Presentations for standard examples

This page contains the documentation for various examples of presentations for finitely presented semigroups and monoids.

Default presentations

For some semigroups and monoids presented on this page, there are multiple presentations. The section below defines some default functions; functions that should be used when the particular relations aren’t important, only the semigroup or monoid they define.

`alternating_group`(…)	A presentation for the alternating group.
`brauer_monoid`(…)	A presentation for the Brauer monoid.
`chinese_monoid`(…)	A presentation for the Chinese monoid.
`cyclic_inverse_monoid`(…)	A presentation for the cyclic inverse monoid.
`dual_symmetric_inverse_monoid`(…)	A presentation for the dual symmetric inverse monoid.
`fibonacci_semigroup`(…)	A presentation for a Fibonacci semigroup.
`full_transformation_monoid`(…)	A presentation for the full transformation monoid.
`hypo_plactic_monoid`(…)	A presentation for the hypoplactic monoid.
`monogenic_semigroup`(…)	A presentation for a monogenic semigroup.
`motzkin_monoid`(…)	A presentation for the Motzkin monoid.
`not_renner_type_B_monoid`(…)	A presentation that incorrectly claims to be the Renner monoid of type B.
`not_renner_type_D_monoid`(…)	A presentation that incorrectly claims to be the Renner monoid of type D.
`not_symmetric_group`(…)	A non-presentation for the symmetric group.
`order_preserving_cyclic_inverse_monoid`(…)	A presentation for the order preserving part of the cyclic inverse monoid.
`order_preserving_monoid`(…)	A presentation for the monoid of order preserving mappings.
`orientation_preserving_monoid`(…)	A presentation for the monoid of orientation preserving mappings.
`orientation_preserving_reversing_monoid`(…)	A presentation for the monoid of orientation preserving or reversing mappings.
`partial_brauer_monoid`(…)	A presentation for the partial Brauer monoid.
`partial_isometries_cycle_graph_monoid`(…)	A presentation for the monoid of partial isometries of a cycle graph.
`partial_transformation_monoid`(…)	A presentation for the partial transformation monoid.
`partition_monoid`(…)	A presentation for the partition monoid.
`plactic_monoid`(…)	A presentation for the plactic monoid.
`rectangular_band`(…)	A presentation for a rectangular band.
`renner_type_B_monoid`(…)	A presentation for the Renner monoid of type B.
`renner_type_D_monoid`(…)	A presentation for the Renner monoid of type D.
`sigma_plactic_monoid`(…)	A presentation for the $\sigma$-plactic monoid.
`singular_brauer_monoid`(…)	A presentation for the singular part of the Brauer monoid.
`special_linear_group_2`(…)	A presentation for the special linear group $\mathrm{SL}(2, q)$.
`stellar_monoid`(…)	A presentation for the stellar monoid.
`stylic_monoid`(…)	A presentation for the stylic monoid.
`symmetric_group`(…)	A presentation for the symmetric group.
`symmetric_inverse_monoid`(…)	A presentation for the partial transformation monoid.
`temperley_lieb_monoid`(…)	A presentation for the Temperley-Lieb monoid.
`uniform_block_bijection_monoid`(…)	A presentation for the uniform block bijection monoid.
`zero_rook_monoid`(…)	A presentation for the $0$-rook monoid.

Specific presentations

The functions documented below provide specific presentations for various semigroups and monoids, usually accompanied by a reference to the source of the presentation. There may be several presentations for any semigroup or monoid.

For each semigroup or monoid, there is a corresponding default function that is documented in the Default presentations section.

