homomorphisms

简明释义

英[/ˌhɒməˈmɔːfɪzəmz/]美[/ˌhoʊməˈmɔrfɪzəmz/]

[数] 同态

英英释义

单词用法

同义词

反义词

例句

1.ResultsSome related properties of topological subalgebras, topological ideals and topological homomorphisms in topological BCI-algebras are obtained.

结果得到了拓扑bci代数的拓扑子代数、拓扑理想和拓扑同态的一些相关性质。

2.ResultsSome related properties of topological subalgebras, topological ideals and topological homomorphisms in topological BCI-algebras are obtained.

结果得到了拓扑bci代数的拓扑子代数、拓扑理想和拓扑同态的一些相关性质。

3.Bascd on this, al - most continuous, almost open and closed order-homomorphisms are intro - duced and studied.

在此基础上，最后引入并研究了几乎连续、几乎开和几乎闭序同态。

4.This paper gives the basic theorems of homomorphisms for G-unions and a characterization of the G-union of matrices within G-unions.

本文给出关于G -盟的一些基本结构定理，并在G -盟类中给出矩阵g -盟的一个特征。

5.AimTo study the notions of topological subalgebras, topological ideals and topological homomorphisms in topological BCI-algebras.

目的为研究拓扑bci代数的拓扑子代数、拓扑理想和拓扑同态的概念。试图在代数结构中嵌入拓扑结构。

6.In category theory, homomorphisms 同态 are used to define morphisms between objects.

在范畴理论中，homomorphisms 同态 用于定义对象之间的态射。

7.When studying rings, we often explore homomorphisms 同态 that preserve ring operations.

在研究环时，我们常常探讨保持环运算的 homomorphisms 同态。

8.The concept of homomorphisms 同态 is crucial in group theory, where we analyze the relationships between groups.

在群论中，homomorphisms 同态 的概念至关重要，因为我们分析群之间的关系。

9.In algebra, we study the properties of homomorphisms 同态 to understand the structure of different algebraic systems.

在代数中，我们研究 homomorphisms 同态 的性质，以理解不同代数系统的结构。

10.Many mathematical theorems rely on the existence of homomorphisms 同态 to establish equivalences between structures.

许多数学定理依赖于 homomorphisms 同态 的存在，以建立结构之间的等价性。

作文

In the field of mathematics, particularly in abstract algebra, the concept of homomorphisms plays a crucial role in understanding the structure of algebraic systems. A homomorphism is a function between two algebraic structures that preserves the operations defined on them. This means that if you have two groups, rings, or other algebraic entities, a homomorphism allows you to map elements from one structure to another while maintaining the integrity of their operations. Understanding homomorphisms can significantly enhance one’s comprehension of how different algebraic systems relate to each other.To illustrate this concept, let’s consider two groups, G and H. If there is a function f: G → H that satisfies the property f(g1 * g2) = f(g1) * f(g2) for all elements g1, g2 in G (where * denotes the group operation), then f is called a homomorphism. This property ensures that the image of the operation in G corresponds to the operation in H, making homomorphisms essential for studying the relationships between different groups.One of the most significant implications of homomorphisms is the ability to classify algebraic structures. By examining the homomorphisms between various groups, mathematicians can identify isomorphic groups, which are essentially the same in structure but may differ in their presentation. This classification helps in simplifying complex problems by allowing the use of known results from one group to infer properties about another.Moreover, homomorphisms are not limited to groups; they also exist in rings and vector spaces. In ring theory, for instance, a ring homomorphism is a function between two rings that respects both the addition and multiplication operations. This broad applicability of homomorphisms across different mathematical structures highlights their fundamental nature in algebra.In addition to their theoretical importance, homomorphisms have practical applications in various fields such as computer science, physics, and engineering. For example, in computer science, understanding homomorphisms can aid in designing algorithms that manipulate data structures efficiently. In physics, homomorphisms can help describe symmetries in physical systems, leading to a deeper understanding of conservation laws and invariances.Furthermore, the study of homomorphisms opens up avenues for further research and exploration. Researchers often investigate properties of homomorphisms, such as kernels and images, which provide insights into the structure of the original algebraic entities. The kernel of a homomorphism is the set of elements in the domain that map to the identity element in the codomain, while the image is the set of all outputs of the homomorphism. These concepts are vital for understanding the behavior of algebraic structures under homomorphisms and contribute to the development of more advanced theories in mathematics.In conclusion, homomorphisms are a fundamental concept in abstract algebra that facilitates the understanding of the relationships between different algebraic structures. Their ability to preserve operations makes them invaluable for classifying groups, rings, and other mathematical entities. The implications of homomorphisms extend beyond pure mathematics, influencing various scientific fields and promoting further research into the nature of algebraic systems. By mastering the concept of homomorphisms, one can gain a deeper appreciation for the interconnectedness of mathematical ideas and their applications in the real world.

在数学领域，特别是在抽象代数中，同态的概念在理解代数系统的结构方面起着至关重要的作用。同态是两个代数结构之间的一种函数，它保持定义在它们上的运算。这意味着，如果你有两个群、环或其他代数实体，同态允许你将一个结构中的元素映射到另一个结构，同时保持它们运算的完整性。理解同态可以显著增强人们对不同代数系统之间关系的理解。为了说明这一概念，让我们考虑两个群G和H。如果存在一个函数f: G → H，满足对于G中的所有元素g1和g2都有f(g1 * g2) = f(g1) * f(g2)（其中*表示群运算），那么f被称为同态。这个性质确保了G中运算的像对应于H中的运算，使得同态在研究不同群之间的关系时至关重要。同态的一个重要意义是能够对代数结构进行分类。通过检查各种群之间的同态，数学家可以识别同构群，这些群在结构上基本相同，但可能在表现形式上有所不同。这种分类有助于通过允许使用已知结果来简化复杂问题，从而推断出关于另一个群的性质。此外，同态不仅限于群；它们在环和向量空间中也存在。例如，在环论中，环同态是两个环之间的一个函数，它尊重加法和乘法运算。这种同态在不同数学结构中的广泛适用性突显了它们在代数中的基本性质。除了理论重要性之外，同态在计算机科学、物理学和工程等多个领域也具有实际应用。例如，在计算机科学中，理解同态可以帮助设计有效操作数据结构的算法。在物理学中，同态可以帮助描述物理系统中的对称性，从而深入理解守恒定律和不变性。此外，对同态的研究开辟了进一步研究和探索的途径。研究人员常常调查同态的性质，例如核和像，这些性质提供了对原始代数实体结构的洞察。同态的核是映射到余域中单位元素的定义域中的元素集合，而像是同态所有输出的集合。这些概念对于理解代数结构在同态下的行为至关重要，并有助于更高级理论的发展。总之，同态是抽象代数中的一个基本概念，促进了对不同代数结构之间关系的理解。它们保持运算的能力使它们在对群、环和其他数学实体进行分类时不可或缺。同态的影响超越了纯数学，影响着多个科学领域，并促进了对代数系统本质的进一步研究。通过掌握同态的概念，人们可以更深刻地欣赏数学思想之间的相互联系及其在现实世界中的应用。