For detail, we studied the Incidence Algebra I defined by finite Partially Ordered Set X, and sheaf F based on the topology T_x defined by the partial order; By studying the relation between finite module category mod I and the sheaf category Sh_x A, we proved that they are equivalent.
特别的，对给定的有限偏序集X确定的Incidence代数Ⅰ，及由X的序关系定义的拓扑结构T_X上的层F，通过考查有限模范畴modⅠ与层范畴Sh_XA之间的关系，证明了Incidence代数上的模范畴与层范畴的等价性。
A join is one of the set operators of relational algebra used to concatenate data found in different tables.
一加入，是一个集商关系代数用来串数据发现，在不同的表。
I received another paper and a table of signs by return mail, and I set to work to learn the notation. But on the night before the algebra examination, while I was struggling over some very complicated examples, I could not tell the combinations of bracket, brace and radical.
在考代数的头天夜里我忙于运算复杂的习题，对于那些括号、大括号和方根的联合使用老是分辨不清。
150 Chapter 1 The History and Future of Computers 3.2 Boolean Algebra Table 3-2 Distributivity Idempotency Absorption laws 分配律同一律吸收律 a=ab+ac a+= a+a=a aa=a a+ab=a a=a'=a'b''=a'+b'DeMorgan's laws德摩根定理计算机专业英语 1-151 Chapter 1 The History and Future of Computers 3.2 Boolean Algebra Since a finite set of n elements has exactly 2n subsets, and it can be shown that the finite Boolean algebras are precisely the finite set algebras, each finite Boolean algebra consists of exactly 2n elements for some integer n.
由于n个元素的有限集有且只有个子集由于个元素的有限集有且只有2n个子集，而且很显然有限布个元素的有限集有且只有个子集，尔代数一定是有限集合代数，所以对某个整数n而言而言，尔代数一定是有限集合代数，所以对某个整数而言，每个有限布尔代数也有且只有2n个元素。例如，上文定义的集合T的限布尔代数也有且只有个元素。
31 Chapter 3 Number Systems and Boolean Algebra 3.2 Boolean Algebra Table 3-2 Distributivity Idempotency Absorption laws 分配律同一律吸收律 a=ab+ac a+= a+a=a aa=a a+ab=a a=a'=a'b''=a'+b'DeMorgan's laws德摩根定理计算机专业英语 3-32 Chapter 3 Number Systems and Boolean Algebra 3.2 Boolean Algebra Since a finite set of n elements has exactly 2n subsets, and it can be shown that the finite Boolean algebras are precisely the finite set algebras, each finite Boolean algebra consists of exactly 2n elements for some integer n.
由于n个元素的有限集有且只有个子集由于个元素的有限集有且只有2n个子集，而且很显然有限布个元素的有限集有且只有个子集，尔代数一定是有限集合代数，所以对某个整数n而言而言，尔代数一定是有限集合代数，所以对某个整数而言，每个有限布尔代数也有且只有2n个元素。例如，上文定义的集合T的限布尔代数也有且只有个元素。
Today, Fuzzy set theories, rough set theory was combined with algebra structures, which gave birth to Fuzzy rough algebra. Zhang and Xiao popularized rough structures from semigroups to Fuzzy semigroups, showing the definition of Fuzzy rough semigroup, Fuzzy rough sub ring and Fuzzy rough ideals by congruence, and then studied their properties.
继Kuroki N之后，张金玲、肖旗梅把半群中的粗糙结构推广到模糊半群、模糊环中，通过普通同余关系定义了模糊半群上的模糊粗子集，如模糊粗半群、模糊粗子环、模糊粗理想的概念，并研究了与他们相应的一些性质。
In this paper, by applying the fuzzy set concept to lattice implication algebra, we introduced the concept of fuzzy lattice implication algebras and discussed some of its basic properties. After that, we proposed the concept of fuzzy universal algebra and studied the properties of fuzzy universal algebra and its category 〓, we also gave the structures of product、equalizer and pullback square of the objects of 〓.
本文把Zadeh模糊集合论应用于格蕴涵代数，较系统地研究了模糊格蕴涵代数的性质，并在此基础上提出并研究了一般的模糊泛代数及其范畴的若干性质，给出了模糊泛代数范畴中对象的乘积、相等子及拉回方块的刻划。
The classic concept of fuzzy convex, set theory methods will be used by the characterization of the classical algebra convex membership function with the theoretical approach to the promotion of fuzzy math to start a fuzzy convex studies.
