Embedding theorems for proper inverse semigroups
Embedding theorems for proper inverse semigroups
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DOI：
10.1016/0021-8693(76)90023-5
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发表时间：
1976-09
影响因子：
0.9
通讯作者：
L. O'Carroll
中科院分区：
文献类型：
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作者：
L. O'Carroll
设 S 为具有幂等半格 E 的逆半群，并令 ϱ (S) 或 ϱ（如果不存在歧义危险）为 S 上的最小群同余。如果 Eϱ= E，则称 S 为真群，或者，如果方程 ex= e 对于某些 e ϵ E 和 x ϵ S 意味着 x ϵ E。例如，自由逆半群和基本 ω-逆半群是正确的。在最近的一篇论文中，McAlister 给出了任意真逆半群的显着结构定理，并使用该定理我们证明（定理 1.3）任何真逆半群 P 都可以嵌入到半格和群的半直积 P ̄ 中。给出了该结果的一些后果；例如，如果 P 是恒等双简单的，则 P ̄ 是简单的（参见定理 1.6）。 Reilly证明了任意逆半群可以嵌入到具有恒等性的双单逆半群中。给定一个真逆半群 P，表明通过放大 P g9 由 P 产生的真逆半群 P' 确实可以嵌入到具有恒等性的双单真逆半群中（定理 2.4）。我们还研究了本身正确的双单真逆半群的同态图像。在最后一节中，描述了真逆半群的里斯商图像（定理 3.4），并考虑了半格和群的（简单）半直积的一些其他构造。
Let S be an inverse semigroup with semilattice of idempotents E, and let ϱ (S), or ϱ if there is no danger of ambiguity, be the minimum group congruence on S. Then S is said to be proper if Eϱ= E, or alternatively, if the equation ex= e for some e ϵ E and x ϵ S implies that x ϵ E. For example, free inverse semigroups and fundamental ω-inverse semigroups are proper. In a recent paper, McAlister has given a remarkable structure theorem for an arbitrary proper inverse semigroup, and using this theorem we show (Theorem 1.3) that any proper inverse semigroup P can be embedded in a semidirect product P ̄ of a semilattice and a group. Some consequences of this result are given; for example, if P is bisimple with identity then P ̄ is simple (see Theorem 1.6). Reilly has proved that an arbitrary inverse semigroup can be embedded in a bisimple inverse semigroup with identity. Given a proper inverse semigroup P it is shown that a proper inverse semigroup P′, arising from P by blowing up P g9, can indeed be embedded in a bisimple proper inverse semigroup with identity (Theorem 2.4). We also investigate those homomorphic images of bisimple proper inverse semigroups which are themselves proper. In the final section, Rees quotient images of proper inverse semigroups are characterised (Theorem 3.4), and some other constructions for (simple) semidirect products of semilattices and groups are considered.