homomorphism

简明释义

英[ˌhəʊməʊˈmɔːfɪzəm]美[ˌhoməˈmɔrfɪzəm]

n. [数] 同态，同形；异质同形

英英释义

单词用法

同义词

反义词

例句

1.The concept of lattice implication homomorphism image, which is proved to be a lattice implication algebra, is introduced.

提出了格蕴涵同态像的概念，证明了格蕴涵同态像是格蕴涵代数；

2.Finally, the homomorphism issues on Rough sets are discussed.

最后，研究了粗糙集的同态问题。

3.In this paper, the concept of natural homomorphism for graph is presented. Furthermore, the method of reducing information graph and keeping the capability of processing information is given.

本文引入图的自然同态概念，并据此给出对信息图进行归约和保持信息处理能力的方法。

4.Finally this paper shows that two kinds of the improved schemes proposed by Qi can not resist the homomorphism attack, and does not have the security as Qi said.

本文最后还证明了他们提出的两类改进型方案也不能抵御同态攻击，因而并不具有所说的安全性。

5.Is the concept of an "interleaved homomorphism" a real thing?

“交叉同态”的概念是一个真实的东西吗？

6.By using the concept of fuzzy spaces, we define the fuzzy modules(submodules)and fuzzy homomorphism of fuzzy modules.

利用模糊空间理论定义模糊模及其子模，并初步研究了模糊模同态。

7.The concept of homomorphism 同态 is crucial in group theory.

在群论中，homomorphism 同态的概念至关重要。

8.In algebra, a function that preserves structure is called a homomorphism.同态

在代数中，保持结构的函数称为homomorphism.同态

9.A homomorphism 同态 maps elements from one structure to another while preserving operations.

一个homomorphism 同态将元素从一个结构映射到另一个结构，同时保持运算。

10.In computer science, a homomorphism 同态 can be used to simplify complex data transformations.

在计算机科学中，homomorphism 同态可以用来简化复杂的数据转换。

11.We can define a homomorphism 同态 between two rings to show their relationship.

我们可以定义两个环之间的homomorphism 同态来展示它们的关系。

作文

In the realm of mathematics, particularly in abstract algebra, the concept of homomorphism plays a crucial role in understanding the structure and relationships between different algebraic systems. A homomorphism is essentially a structure-preserving map between two algebraic structures, such as groups, rings, or vector spaces. This means that it respects the operations defined within those structures. For instance, if we have two groups, G and H, a function f: G → H is called a homomorphism if for any two elements a and b in G, the equation f(a * b) = f(a) * f(b) holds true, where * represents the group operation in both G and H.The significance of homomorphism lies in its ability to simplify complex algebraic problems by allowing mathematicians to study the properties of one structure through another. By establishing a homomorphism, one can transfer knowledge and results from one algebraic system to another, making it easier to analyze their characteristics. For example, if we can show that a certain property holds for the group G, and we have a homomorphism from G to H, then we can often conclude that the same property holds for H as well.Moreover, homomorphisms are not just limited to groups; they also apply to other algebraic structures. In ring theory, a homomorphism must preserve both addition and multiplication operations. Similarly, in linear algebra, a linear transformation between vector spaces is a specific type of homomorphism. This versatility across different mathematical domains illustrates the foundational nature of homomorphisms in abstract algebra.Understanding homomorphisms also leads to the exploration of more advanced concepts such as isomorphisms and automorphisms. An isomorphism is a special kind of homomorphism that is bijective, meaning it has an inverse that is also a homomorphism. This indicates that the two algebraic structures are essentially the same in terms of their algebraic properties. Automorphisms, on the other hand, are homomorphisms from a structure to itself and reveal symmetries within the structure.In practical applications, homomorphisms can be seen in various fields including computer science, cryptography, and coding theory. For instance, in computer science, when designing algorithms that operate on data structures, leveraging homomorphisms can lead to more efficient solutions. In cryptography, certain encryption methods rely on the properties of homomorphisms to allow computations on ciphertexts without needing to decrypt them first, thus enhancing security.In conclusion, the concept of homomorphism serves as a bridge connecting different algebraic structures, facilitating a deeper understanding of their properties and interrelations. It is a fundamental idea that transcends various branches of mathematics and finds applications in real-world scenarios. Mastering the notion of homomorphism not only enriches one’s mathematical knowledge but also equips individuals with powerful tools for problem-solving in diverse fields. As we delve deeper into the world of mathematics, the study of homomorphisms will undoubtedly remain a significant area of exploration, revealing the intricate tapestry of relationships that underlie the mathematical universe.

在数学领域，特别是在抽象代数中，同态的概念在理解不同代数系统之间的结构和关系中发挥着至关重要的作用。同态本质上是两个代数结构（如群、环或向量空间）之间的保持结构的映射。这意味着它尊重这些结构内定义的运算。例如，如果我们有两个群 G 和 H，一个函数 f: G → H 被称为 同态，如果对于 G 中的任何两个元素 a 和 b，方程 f(a * b) = f(a) * f(b) 成立，其中 * 表示 G 和 H 中的群运算。同态的重要性在于它能够通过允许数学家通过另一个结构研究一个结构的性质来简化复杂的代数问题。通过建立一个 同态，可以将知识和结果从一个代数系统转移到另一个，从而更容易分析它们的特征。例如，如果我们能证明某个性质适用于群 G，并且我们有一个从 G 到 H 的 同态，那么我们通常可以得出结论，这一性质也适用于 H。此外，同态不仅限于群；它们也适用于其他代数结构。在环论中，同态必须保持加法和乘法运算。同样，在线性代数中，向量空间之间的线性变换是一种特定类型的 同态。这种跨越不同数学领域的多功能性说明了 同态 在抽象代数中的基础性。理解 同态 还引导我们探索更高级的概念，如同构和自同构。同构是一种特殊的 同态，它是双射的，意味着它有一个也是 同态 的逆。这表明这两个代数结构在代数性质上本质上是相同的。自同构则是从一个结构到其自身的 同态，揭示了结构内部的对称性。在实际应用中，同态 可以在计算机科学、密码学和编码理论等多个领域中看到。例如，在计算机科学中，在设计操作数据结构的算法时，利用 同态 可以导致更高效的解决方案。在密码学中，某些加密方法依赖于 同态 的性质，以允许在不需要首先解密的情况下对密文进行计算，从而增强安全性。总之，同态 的概念作为连接不同代数结构的桥梁，促进了对其性质和相互关系的更深理解。它是一个基本的思想，超越了数学的各个分支，并在现实世界的场景中找到了应用。掌握 同态 的概念不仅丰富了人们的数学知识，还为个人提供了在各种领域中解决问题的强大工具。当我们深入探讨数学的世界时，同态 的研究无疑将仍然是一个重要的探索领域，揭示出构成数学宇宙的复杂关系网络。