GROUPS AND SEMIGROUPS: CONNECTIONS AND CONTRASTS
GROUPS AND SEMIGROUPS: CONNECTIONS AND CONTRASTS
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群体和半群体：联系和对比
DOI：
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发表时间：
2001
期刊：
影响因子：
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通讯作者：
G. C. Smith
中科院分区：
文献类型：
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作者：
E. F. R. M. R. Quick;G. C. Smith
在过去的几十年里，群论和半群理论在不同的方向上发展。虽然Cayley定理使我们能够将群视为某个集合的置换群，但半群理论中的类似结果将半群表示为从集合到自身的函数的半群。当然，群论和半群理论已经大大超越了这些早期的观点，这两个学科现在都被完整地编织到现代数学的结构中，在广泛的领域有联系和应用。然而，将群视为置换群，将半群视为函数半群的早期观点确实渗透到现代文献中：例如，当群作用于集合或空间时，它们通过置换（或等距，或自同构等）起作用，而半群则通过函数（或自同态，或部分等距等）起作用。群的有限维线性表示是可逆矩阵的表示，而半群的有限维线性表示是任意（不一定可逆）矩阵的表示。群和半群的基本结构理论是完全不同的-例如，人们使用半群的理想结构来给出关于半群的信息-半群之间同态的研究是复杂的，因为半群上的同余通常不像群那样由一个同余类决定。因此，这两门学科的发展方向有所不同也就不足为奇了。然而，现代半群理论的几个领域与群论密切相关，有时以相当令人惊讶的方式。例如，有限半群理论（与自动机理论和形式语言理论密切相关）的中心问题与无限群的问题是等价的，或者至少是非常密切相关的。线性代数一元群具有丰富的结构，与单位群的子群结构密切相关，这与（冯·诺伊曼）正则半群理论有有趣的联系。逆半群（即偏1 - 1函数的半群）的理论与几何和组合群论的各个方面密切相关。在本文中，我将讨论群论和半群理论之间的一些联系，我还将讨论这些理论之间的一些相当令人惊讶的对比。虽然我将简要地提到有限半群理论，正则半群理论和线性代数单群理论的一些方面，我
Group theory and semigroup theory have developed in somewhat different directions in the past several decades. While Cayley’s theorem enables us to view groups as groups of permutations of some set, the analogous result in semigroup theory represents semigroups as semigroups of functions from a set to itself. Of course both group theory and semigroup theory have developed significantly beyond these early viewpoints, and both subjects are by now integrally woven into the fabric of modern mathematics, with connections and applications across a broad spectrum of areas. Nevertheless, the early viewpoints of groups as groups of permutations, and semigroups as semigroups of functions, do permeate the modern literature: for example, when groups act on a set or a space, they act by permutations (or isometries, or automorphisms, etc.), whereas semigroup actions are by functions (or endomorphisms, or partial isometries, etc.). Finite dimensional linear representations of groups are representations by invertible matrices, while finite dimensional linear representations of semigroups are representations by arbitrary (not necessarily invertible) matrices. The basic structure theories for groups and semigroups are quite different — one uses the ideal structure of a semigroup to give information about the semigroup for example — and the study of homomorphisms between semigroups is complicated by the fact that a congruence on a semigroup is not in general determined by one congruence class, as is the case for groups. Thus it is not surprising that the two subjects have developed in somewhat different directions. However, there are several areas of modern semigroup theory that are closely connected to group theory, sometimes in rather surprising ways. For example, central problems in finite semigroup theory (which is closely connected to automata theory and formal language theory) turn out to be equivalent or at least very closely related to problems about profinite groups. Linear algebraic monoids have a rich structure that is closely related to the subgroup structure of the group of units, and this has interesting connections with the well developed theory of (von Neumann) regular semigroups. The theory of inverse semigroups (i.e., semigroups of partial one-one functions) is closely tied to aspects of geometric and combinatorial group theory. In the present paper, I will discuss some of these connections between group theory and semigroup theory, and I will also discuss some rather surprising contrasts between the theories. While I will briefly mention some aspects of finite semigroup theory, regular semigroup theory, and the theory of linear algebraic monoids, I