semigroup

简明释义

英[ˈsemɪɡruːp]美[ˌsemaɪɡrʊp]

n. [数] 半群

英英释义

单词用法

同义词

反义词

例句

1.In this paper, we described the green relations on matrix semigroup through the vector maximal independent subset of matrix.

文章利用矩阵的行向量组和列向量组的极大无关组刻画了矩阵半群中的格林关系。

2.The latter two published a monograph on semigroup theory in 1961.

后面二人在1961年出版了半群理论的专论。

3.S is a nil-extension of strong semilattice of right semigroup.

为右群强半格的诣零理想扩张。

4.This paper defined anti-part mapping on set and has given a new deseription of dual semigroup of transformation semigroup on set.

定义了集合上的反部分映射，并由此给出了集合上的变换半群的对偶半群的一个新刻划。

5.In this paper we introduce the notion of a normal fuzzy left (resp. right) ideal in a semigroup, and investigate some of its properties.

本文引入了半群的正规左(右)模糊理想的概念，研究了这类模糊理想的性质。

6.The translational hull of a semigroup plays an important role in the theory of ideal extensions of semigroups.

半群平移壳在半群的理想扩张理论中占据重要地位。

7.It is proved that for a partial semigroup g, the category of G-unions and the category of G-graded rings are isomorphic.

同时对盟定义了盟模，并证明了对部分半群g，所有G -盟作成的范畴与所有G -分次环作成的范畴是同构的。

8.Let S be a right simple right cancellative semigroup.

设S是一个右单纯、右可消的半群。

9.Accordingly we have weakly almost periodic of point in a bounded C-semigroup.

相应获得了有界c -半群点的弱概周期。

10.In algebra, a set equipped with an associative binary operation is called a semigroup.

在代数中，配备有一个结合性二元运算的集合称为半群。

11.The set of natural numbers under addition forms a semigroup.

自然数在加法下构成一个半群。

12.In computer science, certain data structures can be modeled as a semigroup for efficient operations.

在计算机科学中，某些数据结构可以建模为半群以实现高效操作。

13.The concatenation of strings is an example of a semigroup.

字符串的连接是一个半群的例子。

14.A semigroup does not require an identity element, unlike a group.

与群体不同，半群不需要单位元素。

作文

In the realm of abstract algebra, the concept of a semigroup is fundamental yet intriguing. A semigroup is defined as a set equipped with an associative binary operation. This means that if you have any three elements in this set, the way you group them during operation does not affect the outcome. For instance, if a and b are elements of a semigroup, and we perform the operation on them, followed by another element c, the result remains consistent regardless of how we group these elements: (a * b) * c = a * (b * c). This property of associativity is what distinguishes a semigroup from other algebraic structures, such as groups, where the existence of an identity element and inverses is also required. The study of semigroups finds its applications across various fields, including computer science, linguistics, and even biology. In computer science, for example, semigroups can model state transitions in automata. When designing algorithms or systems, understanding how different states combine through an associative operation can lead to more efficient solutions. Similarly, in linguistics, semigroups can be used to analyze the structure of language and the formation of words through associative processes. One interesting aspect of semigroups is their ability to be represented visually through diagrams or graphs. This representation can help illustrate how elements interact under the binary operation. For instance, if we consider a simple semigroup consisting of the set {0, 1} with the operation defined as addition modulo 2, we can create a Cayley table to show how each element combines with one another. Such visual tools make the abstract notion of semigroups more tangible and understandable, especially for those new to the subject. Moreover, semigroups can be classified into various types based on additional properties. For example, a semigroup is called a monoid if it contains an identity element, meaning there exists an element e in the set such that for any element a, the equation e * a = a * e = a holds true. This classification leads to further exploration of semigroups within broader mathematical contexts, allowing mathematicians to draw connections between different structures and their properties. In conclusion, the concept of a semigroup is not only vital in the field of mathematics but also serves as a bridge to various real-world applications. By understanding the associative nature of semigroups, we gain insights into complex systems and processes that govern numerous disciplines. Whether in computer science, linguistics, or other areas, the principles derived from studying semigroups contribute to our overall comprehension of structured interactions and operations. As we continue to explore the depths of algebra, the significance of semigroups will undoubtedly remain a focal point of interest and research.

在抽象代数的领域中，半群的概念既基础又引人入胜。半群被定义为一个配备有结合性二元运算的集合。这意味着，如果你在这个集合中有任何三个元素，分组的方式不会影响结果。例如，如果a和b是半群中的元素，我们对它们进行运算，然后再与另一个元素c进行运算，无论我们如何分组，这个结果始终是一致的：(a * b) * c = a * (b * c)。这种结合性特性使得半群与其他代数结构，如群体区分开来，因为群体还要求存在单位元素和逆元素。 半群的研究在计算机科学、语言学甚至生物学等多个领域都有应用。例如，在计算机科学中，半群可以用于建模自动机中的状态转移。在设计算法或系统时，理解不同状态如何通过结合运算相互作用，可以带来更高效的解决方案。同样，在语言学中，半群可以用来分析语言的结构以及通过结合过程形成单词。 半群的一个有趣方面是它们可以通过图表或图形进行可视化表示。这种表示可以帮助说明元素在二元运算下是如何相互作用的。例如，如果我们考虑一个由集合{0, 1}组成的简单半群，其运算定义为模2加法，我们可以创建一个凯莱表来展示每个元素是如何相互结合的。这些视觉工具使得半群的抽象概念变得更加具体和易于理解，尤其是对于那些初学者。 此外，半群可以根据附加属性进行分类。例如，如果半群包含一个单位元素，则称其为单元半群，这意味着在集合中存在一个元素e，使得对于任何元素a，方程e * a = a * e = a成立。这种分类导致了对半群在更广泛的数学背景下的进一步探索，使数学家能够在不同结构及其属性之间建立联系。 总之，半群的概念不仅在数学领域中至关重要，而且作为各种现实世界应用的桥梁。通过理解半群的结合性质，我们获得了对许多学科中复杂系统和过程的洞察。无论是在计算机科学、语言学还是其他领域，从研究半群中得出的原则都为我们整体理解结构化的交互和运算做出了贡献。随着我们继续探索代数的深度，半群的重要性无疑将继续成为关注和研究的焦点。