theory developed.

p.o. semigroups, negatively ordered semigroups, ideals, regularity conditions, nilpotency, Green’s relations.