论变换和分区半群的直径

IF 1 2区 数学 Q1 MATHEMATICS
James East, Victoria Gould, Craig Miller, Thomas Quinn-Gregson, Nik Ruškuc
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To be more precise, for each finite generating set <span></span><math>\n <semantics>\n <mi>U</mi>\n <annotation>$U$</annotation>\n </semantics></math> for the universal right congruence on <span></span><math>\n <semantics>\n <mi>S</mi>\n <annotation>$S$</annotation>\n </semantics></math>, we have a metric space <span></span><math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <mi>S</mi>\n <mo>,</mo>\n <msub>\n <mi>d</mi>\n <mi>U</mi>\n </msub>\n <mo>)</mo>\n </mrow>\n <annotation>$(S,d_U)$</annotation>\n </semantics></math> where <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>d</mi>\n <mi>U</mi>\n </msub>\n <mrow>\n <mo>(</mo>\n <mi>a</mi>\n <mo>,</mo>\n <mi>b</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$d_U(a,b)$</annotation>\n </semantics></math> is the minimum length of derivations for <span></span><math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <mi>a</mi>\n <mo>,</mo>\n <mi>b</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$(a,b)$</annotation>\n </semantics></math> as a consequence of pairs in <span></span><math>\n <semantics>\n <mi>U</mi>\n <annotation>$U$</annotation>\n </semantics></math>; the right diameter of <span></span><math>\n <semantics>\n <mi>S</mi>\n <annotation>$S$</annotation>\n </semantics></math> with respect to <span></span><math>\n <semantics>\n <mi>U</mi>\n <annotation>$U$</annotation>\n </semantics></math> is the diameter of this metric space. 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引用次数: 0

摘要

对于一个普遍右同调有限生成的半群 S $S$(或者,等价于一个满足同调有限性性质的右型半群- F P 1 $FP_1$)来说,S $S$的右直径是一个参数,它表示 S $S$的元素在一定意义上可以彼此 "相距多远"。更准确地说,对于 S $S$ 上普遍右全等的每个有限生成集 U $U$,我们有一个度量空间 ( S , d U ) $(S,d_U)$ 其中 d U ( a , b ) $d_U(a,b)$ 是 ( a , b ) $(a,b)$ 作为 U $U$ 中成对结果的派生的最小长度;S $S$ 相对于 U $U$ 的右直径就是这个度量空间的直径。S $S$ 的右直径是所有相对于有限生成集的右直径集合的最小值。我们建立了一个理论框架,用于确定任意无限集 X $X$ 上的变换或分割半群是否具有有限生成的普遍右/左同余,如果有,则确定其右/左直径。我们以此证明如下结果。X $X$ 上所有二元关系的单体、X $X$ 上所有部分变换的单体、X $X$ 上所有完全变换的单体以及 X $X$ 上的分割单体和部分布劳尔单体都有右直径 1 和左直径 1。X $X$ 上的对称逆单元具有右直径 2 和左直径 2。X $X$ 上所有注入映射的单元具有右直径 4,其最小理想(称为 X $X$ 上的 Baer-Levi 半群)具有右直径 3,但这两个半群都没有有限生成的普遍左同调。另一方面,X $X$ 上所有投射映射的半群有左直径 4,其最小理想有左直径 2,但这两个半群都没有有限生成的普遍右同余。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

On the diameter of semigroups of transformations and partitions

On the diameter of semigroups of transformations and partitions

For a semigroup S $S$ whose universal right congruence is finitely generated (or, equivalently, a semigroup satisfying the homological finiteness property of being type right- F P 1 $FP_1$ ), the right diameter of S $S$ is a parameter that expresses how ‘far apart’ elements of S $S$ can be from each other, in a certain sense. To be more precise, for each finite generating set U $U$ for the universal right congruence on S $S$ , we have a metric space ( S , d U ) $(S,d_U)$ where d U ( a , b ) $d_U(a,b)$ is the minimum length of derivations for ( a , b ) $(a,b)$ as a consequence of pairs in U $U$ ; the right diameter of S $S$ with respect to U $U$ is the diameter of this metric space. The right diameter of S $S$ is then the minimum of the set of all right diameters with respect to finite generating sets. We develop a theoretical framework for establishing whether a semigroup of transformations or partitions on an arbitrary infinite set X $X$ has a finitely generated universal right/left congruence, and, if it does, determining its right/left diameter. We apply this to prove results such as the following. Each of the monoids of all binary relations on X $X$ , of all partial transformations on X $X$ , and of all full transformations on X $X$ , as well as the partition and partial Brauer monoids on X $X$ , have right diameter 1 and left diameter 1. The symmetric inverse monoid on X $X$ has right diameter 2 and left diameter 2. The monoid of all injective mappings on X $X$ has right diameter 4, and its minimal ideal (called the Baer–Levi semigroup on X $X$ ) has right diameter 3, but neither of these two semigroups has a finitely generated universal left congruence. On the other hand, the semigroup of all surjective mappings on X $X$ has left diameter 4, and its minimal ideal has left diameter 2, but neither of these semigroups has a finitely generated universal right congruence.

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来源期刊
CiteScore
1.90
自引率
0.00%
发文量
186
审稿时长
6-12 weeks
期刊介绍: The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.
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