monomorphism

简明释义

英[ˌmɒnəʊˈmɔːfɪzəm]美[ˌmɑnəˈmɔrfɪzəm]

n. 单型性

英英释义

单词用法

同义词

反义词

例句

1.There was a monomorphism on the prealbumin locus.

前白蛋白位点呈单态；

2.The results show that there have 4 monomorphism markers and 13 polymorphism markers among 17 selected microsatellite markers.

结果表明：选择使用17个微卫星标记，其中有4个为单态标记，13个为多态标记。

3.The nature heart disease monomorphism room fast patient most rooms fast to be allowed through the medicine control, certainly not necessarily needs to melt the treatment.

器质性心脏病单形室速的患者多数室速可以通过药物控制，并不一定需要消融治疗。

4.The nature heart disease monomorphism room fast patient most rooms fast to be allowed through the medicine control, certainly not necessarily needs to melt the treatment.

器质性心脏病单形室速的患者多数室速可以通过药物控制，并不一定需要消融治疗。

5.In category theory, a monomorphism is an injective morphism, which means it preserves distinctness.

在范畴论中，单态是一种单射态，它保持了不同性。

6.The monomorphism property ensures that the image of a morphism is uniquely determined by its source.

单态属性确保态的像是由其源唯一确定的。

7.When studying functions, a monomorphism often represents a one-to-one relationship between sets.

在研究函数时，单态通常表示集合之间的一对一关系。

8.In algebra, a monomorphism can be seen as a way to map one algebraic structure into another without losing information.

在代数中，单态可以被视为将一个代数结构映射到另一个结构而不丢失信息的方法。

9.The concept of monomorphism is crucial in understanding how different structures can be embedded into one another.

单态的概念对于理解不同结构如何互相嵌入至关重要。

作文

In mathematics, particularly in the field of category theory, the concept of monomorphism plays a crucial role in understanding the structure and relationships between different objects. A monomorphism, in simple terms, can be thought of as a kind of 'injective' function or mapping. It is a morphism that is left-cancellable, meaning that if two morphisms composed with it yield the same result, then those two morphisms must be equal. This property makes monomorphisms essential for establishing unique representations within mathematical frameworks.To illustrate this concept, consider the example of sets. If we have two sets, A and B, and a function f: A → B that is a monomorphism, it means that each element in A maps to a unique element in B without any overlap. This injective nature ensures that no two different elements in A are sent to the same element in B. Thus, we can conclude that the structure of A is preserved in B, which is a critical aspect when dealing with functions and their properties.The significance of monomorphisms extends beyond mere functions; they are fundamental in various branches of mathematics, including algebra and topology. For instance, in group theory, a subgroup can be seen as a monomorphic image of a group itself. This relationship allows mathematicians to study the properties of groups through their subgroups, leading to deeper insights into their structure and behavior.Moreover, monomorphisms are not limited to sets and functions; they also appear in other mathematical contexts, such as vector spaces, where a linear transformation can be a monomorphism if it is injective. This injectivity ensures that the dimensions of the vector spaces are preserved, which is vital for many applications in linear algebra.Understanding monomorphisms also opens the door to exploring more complex structures, such as functors and natural transformations in category theory. Functors, which map between categories, can preserve the monomorphic properties of objects and morphisms, allowing for a richer understanding of how different mathematical entities interact with one another.In conclusion, the concept of monomorphism is not just a technical term; it represents a foundational idea that permeates various areas of mathematics. By grasping the essence of monomorphisms, one can appreciate the elegance and interconnectedness of mathematical concepts. As we delve deeper into the world of category theory and its applications, the importance of monomorphisms will become increasingly apparent, revealing the beauty of mathematics as a language that describes the relationships between different structures. Ultimately, the study of monomorphisms enriches our understanding of mathematics and enhances our ability to communicate complex ideas effectively.

在数学中，尤其是在范畴论领域，单态的概念在理解不同对象之间的结构和关系方面发挥着至关重要的作用。简单来说，单态可以被视为一种“单射”函数或映射。它是一种左可消的态射，这意味着如果与它组合的两个态射产生相同的结果，那么这两个态射必须相等。这一性质使得单态在建立数学框架内的唯一表示时至关重要。为了说明这一概念，考虑集合的例子。如果我们有两个集合A和B，以及一个函数f: A → B，它是一个单态，这意味着A中的每个元素都映射到B中的一个独特元素，而没有任何重叠。这种单射特性确保A中没有两个不同的元素被发送到B中的同一个元素。因此，我们可以得出结论，A的结构在B中得以保持，这是处理函数及其属性时的关键方面。单态的重要性不仅限于简单的函数；它们在代数和拓扑等多个数学分支中都是基础。例如，在群论中，一个子群可以被视为一个群本身的单态映像。这种关系使得数学家能够通过子群研究群的性质，从而深入了解其结构和行为。此外，单态不仅出现在集合和函数中；它们还出现在其他数学上下文中，例如向量空间，当线性变换是单射时，可以视为单态。这种单射性确保了向量空间的维度得以保留，这对于线性代数中的许多应用至关重要。理解单态也为探索更复杂的结构打开了大门，例如范畴理论中的函子和自然变换。函子在类别之间进行映射，可以保持对象和态射的单态特性，从而使我们对不同数学实体之间的相互作用有更丰富的理解。总之，单态的概念不仅仅是一个技术术语；它代表了一种基础思想，渗透到数学的各个领域。通过掌握单态的本质，人们可以欣赏到数学概念的优雅和相互联系。当我们深入探讨范畴理论及其应用时，单态的重要性将变得越来越明显，揭示数学作为描述不同结构之间关系的语言之美。最终，单态的研究丰富了我们对数学的理解，并增强了我们有效传达复杂思想的能力。