Bands of semigroups
Bands of semigroups
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半群带
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发表时间：
1954
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通讯作者：
A. Clifford
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文献类型：
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作者：
A. Clifford
在研究一般半群S时，一件很自然的事情就是把5(如果可能的话)分解成集合{sa；互不相交的子半群Sa，使得(1)每个Sa都属于某种或多或少的限制性类型13的半群，以及(2)它们中任意两个的乘积SaSfi完全包含在第三类半群中：SaSpClSy，对于依赖于a和/3的某个yCI，我们称5是13型半群的带。如果对I中的任意a和/3，SASp和SpSa都包含在同一Sy中，则我们称S为13型半群的半格。我们还将讨论下面关于带的概念的特化。设f是两类f和k的直积Jxk，则子半群Sa用两个下标来刻画：5，-<；(ICj，KCK)。此外，假设SitSjxCSik对于所有的i，jcj和所有的k，X GB i GB。我们称5为13型半群的矩阵。本文的主要目的是证明(定理4)13型半群的带是半群的半格，其中每个半群都是13型半群的矩阵。本文的其余部分致力于给出半群5是(1)单半群、(2)完全单半群和(3)群的带或半格的充要条件。(在整篇文章中，我们使用术语简单来表示简单而不是零，即单半群不包含任何真的双边理想。)对于(1)，由于1 Olaf Andersen[L]，我们有一个优雅的条件，对所有的ACS都是CSa2S。如果半群5是[完全]单半群的类和，则它也是[完全]单半群的半格。但群的类和不一定是群的带，群的带也不一定是群的半格；这三个范畴分别由定理6、7和8刻画。我们注意到半群5是“一阶群的带”当且仅当S1的每个元素是幂等元。在这种情况下，我们将5简单地称为“带”，并由此定义：带是其每个元素都是幂等元的半群。同样，我们将半格定义为交换带。一个“群的矩阵”
In studying a general semigroup S, a natural thing to do is to decompose 5 (if possible) into the class sum of a set {Sa; aCl} of mutually disjoint subsemigroups Sa such that (1) each Sa belongs to some more or less restrictive type 13 of semigroup, and (2) the product SaSfi of any two of them is wholly contained in a third: SaSpClSy, for some yCI depending upon a and /3. We shall then say that 5 is a band of semigroups of type 13. If, for every a and /3 in I, SaSp and SpSa are both contained in the same Sy, then we shall call S a semilattice of semigroups of type 13. We shall also be concerned with the following specialization of the notion of band of semigroups. Suppose that / is the direct product JXK of two classes / and K. The subsemigroups Sa are then described by two subscripts: 5,-< (iCJ, kCK). Suppose moreover that SitSjxCSiK for all i, jCJ and all k, X£i£. We shall then call 5 a matrix of semigroups of type 13. The primary purpose of the present paper is to show (Theorem 4) that a band of semigroups of type 13 is a semilattice of semigroups each of which is a matrix of semigroups of type 13. The rest of the paper is devoted to giving necessary and sufficient conditions on a semigroup 5 that it be a band or a semilattice of (1) simple semigroups, (2) completely simple semigroups, and (3) groups. (Throughout this paper we use the term simple to mean simple without zero, i.e. a simple semigroup is one containing no proper two-sided ideal whatever.) For (1), we have the elegant condition, aCSa2S for all aCS, due to1 Olaf Andersen [l ]. If a semigroup 5 is a class sum of [completely] simple semigroups, it is also a semilattice of [completely] simple semigroups. But a class sum of groups need not be a band of groups, nor need a band of groups be a semilattice of groups; these three categories are characterized by Theorems 6, 7, and 8, respectively. We note that a semigroup 5 is a "band of groups of order one" if and only if each element of S1 is idempotent. In this case we call 5 simply a "band," and consequently make the definition: a band is a semigroup every element of which is idempotent. By the same token, we define a semilattice to be a commutative band. A "matrix of groups