根据这些算法和关系代数等价定理，给出了对关系代数查询树进行逻辑优化的规则，并证明了逻辑优化的正确性。
With these algorithms and the equivalence theory, the several rules to logically optimize the accessing trees were presented and the correctness for the logical optimization was proved.
并举例说明，对于不一致决策表，其属性约简的代数表示不能用条件信息量来等价表示。
Through examples, it shows that attribute reduction of an inconsistent decision table cannot entirely be represented by conditional information quantity.
最后证明了属性约简在布尔矩阵和代数两种不同表示下是等价的。
Finally, the equivalence properties between Boolean matrix representation and algebra representation of attribute reduction are proved.
同胚映射和同伦等价是代数拓扑学中的两个重要概念。
Homeomorphic morphism and homotopy equivalence are two important concepts in the theory of algebraic topology.
本文通过对合交换半群概念的引入，对MV-代数建立了几组等价公理系，对原有的公理系进行了较好的简化。
In this paper, we introduce the notion of involution commutative semi-group, and give some equivalence axioms of MV - algebras.
通过一个简单有效的代数变换，将具有未知输入的系统变换成一个不含未知输入的等价系统；
The system with unknown input is translated into an equivalent system without unknown input by a simple algebraic transformation.
本文中，我们证明了作为泛代数的半格的定义与作为偏序集的半格的定义是等价的。
In this paper, we proved that the definition of a semilattice as a universal algebra and the definition of a semilattice as a partial ordered set are equivalent.
由此，以多项式符号代数为理论基础，提出了一个高层次数据通路的等价验证算法。
So in this paper, a new algorithm for the equivalence verification of high-level data paths based on polynomial symbolic algebra is proposed.
这些能产生等价s盒的矩阵的发现，很可能有助于代数攻击法的实施。
The discovery of the new way of generating the s -box may be a great help to the algebraic attack.
还证明了很多有限次代数数域不与上述的超积初等等价。
Also, it is proved that many algebraic number fields of finite degree are not elementarily equivalent to the above-mentioned ultraproducts.
基于等价类的划分、线性方程组的求解和标准基之对偶基的计算，提出了域元素分量代数表达式的三种求法。
Based on the partition of equivalent classes, the resolving of a linear system of equations and the calculation of the dual basis of the standard basis, three methodologies are presented.
本文主要研究了等价关系的交并运算，建立了等价关系对于交并运算的代数结构。
This Paper study the intersection and union operations, with it establishes the algebraic structure of equivalence relations.
ASQL是基于对象代数的查询语言，本文讨论了对象代数表达式的等价变换规则，利用这些规则进行查询优化。
ASQL is a query language based on object algebra. Also we discuss the rules of equivalence transform and can use these rules to process query optimization.
布尔代数上保序对合映射与对称运算等价；
The order-preserving doubly mapping on a Boolean algebra equivalents to symmetry operation;
本文考虑关联BCK-代数的半单性，得到这类代数为半单的若干等价条件。
In this paper we consider the semisimplicity of implicative BCK-algebras and obtain several equivalent conditions which an implicative BCK-algebra is semisimple.
接着对矩阵代数m_3 (C)的子代数上的2 -上循环进行了等价刻画，得到了其上的双线性映射是2 -上循环的充要条件。
Subsequently, we character and study the 2-cocycles on a subalgebra of the algebra M3 (c) and obtain the necessary and sufficient conditions that a bilinear mapping is a 2-cocycle on this algebra.
接着对矩阵代数m_3 (C)的子代数上的2 -上循环进行了等价刻画，得到了其上的双线性映射是2 -上循环的充要条件。
Subsequently, we character and study the 2-cocycles on a subalgebra of the algebra M3 (c) and obtain the necessary and sufficient conditions that a bilinear mapping is a 2-cocycle on this algebra.
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