Enlarging the Munn representation of inverse semigroups
Enlarging the Munn representation of inverse semigroups
复制标题
扩大逆半群的 Munn 表示
DOI：
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发表时间：
1977
影响因子：
0.7
通讯作者：
N. Reilly
中科院分区：
文献类型：
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作者：
N. Reilly
Munn（1966，1970）引入并研究了E的主理想到E的主理想上的同构的逆半群TE，其中E是半格.他表明，任何逆半群S与半格E，有一个代表S的逆子半群TE。然而，穆恩表象并不总是忠实的。在本文中，被认为是扩大的载体集E的Munn表示，以获得一个忠实的表示S作为逆子半群的结构类似TE在许多方面的可能性。结构X通过用集合替换E的每个元素而获得。则X ={：e ∈ E}，其中X，表示某个集合，具有从E继承的自然预序关系≤（其中x ≤ y当且仅当x ∈ X，y ∈ Xf且e ∈ Xf），使得如果T = {（x，y）∈X × X;x ≤ y且y ≤ x}则X/T同构于E.这样的集合X称为具有半格E的预半格。如果Tx表示X的主理想到X的主理想上的所有同构的集合，则Tx是逆半群。Tx的基本属性被认为是。证明了当X是局部一致时，即当|XE| = |XF|，对于所有的e，f ∈ E，Tx可以被描述为置换群与TE的圈积。集合s本身是一个具有半格E的预半格，关于由a ≤ B定义的预序≤当且仅当a− 1 a B− 1 B。然后证明了Vagner-Preston表示嵌入S作为Ts的全逆子半群。作为这些概念的应用，建立了以下结果。设R和S是逆半群，θ1（θ2）是半格E到R（S）的半格上的同构.则存在局部一致预半格W以及R和S的嵌入<$1，<$2作为Tw的全逆子半群使得（1）θ1 <$1 = θ2 <$2和（2）（eθ1 <$1，eθ2 <$2）∈当且仅当Ee与Ef同构.
Abstract The inverse semigroup TE of isomorphisms of principal ideals of E onto principal ideals of E, where E is a semilattice, has been introduced and studied by Munn (1966, 1970). He showed that, for any inverse semigroup S with semilattice E, there is a representation of S by an inverse subsemigroup of TE. The Munn representation, however, is not always faithful. In this paper, the possibility is considered of enlarging the carrier set E of the Munn representation in order to obtain a faithful representation of S as an inverse subsemigroup of a structure resembling TE in many ways. A structure X is obtained by replacing each element of E by a set. Then X = ∪{Xe: e ∈ E}, where Xe, denotes some set, has a natural pre-order relation ≤ (where x ≤ y if and only if x ∈ Xe, y ∈ Xf and e ≦ f ) inherited from E such that if T = {(x, y)∈X × X;x ≤ y and y ≤ x} then X/T is isomorphic to E. Such a set X is referred to as a pre-semilattice with semilattice E. If Tx denotes the set of all isomorphisms of principal ideals of X onto principal ideals of X then Tx is an inverse semigroup. Basic properties of Tx are considered. It is shown that when X is locally uniform, that is, when |Xe| = |Xf|, for all e, f ∈ E, Tx may be described as a wreath product of a permutation group with TE. The set s itself is a presemilattice with semilattice E with respect to the pre-order ≤ defined by a ≤ b if and only if a−1a ≦ b−1b. It is then shown that the Vagner-Preston representation embeds S as a full inverse subsemigroup of Ts. As an application of these concepts the following result is established. Let R and S be inverse semigroups and let θ1(θ2) be an isomorphism of a semilattice E onto the semilattice of R(S). Then there exists a locally uniform presemilattice W and embeddings ϕ1, ϕ2 of R and S, respectively, as full inverse subsemigroups of Tw such that (1) θ1ϕ1 = θ2ϕ2 and (2) (eθ1ϕ1, eθ2ϕ2) ∈ if and only if Ee is isomorphic to Ef.