查询词典 commutative algebra

Based on the outcome of Xu Yang and Qin Keyun about lattice implication algebra and lattice-valued prepositional logic LP with truth-value in a lattice implication algebra, the author studied the properties of lattice implication algebra and the α-automated reasoning method based on α-resolution principle of LP. The specific contents are as follows: The Study of Lattice Implication Algebra On the basis of previous results of lattice implication algebra, this part consists of the following three points: 1. Some properties of lattice implication algebra L were discussed, and some important results were given if L was a complete lattice implication algebra. 2. The properties of left idempotent elements of lattice implication algebras were discussed, and the conclusion that lattice implication algebra L was equals of the directed sum of the range and dual kernel of a left map constructed by a left idempotent element was proved. 3. The properties of the filters of lattice implication algebra were discussed, the theorem was shown that they satisfy the hypothetical syllogism and substitute theorem of the propositional logic. 4. The concept of weak niters of lattice implication algebras and their properties and structures are discussed. It is proved that all weak filters of a lattice implication algebra form a topology and the the implication isomorphism betweem two lattice implication algebras is a topological mapping between their topological spaces. The Study of α-automated reasoning method based on the lattice-valued propositional logic LP In this part, the author given an a-automated reasoning method based on the lattice-valued propositional logic LP.

本文基于徐扬和秦克云的关于格蕴涵代数和以格蕴涵代数为真值域的格值命题逻辑系统LP的研究工作，对格蕴涵代数以及格值命题逻辑系统LP中基于α-归结原理的自动推理方法进行了系统深入的研究，主要有以下两方面的研究成果：一、关于格蕴涵代数的研究 1、对格蕴涵代数的格论性质进行了研究，得到了当L为完备格蕴涵代数时，关于∨，∧，→运算的一些结果； 2、对格蕴涵代数的左幂等元进行了研究，证明了格蕴涵代数L可以分解为任何一个左幂等元所对应的左映射的像集合与其对偶核的直和； 3、对格蕴涵代数的滤子的性质进行了研究，证明了滤子的结构相似于逻辑学中的Hypothetical syllogism规则和替换定理； 4、给出了格蕴涵代数中弱滤子的概念，对弱滤子的性质个结构进行了研究，证明了格蕴涵代数的全体弱滤子构成一个拓扑结构，格蕴涵代数之间的蕴涵同构是相应的拓扑空间之间的拓扑映射。

Chapter three: Define fuzzy congruence relation of MTL-algebra, prove that fuzzy fiter and fuzzy congruence relation is a bijective function in MTL-algebra, quotient algebra induced by congruence relation still forms a MTL-algebra; Introduce the relation between some kinds of fiters and fuzzy filters maitained above in IMTL-algebra,i.e. BR_0 algebra, which is a MTL-algebra satisfied inversely odering and involutive relation.

第三章：定义了MTL-代数中的Fuzzy同余关系，证明了MTL-代数中Fuzzy滤子与Fuzzy同余关系是——对应的，由同余关系所诱导的商代数依然构成一个MTL-代数；介绍了在满足逆序对合对应的MTL-代数-IMTL-代数，即BR_0-代数中上述几中特殊滤子，Fuzzy滤子之间的关系。

The most famous rough algebras are Rough Double Stone Algebra, Rough Nelson Algebra and Approximation Space Algebra, and their corresponding general algebra structures are regular double Stone algebra, semi-simple Nelson algebra and pre-rough algebra respectively.

其中最有影响的粗代数分别是粗双Stone代数、粗Nelson代数和近似空间代数，它们对应的一般代数结构分别是正则双Stone代数、半简单Nelson代数和预粗代数。

The third part proposes in Luk the concept of lattice-valued truth degrees based on the second part by generalizing truth values from the MV-unit interval(a special MV-algebra) to a general MV-algebra. And the properties of lattice-valued truth degrees suggest that this generalization is natural and reasonable. Because of the equivalence between the two algebra systems, i.e. MV-algebra and lattice implication algebra, it's natural and easy to grade the truth of formulas in lattice-valued propositional logic whose basic algebra system is the lattice implication algebra.

