【作者】 杜守旭；

【导师】 许新斋；

【作者基本信息】 山东师范大学 ， 基础数学， 2007， 硕士

【摘要】 本文讨论了几类正则序半群的一些重要性质。第一节给出了本文的引言及一些基本定义。第二节讨论了纯正的自然序Dubreil-Jacotin半群的结构。主要是利用无序半群理论中的一个重要的结构定理-Yamada定理，并借助Blyth和McFadden的方法给出了纯正的自然序Dubreil-Jacotin半群的结构。这一节的主要结果是定理2．5设S为逆NODJ半群且其上的格林关系L，R是正则的。设L为偏序左正规带且有最大元1_L并且1_L为右单位，设R为偏序右正规带且有最大元1_R并且1_R为左单位。那么L=（?） L_α是尖左零半群的尖半格，且R=（?） R₃是尖右零半群的尖半格。设S=（L（?）S（?）R）。表示带有笛卡儿序和运算的集合L（?）S（?）R={（e，x，f）；x∈S，e∈L_xx-1，f∈R_x^-1x}，其中L_α^*为L_α中的最大元，R_α^*为R_α中的最大元。那么（L（?）S（?）R）。为纯正的NODJ半群且其上的格林关系L，R为正则的。定理2．7设T为纯正的NODJ半群。设ξ为T的最大的幂等元且E为T的幂等元带。那么Eξ为一个有最大元的序左正规带且该最大元为其右单位，同时ξE为一个有最大元的序右正规带且该最大元为其左单位。另外，ξTξ为逆NODJ半群且其幂等元半格ξEξ是Eξ和ξE的结构半格。若T上的格林关系L，R是正则的，则有序半群同构T（?）（Eξ（?）ξTξ（?）ξE）_c。第三节讨论了主序正则半群上n个可比较的幂等元生成的子半群。在映射x（?）x^*弱保序的条件下，考虑了这样的子半群其元素的形式，元素的个数以及该子半群的哈斯图。这一节的主要结果是定理3．4 C_n有如下的哈斯图（为了简单起见，我们用*表示e₁^*=e₂^*=…=e_n^*，用0表示e°_n，用i表示e_i，i=1，2，…，n），其中，斜率为正的直线连接的元素之间有R关系，而斜率为负的直线连接的元素之间有L关系。定理3．5假设S是主序正则半群且x（?）x^*是弱保序的。设e₁，e₂，e₃∈E（S）使得e₁≥e₂≥e₃且e₁^*=e₂^*=e₃^*。那么由{e₁，e₂，e₃}生成的*子半群B₃是一个格序的正规带，且至多含有30个元素，其哈斯图如下（为了简单起见，用*表示e₁^*=e₂^*=…=e_n^*，用i表示e_i，i=1，2，…，n）：其中，由斜率为正的直线连接的元素之间有R关系，由斜率为负的直线连接的元素之间有L关系，而竖线表示自然序（?）。第四节讨论了一类主序正则半群（文中称之为o-反保序半群）的性质。给出了一个主序正则半群成为o-反保序半群的充要条件，并证明该半群既是compact半群，又是strong Dubreil-Jacotin半群，同时又是Perfect Dubreil-Jacotin半群。这一节的主要结果是定理4．9设S是主序正则半群，则下列两命题等价：（1） S是o-反保序半群；（2） S是compact半群且是*-反保序半群。更多还原

【Abstract】 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.更多还原