`alternating_group_Moo97`(…)	A presentation for the alternating group.
`brauer_monoid_KM07`(…)	A presentation for the Brauer monoid.
`chinese_monoid_CEKNH01`(…)	A presentation for the Chinese monoid.
`cyclic_inverse_monoid_Fer22_a`(…)	A presentation for the cyclic inverse monoid.
`cyclic_inverse_monoid_Fer22_b`(…)	A presentation for the cyclic inverse monoid.
`dual_symmetric_inverse_monoid_EEF07`(…)	A presentation for the dual symmetric inverse monoid.
`fibonacci_semigroup_CRRT94`(…)	A presentation for a Fibonacci semigroup.
`full_transformation_monoid_Aiz58`(…)	A presentation for the full transformation monoid.
`full_transformation_monoid_II74`(…)	A presentation for the full transformation monoid.
`full_transformation_monoid_MW24_a`(…)	A presentation for the full transformation monoid.
`full_transformation_monoid_MW24_b`(…)	A presentation for the full transformation monoid.
`hypo_plactic_monoid_Nov00`(…)	A presentation for the hypoplactic monoid.
`motzkin_monoid_PHL13`(…)	A presentation for the Motzkin monoid.
`not_renner_type_B_monoid_Gay18`(…)	A presentation that incorrectly claims to be the Renner monoid of type B.
`not_renner_type_D_monoid_God09`(…)	A presentation that incorrectly claims to be the Renner monoid of type D.
`not_symmetric_group_GKKL08`(…)	A non-presentation for the symmetric group.
`order_preserving_cyclic_inverse_monoid_Fer22`(…)	A presentation for the order preserving part of the cyclic inverse monoid.
`order_preserving_monoid_AR00`(…)	A presentation for the monoid of order preserving mappings.
`orientation_preserving_monoid_AR00`(…)	A presentation for the monoid of orientation preserving mappings.
`orientation_preserving_reversing_monoid_AR00`(…)	A presentation for the monoid of orientation preserving or reversing mappings.
`partial_brauer_monoid_KM07`(…)	A presentation for the partial Brauer monoid.
`partial_isometries_cycle_graph_monoid_FP22`(…)	A presentation for the monoid of partial isometries of a cycle graph.
`partial_transformation_monoid_MW24`(…)	A presentation for the partial transformation monoid.
`partial_transformation_monoid_Shu60`(…)	A presentation for the partial transformation monoid.
`partition_monoid_Eas11`(…)	A presentation for the partition monoid.
`partition_monoid_HR05`(…)	A presentation for the partition monoid.
`plactic_monoid_Knu70`(…)	A presentation for the plactic monoid.
`rectangular_band_ACOR00`(…)	A presentation for a rectangular band.
`renner_type_B_monoid_Gay18`(…)	A presentation for the Renner monoid of type B.
`renner_type_D_monoid_Gay18`(…)	A presentation for the Renner monoid of type D.
`sigma_plactic_monoid_AHMNT24`(…)	A presentation for the $\sigma$-plactic monoid.
`singular_brauer_monoid_MM07`(…)	A presentation for the singular part of the Brauer monoid.
`special_linear_group_2_CR80`(…)	A presentation for the special linear group $\mathrm{SL}(2, q)$.
`stellar_monoid_GH19`(…)	A presentation for the stellar monoid.
`stylic_monoid_AR22`(…)	A presentation for the stylic monoid.
`symmetric_group_Bur12`(…)	A presentation for the symmetric group.
`symmetric_group_Car56`(…)	A presentation for the symmetric group.
`symmetric_group_Moo97_a`(…)	A presentation for the symmetric group.
`symmetric_group_Moo97_b`(…)	A presentation for the symmetric group.
`symmetric_inverse_monoid_Sol04`(…)	A presentation for the symmetric inverse monoid.
`symmetric_inverse_monoid_MW24`(…)	A presentation for the symmetric inverse monoid.
`symmetric_inverse_monoid_Shu60`(…)	A presentation for the symmetric inverse monoid.
`temperley_lieb_monoid_Eas21`(…)	A presentation for the Temperley-Lieb monoid.
`uniform_block_bijection_monoid_Fit03`(…)	A presentation for the uniform block bijection monoid.
`zero_rook_monoid_Gay18`(…)	A presentation for the $0$-rook monoid.

Full API

This page contains the documentation for the presentation.examples subpackage, that contains functions for generating various examples of presentations for finitely presented semigroups, monoids, and groups.

presentation.examples.alternating_group(n: int) → Presentation

A presentation for the alternating group.

This function returns a monoid presentation defining the alternating group of degree n, as in Theorem B of [Moo97].

Parameters:: n (int) – the degree of the alternating group.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 4.

