; bothsides in [ 5 ] , [ 6 ] , the authors determine structures and the least group congruences . these results make us have a more explicit conception about semidirect products and structures and congruences on them 这些结果使我们对以上这些幺半群的半直积及其结构和同余有了一个比较明确的认识。
Furthermore , in the second chapter semidirect product of 5 and te is discussed . we have the result that it is also clifford quasi - regular semigroup . besides semidirect product of s and te is semilattice of quasi - groups 进一步，论文又在第h章中讨论了半群s和t的子半群te的半直积及其结构，得出了s和t ”的半直积也是clvj 。
In order to break the confinement , in this paper we manage to study semidirect products of semigroups regardless of identity elements . that is , we get rid of the especially important condition that semigroups have identity elements 为了突破这一局限性，本文就力争在一般的半群上研究其半直积，即去掉单位元这个特别重要的条件。
By describing semidirect products and structures and congruences , we can manage to obtain some structures and congruences on this kind of semigroup . it is under the condition that semigroups have identity elements that most of researches on semidirect products 因此我们可以通过去刻划半群的半直积及其结构与同余，来刻划这类半群的某些结构特点及其上的同余。
In the fourth chapter the paper discusses the following relation between the least group congruences on the e - unitary inverse semigroup and the semidirect product : moreover we describe the following structure of the semidirect product of two e - unitary semigroups by mcalistertriple : T ny 、 tys陀xdt op这佯的关系；以及通过me人hater三元组刻划了e酉逆半群的半直积的如下的结构： sx 。
By this means , we get the descriptions of semidirect products to be strongly - inverse semigroup , clifford quasi - regular semigroup , strongly - e - unitary inverse semigroup and e - unitary inverse semigroup . for strongly - e - unitary inverse semigroup , we define it in the third chapter 主要给出了保持半直积的封闭性的充要条件，某些半直积的结构以及半直积上的某些同余与半群上的这类同余之间的关系
This paper discusses the semidirect products of two semigroups in the range of - regular semigroups . giving necessary and sufficient conditions for the semidirect products of two semigroups with strong left - inverse semigroups . providing a new method to the study of the structures of semigroups 在-正则半群范围内讨论半群的半直积.给出了两个半群的半直积为强左-逆半群的充要条件,给半群结构的研究提供了一种新的方法
What we do at this aspect are : firstly , we describe the permutation symmetry of the structure of some special networks and the corresponding attractor sets with some geometric graphs in euclidean space , which are called attractors graph and geometrized structure graph of the networks respectively ; the geometrizing conditions are also given ; we study the dynamical behavior of the networks using the geometrized structure graph and attractors graph of the network ; moreover , we propose an approach to construct a big - size network with some small - size network with symmetry by the method of direct - sum , direct - produce and semidirect - produce . we also study the dynamical properties " relation between the big - size network and the small - size networks . all those results will provide some theoretical basis for designing a special large - scale network 本文在这方面所做的工作如下：首次将一些特殊网络的结构和吸引子集的置换对称性用三维欧氏空间中的一些几何图来表示，分别称之为几何结构图和吸引子图；给出了网络对称性的几何化条州即相应的对称性群为可迁群） ：并惜助网络的几何结构图和吸弓吁图分析网络的动力学性质；此外，我们提出了用简单的具有一定对称性的小网络按照群的直和、半直积和直积的方式组合成较大的网络的方法，探讨了这些小网络和所组成的大网络的一些动力学性质的关系，如稳定态的个数、各稳定态的回忆性质等，为较大网络的设计提供一些理论依据。
In the second section , we structure a kind of semirings , namely semidirect product of semirings , and prove an isomorphic theorem of semidirect products . in the third section , we give the characterizations of the relations of all kinds of regular semirings and introduce the concept of pseudo - inverse and the necessary and sufficient conditions of pseudo - invertible element . in the fourth section , we define an equivalence relation on the cartersian product of commutative semiring and its multiplicative subset 第二部分，先构造一类半环，半环半直积，然后证明半直积的同构定理第三部分，刻划了半环各类正则元之间关系，引入伪逆的定义，给出了可伪逆元素的充要条件第四部分，在交换半环和乘法集的卡氏积上定义等价关系，进而构造了一类交换半环：分式半环
