endomorphism

简明释义

英[ˌendəʊˈmɔːfɪzəm]美[ˌendoˈmɔrfɪzəm]

n. [数] 自同态；内变质作用

英英释义

单词用法

同义词

反义词

例句

1.Bipartite graphs with P-regular endomorphism monoids are characterized. P-regularity of the endomorphism monoid of lexicographic product of graphs is discussed.

刻划了具有P -正则自同态幺半群的二分图，讨论了字典序积图的自同态幺半群的P -正则性。

2.Bipartite graphs with P-regular endomorphism monoids are characterized. P-regularity of the endomorphism monoid of lexicographic product of graphs is discussed.

刻划了具有P -正则自同态幺半群的二分图，讨论了字典序积图的自同态幺半群的P -正则性。

3.Finally, the endomorphism ring of radical-projective modules is discussed.

最后对根投射模的自同态环进行了讨论。

4.In algebra, an endomorphism is a function that maps a mathematical object to itself, which can be represented as a 自同态.

在代数中，endomorphism 是一个将数学对象映射到自身的函数，可以表示为 自同态。

5.The study of endomorphisms helps in understanding the structure of vector spaces, particularly in linear transformations, known as 自同态.

对endomorphisms的研究有助于理解向量空间的结构，特别是在被称为 自同态 的线性变换中。

6.In category theory, every morphism can have an associated endomorphism, which is crucial for defining identities, referred to as 自同态.

在范畴论中，每个态射都可以有一个相关的 endomorphism，这对于定义身份是至关重要的，被称为 自同态。

7.An endomorphism can be represented as a matrix when dealing with finite-dimensional vector spaces, which makes it easier to analyze, known as 自同态.

在处理有限维向量空间时，endomorphism 可以表示为一个矩阵，这使得分析更容易，被称为 自同态。

8.In group theory, the concept of endomorphisms allows us to explore the internal symmetries of groups, also known as 自同态.

在群论中，endomorphisms 的概念使我们能够探索群的内部对称性，也称为 自同态。

作文

In mathematics, the concept of endomorphism plays a crucial role in the study of algebraic structures. An endomorphism is defined as a morphism (or mapping) from a mathematical object to itself that preserves the structure of that object. This means that if we have a set, vector space, or group, an endomorphism will map elements of that set back into the same set while maintaining the operations defined on it. Understanding endomorphisms is essential for anyone delving into fields such as linear algebra, abstract algebra, and even topology. To illustrate the importance of endomorphisms, let’s consider a simple example involving vector spaces. Suppose we have a vector space V over a field F. A linear transformation T: V → V is an endomorphism if it satisfies the properties of linearity, meaning that T(v1 + v2) = T(v1) + T(v2) for all vectors v1, v2 in V, and T(cv) = cT(v) for any scalar c in F. This self-mapping property allows us to analyze the behavior of the vector space under various transformations, which can lead to deeper insights into its structure. Moreover, endomorphisms are not limited to vector spaces. In group theory, a group homomorphism that maps a group G to itself is also classified as an endomorphism. This self-mapping property enables mathematicians to explore the internal symmetries and characteristics of groups. For instance, consider the group of integers under addition. The function f: Z → Z defined by f(n) = 2n is an endomorphism of the group because it takes any integer n and maps it to another integer within the same group while preserving the operation of addition. The study of endomorphisms leads to various interesting concepts, such as the notion of eigenvalues and eigenvectors in linear algebra. An eigenvector of an endomorphism is a non-zero vector that remains proportional to itself after the transformation is applied, allowing us to define eigenvalues that provide valuable information about the transformation's effects on the vector space. This connection between endomorphisms and eigenvalues is fundamental in many applications, including systems of differential equations, stability analysis, and quantum mechanics. In summary, the concept of endomorphism is a powerful tool in mathematics that facilitates the understanding of various algebraic structures. Whether one is dealing with vector spaces, groups, or other mathematical entities, endomorphisms offer insight into the internal workings and symmetries of these structures. By studying endomorphisms, mathematicians can uncover patterns and relationships that may not be immediately apparent, leading to advancements in both theoretical and applied mathematics. Thus, a solid grasp of endomorphisms is invaluable for anyone aspiring to excel in the mathematical sciences.

在数学中，自同态的概念在代数结构的研究中起着至关重要的作用。自同态被定义为从一个数学对象到自身的映射（或变换），并保持该对象的结构。这意味着如果我们有一个集合、向量空间或群体，那么自同态将把该集合的元素映射回同一集合，同时保持其上定义的运算。理解自同态对于任何深入研究线性代数、抽象代数甚至拓扑学的人来说都是必不可少的。 为了说明自同态的重要性，让我们考虑一个涉及向量空间的简单例子。假设我们有一个定义在域F上的向量空间V。若线性变换T: V → V是自同态，则它满足线性性质，即对于V中的所有向量v1和v2，有T(v1 + v2) = T(v1) + T(v2)，并且对于域F中的任意标量c，有T(cv) = cT(v)。这种自映射属性使我们能够分析向量空间在各种变换下的行为，从而深入了解其结构。 此外，自同态并不限于向量空间。在群论中，将群G映射到自身的群同态也被归类为自同态。这种自映射属性使数学家能够探索群体的内部对称性和特征。例如，考虑整数在加法下的群体。定义f: Z → Z的函数f(n) = 2n是该群体的自同态，因为它将任何整数n映射到同一群体内的另一个整数，同时保留加法运算。 对自同态的研究引出了许多有趣的概念，例如线性代数中的特征值和特征向量的概念。自同态的特征向量是一个在应用变换后仍保持比例关系的非零向量，从而允许我们定义特征值，这为变换对向量空间的影响提供了宝贵的信息。自同态与特征值之间的这种联系在许多应用中都是基础，包括微分方程组、稳定性分析和量子力学。 总之，自同态的概念是数学中的一种强大工具，有助于理解各种代数结构。无论是在处理向量空间、群体还是其他数学实体时，自同态都提供了对这些结构内部工作和对称性的洞察。通过研究自同态，数学家可以发现可能不易察觉的模式和关系，从而推动理论和应用数学的发展。因此，对于任何希望在数学科学中取得成功的人来说，扎实掌握自同态是非常宝贵的。