On Several Classes Of Regular Ordered Semigroup

In this paper we study the important properties of several classes of ordered regular semigroup.In section one, we give the introduction and some basic definitions.In section two, we consider the structure of orthodox naturally ordered DubrcilJacotin semigroup. Wc mainly use an important structure theorem (in non-ordered scmigroup thcory)-Yamada Theorem, and give the structure of orthodox naturally ordered Dubreil-Jacotin regular semigroup with Blyth and McFaddcn's method. The main conclusions of this section areTheorem2. 5 Let S be an inverse NODJ scmigroup and the Green's relations (?), R are regular on S. Let L bc an ordered left normal band with a greatest clement 1_L that is a right identity, and let R be an ordered right normal band with a greatest element 1_R that is a left identity. Then L = (?) L_αis a pointed scmilatticc of pointed left zero scmigroups, and R = (?) R_βis a pointed semilatticc of pointed right zero scmigroups. Suppose S = (L (?) S (?) R)_c denote the set together with the cartesian order and the multiplication where L^*_αis the greatest element of L_α, R^*_αis the greatest element of R_α. Then is an orthodox NODJ semigroup on which (?), R are regular. Theorem2. 7 Let T be an orthodox NODJ semigroup. Ifξis the greatest idempotent of T and E is the band of idempotents of T. Then Eξis an ordered left normal band with a greatest element that is a right identity, andξE is an ordered right normal band with a greatest element that is a left identity. Moreover,ξTξis an inverse NODJ semigroup whose semilattice of idempotentsξEξis the structure scmilatticc of EξandξE. If Green's relations (?), R are regular on T then there is an ordered semigroup isomorphismIn section three, we consider, on principally ordered regular semigroup, the subscmigroup generated by n idempotents which are comparable. And with the condition that the mapping x (?) x°is antitone, we consider, in such a subsemigroup, the shape of the clcmcnts, the number of the elements, and the Hasse diagram. The main conclusions of this section areTheorem3. 4 C_n has Hasse diagram below (for the sake of simplicity, we denotc and e_i by i, i = 1, 2,…, n) in which elements joined by lines of positive gradient are R-related, those joined by lines of negative gradient are L-related.Theorem3.5 Let S be a principally ordered regular semigroup on which x(?)x^* is weakly isotone. If e₁, e₂, e₃∈E(S) are such that e₁≥e₂≥e₃ and e₁^*=e₂^*=e₃^* then B₃, the *-subsemigroup that is generated by {e₁, e₂, e₃}, is a lattice-ordered band with at most 30 elements, and has Hasse diagram (for the sake of simplicity, we denote e₁^*=e₂^*=e₃^* by *, e_i by i, i = 1, 2, 3) in which elements joined by lines of positive gradient are (?)-related, those joined by lines of negative gradient are (?)-related, and vertical lines also indicate (?).In section four, we consider the basic properties of a class of principally ordered regular semigroup(we call it o- antitone semigroup in this article). We give a necessary and sufficient condition for a principally ordered regular semigroup to be a o-antitone semigroup. And show that such a semigroup is a compact semigroup, an orthodox strong Dubreil-Jacotin semigroup, and also a Perfect Dubreil-Jacotin semigroup. The main conclusion of this section isTheorem4. 9 Let S be a principally ordered regular semigroup, then the following are equivalent:(1)S is a o-antitone semigroup;(2)S is compact and also a *-antitone semigroup.