groupoid
简明释义
n. [数] 广群
英英释义
单词用法
同义词
反义词
例句
1.In this paper, in accordance with the knowledge of Groupoid, we proved that the nature life of Lie Group on a Fundamental Groupoid has a coadjoint equivariant momentum mapping.
本文利用群胚的有关知识证明了李群在基本群胚上的提升作用有余伴随等变的动量映射这一结论，进而刻划了辛群胚的几何性质。
2.In this paper, in accordance with the knowledge of Groupoid, we proved that the nature life of Lie Group on a Fundamental Groupoid has a coadjoint equivariant momentum mapping.
本文利用群胚的有关知识证明了李群在基本群胚上的提升作用有余伴随等变的动量映射这一结论，进而刻划了辛群胚的几何性质。
3.Through these studies, the study of rough set's algebraic properties are spreaded to right involution groupoid.
这些结果将粗糙集代数性质的研究扩展到右对合广群这个代数系统中。
4.The notions of fuzzy prime relations, fuzzy strongly(weakly) prime relations on a groupoid S are introduced, and their fundamental properties and relations are discussed.
在广群中引入了模糊素关系、模糊强素关系及模糊弱素关系的概念，讨论了它们之间的关系及其基本性质。
5.Starting with a subset s of a juxtaposed groupoid one may generate the smallest juxtaposed subgroupoid containing s by applying the compositions over and over again.
从这个集的子集s出发，利用二元合成可以生成一个包含s的最小并立子群胚。
6.In category theory, a groupoid is a generalization of a group that allows for the existence of inverses only for certain elements.
在范畴论中，groupoid 是一种群的广义化，允许只有某些元素存在逆元素。
7.In algebraic topology, the fundamental groupoid captures more information than just the fundamental group.
在代数拓扑中，基本groupoid 捕捉的信息比基本群要多。
8.The concept of a groupoid is useful in studying symmetries in various mathematical structures.
groupoid 的概念在研究各种数学结构中的对称性时非常有用。
9.A topological groupoid can be used to model the relationships between different spaces.
拓扑groupoid 可用于建模不同空间之间的关系。
10.A groupoid can be thought of as a category where every morphism is invertible.
groupoid 可以被视为一个每个态射都是可逆的范畴。
作文
In the realm of mathematics, particularly in category theory and algebra, the concept of a groupoid plays a significant role in understanding symmetries and transformations. A groupoid can be thought of as a generalization of a group, where instead of having a single set of elements that combine in a certain way, we have a collection of objects and morphisms between them. This structure allows mathematicians to study not only the elements themselves but also the relationships between different objects. The idea of a groupoid is particularly useful in various fields such as topology, geometry, and even theoretical physics, where the notion of symmetry is crucial.To illustrate the importance of a groupoid, consider the example of a set of points on a surface, such as a sphere. Each point can be thought of as an object, and the transformations that map one point to another can be seen as morphisms. In this context, a groupoid captures the idea that certain transformations are only defined between specific pairs of points, rather than for all points in the space. This localized perspective enables mathematicians to analyze complex systems more effectively.Furthermore, groupoids have applications beyond pure mathematics. In computer science, for instance, they can be utilized in programming languages that emphasize functional programming and type theory. By modeling data and functions through groupoids, developers can create more robust and flexible software systems. This cross-disciplinary relevance highlights how foundational mathematical concepts often find unexpected applications in technology and other fields.Another fascinating aspect of groupoids is their ability to represent equivalence relations. When two objects can be transformed into one another through a series of morphisms, they can be considered equivalent within the context of a groupoid. This perspective is invaluable in many areas of mathematics, allowing for the abstraction of complex relationships into manageable forms. For example, in algebraic topology, groupoids can help classify spaces based on their fundamental group, providing insights into their topological properties.As we delve deeper into the study of groupoids, we begin to appreciate their intricate structure and the rich tapestry of ideas they encompass. The interplay between objects and morphisms invites us to think critically about the nature of mathematical relationships and the ways in which we can categorize and analyze them. Moreover, the versatility of groupoids encourages mathematicians to explore new avenues of thought, leading to innovative discoveries and connections across diverse fields.In conclusion, understanding the concept of a groupoid is essential for anyone interested in advanced mathematics. Its ability to generalize the notion of groups and provide a framework for analyzing relationships between objects opens up a world of possibilities. Whether in theoretical explorations or practical applications, the study of groupoids enriches our comprehension of symmetry and transformation, making it a vital topic in contemporary mathematics. As we continue to uncover the mysteries of groupoids, we not only enhance our mathematical toolkit but also deepen our appreciation for the beauty and complexity of the mathematical universe.
在数学领域，特别是在范畴理论和代数中，groupoid的概念在理解对称性和变换方面起着重要作用。groupoid可以被视为群的推广，其中不是只有一组以某种方式结合的元素，而是我们有一组对象及其之间的态射。这种结构使数学家能够研究不仅是元素本身，还有不同对象之间的关系。groupoid的概念在拓扑学、几何学甚至理论物理等多个领域中都非常有用，在这些领域中，对称性的概念至关重要。为了说明groupoid的重要性，考虑一个表面上的点集，例如一个球体。每个点可以被视为一个对象，而将一个点映射到另一个点的变换则可以看作是态射。在这个上下文中，groupoid捕捉了某些变换仅在特定点对之间定义的思想，而不是在空间中的所有点上。这种局部化的视角使数学家能够更有效地分析复杂系统。此外，groupoids在纯数学之外还有应用。例如，在计算机科学中，它们可以用于强调函数式编程和类型理论的编程语言。通过通过groupoids建模数据和函数，开发人员可以创建更强大和灵活的软件系统。这种跨学科的相关性突显了基础数学概念如何在技术和其他领域找到意想不到的应用。groupoids的另一个迷人之处在于它们能够表示等价关系。当两个对象可以通过一系列态射相互转换时，它们可以在groupoid的上下文中被视为等价。这种视角在许多数学领域中都是无价的，使得将复杂关系抽象为可管理的形式成为可能。例如，在代数拓扑中，groupoids可以帮助根据基本群对空间进行分类，从而提供对其拓扑性质的洞察。随着我们深入研究groupoids，我们开始欣赏它们复杂的结构和它们所包含的丰富思想的织锦。对象和态射之间的相互作用邀请我们批判性地思考数学关系的性质以及我们可以如何对其进行分类和分析。此外，groupoids的多功能性鼓励数学家探索新的思维途径，从而导致创新的发现和跨越不同领域的联系。总之，理解groupoid的概念对于任何对高级数学感兴趣的人来说都是必不可少的。它能够推广群的概念并提供分析对象之间关系的框架，为我们打开了一个可能性的大门。无论是在理论探索还是实际应用中，groupoids的研究丰富了我们对对称性和变换的理解，使其成为当代数学中的一个重要主题。当我们继续揭开groupoids的神秘面纱时，我们不仅增强了我们的数学工具箱，还加深了我们对数学宇宙的美丽和复杂性的欣赏。