中。 本文第三部分在第二部分的基础上，将Luk中公式的真度值推广到了一般的MV代数上，相对于第二部分的数值真度，给出公式的格值真度，并通过格值真度的性质说明格值真度是数值真度的合理推广。

The number of polynomial function over a finite commutative ring. This paper is on the basis of the research for the first question.With the structure of finite commutative ring,we know that a finite commutative ring can be expressed by direct sum of some finite commutative local rings.

本文主要基于对问题一的研究，并由有限交换环的构造可以知道，有限交换环可以由一些有限交换局部环的直和表示，于是就将问题简化到对有限交换局部环上多项式的判定。

Besides having a some insight into theinternal structure of operator algebras,it gives a greatimpetus to the development of modern mathematics towords thenon-commutative direction,especially to the developement ofnon-commutative geometry,non-commutative algebraical topologyas well as non-commutative algebraical geometry.

除了反映算子代数自身的内在性质之外，它还对于现代数学朝着非交换的方向发展起着积极的推动作用，特别对于&非交换微分几何&，&非交换的代数拓朴&，甚至&非交换的代数几何&等非交换的数学学科的发展具有重要的影响。

In this paper, we discuss effect algebra and related topics. The main results are the following:1. Effect algebra is a commutative algebraic structure. To get a non-commutative algebraic structure.we introduced the concepts of left difference and right difference, and we also introduced the concept of the Bi-difference sets, the concept of pseudo-difference poset, and proved that pseudo-difference poset is equivalent to pseudo-effect algebra.

效应代数描述的是具有交换性的量子系统，我们借助于差运算的思想研究了不可交换的量子逻辑结构，建立了左差、右差、伪差偏序集、双差集等概念，证明了伪差偏序集与伪效应代数等价，并得到了双差集的幺半群表示定理。

The latter is essentially derived from the geometric realization of Happels triangulated equivalence between stable module category of repetitive algebra and bounded derived category of finite dimensional algebra. In terms of this realization, we deduce that the Lie algebra realized by derived category of a finite dimensional algebra is isomorphic to the Lie algebra realized by stable module category of the corresponding repetitive algebra.

后者本质上是Happel关于重复代数的稳定模范畴和导出范畴的三角等价的一个几何实现及其应用，使用这种几何实现，我们可以证明在重复代数的稳定模范畴上定义的李代数同构于相应的导出范畴上实现的李代数。

The questions covering pre-algebra and elementary algebra make up the Pre-Algebra/Elementary Algebra Sub score. The questions covering intermediate algebra and coordinate geometry make up the Intermediate Algebra/Coordinate Geometry sub score. The questions covering plane geometry and trigonometry make up the Plane Geometry/Trigonometry sub score.

包含有基础初等代数和初等代数的考题构成了基础初等代数/初等代数的技能分数，涉及到中等代数和坐标几何的考题构成了中等代数/坐标几何的技能分数，而涵盖平面几何和三角函数的考题则构成了平面几何/三角函数的技能分数。

In chapter 3, the concepts of M-matrix and H-matrix over a commutative Banach algebra have been proposed, and some of their important properties and applications in linear systems over a commutative Banach algebra have been studied. For usual complex matrix, the Bellman inequality with Hadamard product has been proved.

第三章，提出了交换Banach代数上M一矩阵及H一矩阵的概念，研究了其一些重要性质，并给出了在交换Banach代数上线性系统中的应用；对有限维复矩阵，证明了关于 Hadamard乘积的 Bell.an不等式。

For every one of my New Year resolutions, I have been a backslider for years.

说到我的新年决心，我每年都做

With empathy and focus on a feeling level, we can all relate to the common ground of mutuality of emotions.

有了同情心和感觉上一级的重点，我们都可以涉及到相互的情感共同点。