1971年，经典模糊凸集的概念，将用集合论方法所刻画的经典代数中凸集的理论用隶属函数的方法推广到模糊数学中去，开始了模糊凸集的研究。
In this paper, based on network computation and developed from disjoint algebra and tree construc ting algorithm for disjoint minimal path set, is presented a new algorithm ford i rectly constructing disjoint minimal pathset.
本文以网络计算为基础，根据不交化代数及不交化最小路集的树生成算法提出了一种新的直接求解网络系统不交化最小路集的算法。
The MP-filters and fuzzy filters of a Implication Algebra on a partial ordered set are studied with the condition given in chapter 2 which implicative operator should satisfy.
利用上述得到的关于蕴涵代数中蕴涵算子的条件，研究偏序集上具有条件的蕴涵代数的MP滤子及Fuzzy滤子。
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。
The representation theories of MP-filter which is created by an non-empty set of a Implication Algebra on a partial ordered set with condition are obtained at first.
借助于MP-滤子的概念，得到了偏序集上具有条件的蕴涵代数中由非空集合所生成MP滤子的表示定理，证明了由其上所有MP-滤子组成的集合MF是一个完备的分配格；得到了蕴涵代数中Fuzzy滤子与MP滤子的关系，给出了Fuzzy滤子成为Fuzzy素滤子的若干刻画；并利用MP-滤子和Fuzzy滤子，刻画了一类偏序集上蕴涵代数的结构。
The new method gives out a set of equations based on the fundamental algebra rules that principal stress trace obeys and the essential attributes it possess, we then combine these equations with further information described in this paper leading to an exact description of the conic representing the principal stress traces.
该方法是根据主应力迹线满足过任意三条等倾线的主应力都可以用二次曲线来描述这一条件，再利用主应力迹线本身所具有的性质列出方程组解算出描述主应力迹线的一元二次方程，然后根据方程描绘出所求的主应力迹线。
On the generalization of rough set theory, molecular lattice was introduced into rough set theory as the base algebra.
关于粗糙集理论的拓展问题，将分子格引入到粗糙集理论，本文试图将近似算子作进一步抽象与推广。
In mathematical terms, modern algebra is a set of objects with rules for connecting or relating those objects.
在数学术语里，现代袋鼠是一系列带有规则的客体的集合。而这些规则是关联这些客体的。
In mathematical terms, modern algebra is set of objects with rules for connecting orrelating those objects.
解释：a set of是固定词组，表示一套、的一组，后面接可数名词的复数，与题中objects吻合
In this paper, we will study the relationship between the algebra view and information view of rough set theory.
本文对Rough集理论的代数观点和信息论观点进行了研究，分析了二者再相容信息系统中的等价关系，并发现了二者在不相容信息系统中的不等价关系。
For asystem with a limited number of rovibrational constrained states, a spectrumgenerated algebra can be constructed using a set of Boson creation operator〓 and annihilation operator 〓, and all operators can be bilinearly expanded interms of Boson operators 〓 and 〓.
对具有有限数目振转约束态的系统，可以利用一组Boson产生算符〓和消灭算符〓构造一个谱生成代数，所有的算符可以按Boson算符〓和〓进行双线性展开。
Firstly, this paper definited pseudometric d for n -foldproduct of σ-algebra A of subsets of a given set X, and definited indicator function; secondly, gave the concept of differentiable of set function at S° and concept of partial derivative at S° with respect to the ith argument Si through indicator function, and defound the weak efficient solution of and optimality necessary condition; Finally, obtained three weak duality results and strong duality results under generalized convexity conditions.
首先，对已知集X的子集的σ-代数A的n-折积A^n，定义了伪度量d，给出了相应的特征函数〈h，Is〉；其次，通过特征函数给出了集函数在S 可微的定义及集函数在S 关于第i个变量Si的偏导数定义；给出了多目标规划问题的弱有效解概念及的最优性必要条件；最后，分别在目标函数和约束函数的3种较弱凸性条件下，研究n-集函数多目标规划问题的对偶问题，获得了3个弱对偶结果和强对偶结果。
Course Content:Discrete mathematics is a subject which puts an emphasis on the structure and the interrelations of discrete numbers, including:mathematical logic, set theory, graphic theory and algebra structure.
课程内容：离散数学是研究离散量的结构及相互关系的数学学科，它的主要内容包括数理逻辑，集合论，图论和代数结构等。