Congruences on Infinite Partition and Partial Brauer Monoids
Congruences on Infinite Partition and Partial Brauer Monoids
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DOI：
10.17323/1609-4514-2022-22-2-295-372
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发表时间：
2018-09
影响因子：
0.8
通讯作者：
J. East;N. Ruškuc
中科院分区：
文献类型：
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作者：
J. East;N. Ruškuc
本文给出了任意无限集的分划幺半群P_X$和部分Brauer幺半群PB_X$上的同余以及由这些同余构成的格的完整描述。我们的结果补充了East，Mitchell，Ruskuc和Torpey最近的一篇文章，其中涉及有限的情况。作为分类结果的一个推论，我们证明了$P_X$和$PB_X$的同余格是同构的，并且是分配的和良好的拟序的。我们还计算了生成任何同余所需的最小划分对的数量;当这个数量是无限的时，它取决于某些极限基数的共尾性。
We give a complete description of the congruences on the partition monoid $P_X$ and the partial Brauer monoid $PB_X$, where $X$ is an arbitrary infinite set, and also of the lattices formed by all such congruences. Our results complement those from a recent article of East, Mitchell, Ruskuc and Torpey, which deals with the finite case. As a consequence of our classification result, we show that the congruence lattices of $P_X$ and $PB_X$ are isomorphic to each other, and are distributive and well quasi-ordered. We also calculate the smallest number of pairs of partitions required to generate any congruence; when this number is infinite, it depends on the cofinality of certain limit cardinals.