ON SOME CLASSES OF REGULAR ORDER SEMIGROUPS

Here, some classes of regular order semigroups are discussed. We shall consider that the problems of the existences of (multiplicative) inverse $^{\delta}po$ -transversals for such classes of po-semigroups and obtain the following main results: (1) Giving the equivalent conditions of the existence of inverse $^{\delta}po$ -transversals for regular order semigroups (2) showing the order orthodox semigroups with biggest inverses have necessarily a weakly multiplicative inverse $^{\delta}po$ -transversal. (3) If the Green's relation $\cal{R}$ and $\cal{L}$ are strongly regular (see. sec.1), then any principally ordered regular semigroup (resp. ordered regular semigroup with biggest inverses) has necessarily a multiplicative inverse $^{\delta}po$ -transversal. (4) Giving the structure theorem of principally ordered semigroups (resp. ordered regular semigroups with biggest inverses) on which $\cal{R}$ and $\cal{L}$ are strongly regular.