Heights of one- and two-sided congruence lattices of semigroups
Fuente:
arXiv
Guardado en:
| Autores principales: | , , , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2023
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866912167684472832 |
|---|---|
| author | Brookes, M. East, J. Miller, C. Mitchell, J. D. Ruskuc, N. |
| author_facet | Brookes, M. East, J. Miller, C. Mitchell, J. D. Ruskuc, N. |
| contents | The height of a poset $P$ is the supremum of the cardinalities of chains in $P$. The exact formula for the height of the subgroup lattice of the symmetric group $\mathcal{S}_n$ is known, as is an accurate asymptotic formula for the height of the subsemigroup lattice of the full transformation monoid $\mathcal{T}_n$. Motivated by the related question of determining the heights of the lattices of left- and right congruences of $\mathcal{T}_n$, we develop a general method for computing the heights of lattices of both one- and two-sided congruences for semigroups. We apply this theory to obtain exact height formulae for several monoids of transformations, matrices and partitions, including: the full transformation monoid $\mathcal{T}_n$, the partial transformation monoid $\mathcal{PT}_n$, the symmetric inverse monoid $\mathcal{I}_n$, the monoid of order-preserving transformations $\mathcal{O}_n$, the full matrix monoid $\mathcal{M}(n,q)$, the partition monoid $\mathcal{P}_n$, the Brauer monoid $\mathcal{B}_n$ and the Temperley-Lieb monoid $\mathcal{TL}_n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_08229 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Heights of one- and two-sided congruence lattices of semigroups Brookes, M. East, J. Miller, C. Mitchell, J. D. Ruskuc, N. Group Theory Combinatorics Rings and Algebras 20M10, 20M30, 20M20, 08A30, 08A60, 06B05 The height of a poset $P$ is the supremum of the cardinalities of chains in $P$. The exact formula for the height of the subgroup lattice of the symmetric group $\mathcal{S}_n$ is known, as is an accurate asymptotic formula for the height of the subsemigroup lattice of the full transformation monoid $\mathcal{T}_n$. Motivated by the related question of determining the heights of the lattices of left- and right congruences of $\mathcal{T}_n$, we develop a general method for computing the heights of lattices of both one- and two-sided congruences for semigroups. We apply this theory to obtain exact height formulae for several monoids of transformations, matrices and partitions, including: the full transformation monoid $\mathcal{T}_n$, the partial transformation monoid $\mathcal{PT}_n$, the symmetric inverse monoid $\mathcal{I}_n$, the monoid of order-preserving transformations $\mathcal{O}_n$, the full matrix monoid $\mathcal{M}(n,q)$, the partition monoid $\mathcal{P}_n$, the Brauer monoid $\mathcal{B}_n$ and the Temperley-Lieb monoid $\mathcal{TL}_n$. |
| title | Heights of one- and two-sided congruence lattices of semigroups |
| topic | Group Theory Combinatorics Rings and Algebras 20M10, 20M30, 20M20, 08A30, 08A60, 06B05 |
| url | https://arxiv.org/abs/2310.08229 |