التفاصيل البيبلوغرافية
| العنوان: |
On the diameter of semigroups of transformations and partitions. |
| المؤلفون: |
East, James, Gould, Victoria, Miller, Craig, Quinn‐Gregson, Thomas, Ruškuc, Nik |
| المصدر: |
Journal of the London Mathematical Society; Jul2024, Vol. 110 Issue 1, p1-34, 34p |
| مصطلحات موضوعية: |
MONOIDS, DIAMETER, METRIC spaces |
| People: |
MATTHEW, the Apostle, Saint |
| مستخلص: |
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‐FP1$FP_1$), 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,dU)$(S,d_U)$ where dU(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. [ABSTRACT FROM AUTHOR] |
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