Compact semigroups irreducibly connected between two idempotents
Compact semigroups irreducibly connected between two idempotents
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DOI：
10.1090/s0002-9939-1955-0071712-x
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发表时间：
1955-05
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通讯作者：
W. M. Faucett
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文献类型：
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作者：
W. M. Faucett
具有（拓扑）群结构的最简单、最熟悉的紧连通空间是简单闭合曲线C（拓扑圆）。也许紧连通（拓扑）半群最自然的例子是具有通常乘法的闭单位区间I。空间C允许群的唯一结构，而简单的例子表明（见例1、2、3）空间I允许半群的许多结构。这样的乘法不一定是阿贝尔的，可以承认幂零和幂等，也可以没有零元素。在这篇笔记中，我们开始分析可以提供空间I的半群结构。虽然我们的定理是更一般的，下面的推论将给我们的结果一个公平的画面。假设I允许这样一个乘法，它的端点自然地扮演0和单位的角色。如果不存在除零以外的其他幂等元素和幂零元素，则乘法必须是实数的乘法。证明的主要步骤是证明任何正的并进有理q存在唯一的q次根。我们很高兴地感谢A. D. Wallace在准备本文时提出的有益的建议和意见。我们将群定义为一个具有连续关联乘法的Hausdorff空间。氏族是一个紧密相连的有单位的群体。如果S是一个群，那么如果T、F和STC T（TSC T）是一个左（右）理想。双面理想既是左的理想又是右的理想。利用Clifford的术语[1]，我们将表示群S × K的极小双边理想和S × e的幂等元集合。所谓零元素，我们指的是元素0，使得对于所有xCS， x =0 = xO。我们定义一个元素s是幂零的，当Sn= 0时，对于某个正整数n。如果连通空间s中不存在同时包含A和b的连通子集，则连通空间s在两点A和两点b之间是不可约连通的。在这样的连通空间中，每个不同于A和b的点都是切点，将空间划分为恰好两个分量[6]。我们可以引入
The simplest and most familiar compact connected space which can be provided with the structure of a (topological) group is the simple closed curve C (topological circle). Perhaps the most natural example of a compact connected (topological) semigroup is the closed unit interval I with the usual multiplication. The space C admits the unique structure of a group, whereas simple examples show (see Examples 1, 2, 3) that the space I admits many structures of a semigroup. Such multiplications need not be abelian, may admit both nilpotents and idempotents, and may not have a zero element. In this note we initiate the analysis of the semigroup structures with which the space I may be provided. While our theorems are much more general, the following corollary will give a fair picture of our results. Suppose that I admits such a multiplication that its end points play the natural roles of zero and unit. If there are no other idempotents and no nilpotent elements except zero, then the multiplication must be that of the real numbers. The major step in the proof is that of showing that unique qth roots exist for any positive dyadic rational q. It is with pleasure that we acknowledge the helpful suggestions and advice of A. D. Wallace in the preparation of this paper. We define a mob to be a Hausdorff space together with a continuous, associative multiplication. A clan is a compact connected mob with unit. If S is a mob, a set TCS is a left (right) ideal if T,F and STC T(TSC T). A two-sided ideal is both a left and right ideal. Using Clifford's terminology [1], we shall denote the minimal twosided ideal of a mob S by K and the set of idempotents of S by E. By a zero element, we mean an element 0, such that Ox =0 = xO for all xCS. We define an element s to be nilpotent if Sn=O, for some positive integer n. A connected space S is irreducibly connected between two points a and b if no proper connected subset of S contains both a and b. In such a space, every point different from a and b is a cut point, separating the space into exactly two components [6]. We can intro-