Commun. Korean Math. Soc. 2008 Vol. 23, No. 1, 29-40 Printed March 1, 2008
Zhenlin Gao and Guijie Zhang Science College of University of Shanghai for Science and Technology
Abstract : 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 $\mathcal{R}$ and $\mathcal{L}$ 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 $\mathcal{R}$ and $\mathcal{L}$ are strongly regular.
Keywords : regular order semigroup, inverse $^\delta$po-transversals, POR-semigroups, ORB-semigroups