Note

This function returns exactly the same presentation as alternating_group_Moo97, and exists as a convenience function for when a presentation for the alternating group is required, but the specific presentation is not important.

presentation.examples.alternating_group_Moo97(n: int) → Presentation

A presentation for the alternating group.

This function returns a monoid presentation defining the alternating group of degree n, as in Theorem B of [Moo97].

Parameters:: n (int) – the degree of the alternating group.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 4.

presentation.examples.brauer_monoid(n: int) → Presentation

A presentation for the Brauer monoid.

This function returns a monoid presentation defining the Brauer monoid of degree n, as described in Theorem 3.1 of [KM07].

Parameters:: n (int) – the degree.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 1.

Note

This function returns exactly the same presentation as brauer_monoid_KM07, and exists as a convenience function for when a presentation for the alternating group is required, but the specific presentation is not important.

presentation.examples.brauer_monoid_KM07(n: int) → Presentation

A presentation for the Brauer monoid.

This function returns a monoid presentation defining the Brauer monoid of degree n, as described in Theorem 3.1 of [KM07].

Parameters:: n (int) – the degree.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 1.

presentation.examples.chinese_monoid(n: int) → Presentation

A presentation for the Chinese monoid.

This function returns a monoid presentation defining the Chinese monoid with n generators, as in [CEK+01].

Parameters:: n (int) – the number of generators.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 2.

Note

This function returns exactly the same presentation as chinese_monoid_CEKNH01, and exists as a convenience function for when a presentation for the alternating group is required, but the specific presentation is not important.

presentation.examples.chinese_monoid_CEKNH01(n: int) → Presentation

A presentation for the Chinese monoid.

This function returns a monoid presentation defining the Chinese monoid with n generators, as in [CEK+01].

Parameters:: n (int) – the number of generators.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 2.

presentation.examples.cyclic_inverse_monoid(n: int) → Presentation

A presentation for the cyclic inverse monoid.

This function returns a monoid presentation defining the cyclic inverse monoid of degree n, as in Theorem 2.7 of [Fer22].

This presentation has $2$ generators and $\frac{1}{2}\left(n^2 - n + 6\right)$ relations.

Parameters:: n (int) – the degree.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 3.

See also

For a specific presentation of the full transformation monoid, see one of the following functions:

cyclic_inverse_monoid_Fer22_a;

cyclic_inverse_monoid_Fer22_b.

presentation.examples.cyclic_inverse_monoid_Fer22_a(n: int) → Presentation

A presentation for the cyclic inverse monoid.

This function returns a monoid presentation defining the cyclic inverse monoid of degree n, as in Theorem 2.6 of [Fer22]. This has $n + 1$ generators and $\frac{1}{2} \left(n^2 + 3n + 4\right)$ relations.

Parameters:: n (int) – the degree.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 3.

presentation.examples.cyclic_inverse_monoid_Fer22_b(n: int) → Presentation

A presentation for the cyclic inverse monoid.

This function returns a monoid presentation defining the cyclic inverse monoid of degree n, as in Theorem 2.7 of [Fer22]. This presentation has $2$ generators and $\frac{1}{2}\left(n^2 - n + 6\right)$ relations.

Parameters:: n (int) – the degree.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 3.

presentation.examples.dual_symmetric_inverse_monoid(n: int) → Presentation

A presentation for the dual symmetric inverse monoid.

This function returns a monoid presentation defining the dual symmetric inverse monoid of degree n, as in Section 3 of [EEF07].

Parameters:: n (int) – the degree.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 3.

Note

This function returns exactly the same presentation as dual_symmetric_inverse_monoid, and exists as a convenience function for when a presentation for the alternating group is required, but the specific presentation is not important.

presentation.examples.dual_symmetric_inverse_monoid_EEF07(n: int) → Presentation

A presentation for the dual symmetric inverse monoid.

This function returns a monoid presentation defining the dual symmetric inverse monoid of degree n, as in Section 3 of [EEF07].

Parameters:: n (int) – the degree.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 3.

presentation.examples.fibonacci_semigroup(r: int, n: int) → Presentation

A presentation for a Fibonacci semigroup.

