Fuzzy logic is studied with algebraic tools in this paper . a kind of algebraic abstract of fuzzy logic , implication algebra on a partial ordered set , is given . the relations between implication algebra and other algebraic structures , such as mv - algebra and heyting algebra etc . , and the filter and the structure of implication algebra on a partial ordered set are studied 本文的目的是使用代数工具对模糊逻辑进行研究，给出模糊逻辑的一类代数抽象，即偏序集上的蕴涵代数，研究偏序集上蕴涵代数与其它代数结构，如mv -代数， heyting代数之间的关系，以及偏序集上蕴涵代数的滤子与其结构等。
Based on the above knowledge , the core components s - boxes of the rijndael algorithm and the camellia algorithm , which are the only nonlinear components , are analyzed and compared in detail . eight algebraic properties of s - boxes such as balanceness , strict avalanche criterion , differential uniformance , algebraic order , and so on are calculated and proved . the reason that the s - boxes which are constructed by the inverse operation on finite field has the specialty of 4 - difference is analyzed 在此之后，本论文分针对rijndael和camellia算法的核心部件，也是唯一的非线性部件s盒进行了详细的分析和比较，计算并证明了s盒的平衡性、严格雪崩准则、差分均匀度、代数次数等八种代数性质，特别给出了通过有限域上的取逆运算构造的s盒其差分特性为4的原因，最后计算了两个算法s盒的两种代数结构，表明rijndael和camellia算法的s盒均具有良好的代数性质和复杂的代数结构，为今后两种算法在各个领域的应用提供了理论保障。
We can give different structures on a tensor product h q to make it a bialgebra or hopf algebra . in [ 1 ] radford constructed a hopf algebra a x h with the smash algebra structure and the smash coproduct coalgebra structure , and pointed out that if a bialgebra b has a hopf subalgebra h and a projection ! : b - - > h , then there must be a subalgebra a of b such that b @ a x h is a bialgebra isomorphism 在张量积hq上可以给出不同的构造，使它成为双代数或hopf代数， radford在文献[ 1 ]中构造了以smash积为代数结构，余smash积为余代数结构的hopf代数a h ，并指出若b是双代数， h是b的子hopf代数，且存在投射： b h (即是双代数同态，且| _ h = id _ h ) ，则一定存在b的子代数a ，使ba h是双代数同构。
In section 4 , we first show that for two bicomodule algebra a # h and b # h , the cotensor product ( a # h ) h ( b # h ) is still a smash product ( a b ) # h . then we describe the structure on a b and give the corresponding equivalent conditions . in section 5 , we will complete the description of the bialgebra structures we have given , and give the equivalent conditions for such bialgebras to be hopf algebras 在第4部分中，我们得出结论：两个h -双余模代数a # h和b # h的余张量积( a # h ) _ h ( b # h )仍是smash积( ab ) # h ，并给出了ab应具有的结构及相应的等价条件。在第5部分中我们得出了a # h上双代数结构的完整描述，并刻划了这个双代数结构何时是一个hopf代数。
2 . in chapter three , the results show that , on condition that the clock - controlled sequence is m sequences , the generalized shrinking sequences have large period and linear complexity , good correlation property , and the generalized shrinking sequences family has rich algebra structure 在第三章提出并研究了广义互缩序列，研究表明生成的序列具有大的周期和线性复杂度，其中在被控序列为n级m序列的情况下证明了如下的结果：族内序列具有良好的互相关性质以及丰富的代数结构。
In the second part , several fuzzy algebra structures , such as l - fuzzy subfield , l - fuzzy linear spaces over l - fuzzy subfields , l - fuzzy algebra over l - fuzzy subfields , l - fuzzy sublattice group etc , are defined . subsequently , based on several kinds of level cut sets in [ 8 ] , their some nature properties are discussed 第二部分的主要内容如下：第二部分这一部分给出了l - fuzzy代数中的l - fuzzy子域， l - fuzzy子域上的l - fuzzy线性空间， l - fuzzy子域上的l - fuzzy代数， l - fuzzy子格群等代数结构的定义并借助于[ 8 ]中的几种水平截集讨论了它们的若干特征性质。
In the charter five , the characters and mathematical structures of r - iago generation space are researched by the mathematical methods , the conception of mapping in grey generating space put forward , and the properties that ago and iago are not only the linear transformation , but also the power transformation 第五章通过代数方法研究卜iago生成空间的性质和代数结构，提出了生成空间中映射的概念，并从数学上证明了累加生成与累减生成既为线性逆变换，又为幂变换，并得到它们的变换矩阵。
Difference from other algebraic structures which are introduced for some logic system , implication algebra is a abstraction of one logic connective , i . e . implicative operator , and other operators in it are all introduced by implicative operator . the main results of this paper is given as the following : 1 特别值得提出的是，与其它为研究逻辑系统而引入的代数结构不同，蕴涵代数是对一个逻辑联结词，即蕴涵算子(蕴涵逻辑联结词)抽象而得到的，其它算子均是由蕴涵算子诱导而得到。
An investigation of the rijndael algorithm which is the advanced encryption standard of usa is taken in this thesis . we have focused on developing the cryptographic properties of the rijndael sbox from the viewpoint of boolean function , walsh spectrum and algebraic structures , on the attacks against the reduced variants of rijndael , and on the optimized implementations of rijndael . the key contributions follow below 本文对美国高级加密标准rijndael算法进行了比较深入的研究，内容包括：从布尔函数、 walsh谱和代数结构的角度对其s盒密码性质进行的研究，简化算法的攻击方法以及算法的优化实现问题，主要成果有： 1 、提出求解布尔函数表达式的两种新方法，具有简洁、易于编程实现、准确而快速的特点，应用于des算法获得与公开文献相符的结果，应用于rijndael算法首次求出其s盒布尔函数表达式。
The thesis consists of four sections . in section one , we introduce some background of the topic , in section two we review some basic and recent results about the structure and hierarchies of the computably enumerable degrees which are closely related to our topic - the algebraic structure of the plus cupping turing degrees , in section three , we outline the basic principles of the priority tree argument , one of the main frameworks and tools of theorem proving in computability theory , and in section four , we prove a new result concerning the algebraic structure of the plus cupping turing degrees that there exist two computably enumerable degrees a , b such that a , b ? pc , and the join a v b of a and b is high 本篇论文分为4个部分：第一部分介绍了这个领域的一些背景知识；第二部分主要回顾了前人在研究可计算枚举度的结构和层谱时所取得的一些基本和最新结果，这些结果与我们的主题?加杯图灵度的代数结构密切相关；在第三部分中，我们概要的描述了优先树方法的基本原理，此方法是可计算性理论中定理证明的一个重要框架和工具；第四部分证明了一个加杯图灵度代数结构的新结果：存在两个可计算枚举度a ， b ，满足a ， b pc ，而且a和b的并a b是一个高度。