This function returns a semigroup presentation defining the Fibonacci semigroup $F(r, n)$, where $r$ is r and $n$ is n, as described in [CRRT94].

Parameters:

r (int) – the length of the left hand sides of the relations.
n (int) – the number of generators.

Returns:

The specified presentation.

Return type:

Presentation

Raises:

LibsemigroupsError – if n < 1.
LibsemigroupsError – if r < 1.

Note

This function returns exactly the same presentation as fibonacci_semigroup_CRRT94, and exists as a convenience function for when a presentation for the alternating group is required, but the specific presentation is not important.

presentation.examples.fibonacci_semigroup_CRRT94(r: int, n: int) → Presentation

A presentation for a Fibonacci semigroup.

This function returns a semigroup presentation defining the Fibonacci semigroup $F(r, n)$, where $r$ is r and $n$ is n, as described in [CRRT94].

Parameters:

r (int) – the length of the left hand sides of the relations.
n (int) – the number of generators.

Returns:

The specified presentation.

Return type:

Presentation

Raises:

LibsemigroupsError – if n < 1.
LibsemigroupsError – if r < 1.

presentation.examples.full_transformation_monoid(n: int) → Presentation

A presentation for the full transformation monoid.

This function returns a monoid presentation defining the full transformation monoid of degree n, corresponding to $\mathcal{T}$ in Theorem 1.5 of [MW24]. For n >= 4 this presentation has five non-symmetric-group relations.

Parameters:: n (int) – the degree of the full transformation monoid.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 2.

See also

For a specific presentation of the full transformation monoid, see one of the following functions:

full_transformation_monoid_Aiz58;

full_transformation_monoid_II74;

full_transformation_monoid_MW24_a;

full_transformation_monoid_MW24_b.

presentation.examples.full_transformation_monoid_Aiz58(n: int) → Presentation

A presentation for the full transformation monoid.

This function returns a monoid presentation defining the full transformation monoid of degree n, as in Section 5, Theorem 2 of [Aiz58] (Russian) and Chapter 3, Proposition 1.7 of [Rus95] (English).

Parameters:: n (int) – the degree of the full transformation monoid.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 4.

presentation.examples.full_transformation_monoid_II74(n: int) → Presentation

A presentation for the full transformation monoid.

This function returns a monoid presentation defining the full transformation monoid of degree n due to Iwahori and Iwahori [II74], as in Theorem 9.3.1 of [GM09].

Parameters:: n (int) – the degree of the full transformation monoid.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 4.

presentation.examples.full_transformation_monoid_MW24_a(n: int) → Presentation

A presentation for the full transformation monoid.

This function returns a monoid presentation defining the full transformation monoid of degree n, corresponding to $\mathcal{T}$ in Theorem 1.5 of [MW24]. For n >= 4 this presentation has five non-symmetric-group relations.

Parameters:: n (int) – the degree of the full transformation monoid.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 2.

presentation.examples.full_transformation_monoid_MW24_b(n: int) → Presentation

A presentation for the full transformation monoid.

This function returns a monoid presentation defining the full transformation monoid of degree n, corresponding to $\mathcal{T}'$ in Theorem 1.5 of [MW24]. This presentation is only valid for odd values of n, and for n >= 5 this presentation has four non-symmetric-group relations.

Parameters:

n (int) – the degree of the full transformation monoid.

Returns:

The specified presentation.

Return type:

Presentation

Raises:

LibsemigroupsError – if n < 3.
LibsemigroupsError – if n is not odd.

presentation.examples.hypo_plactic_monoid(n: int) → Presentation

A presentation for the hypoplactic monoid.

This function returns a presentation for the hypoplactic monoid with n generators, as in Definition 4.2 of [Nov00].

This monoid is a quotient monoid of the plactic monoid, and this presentation includes the rules from plactic_monoid_Knu70.

Parameters:: n (int) – the number of generators.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 1.

Note

This function returns exactly the same presentation as hypo_plactic_monoid_Nov00, and exists as a convenience function for when a presentation for the alternating group is required, but the specific presentation is not important.

presentation.examples.hypo_plactic_monoid_Nov00(n: int) → Presentation

A presentation for the hypoplactic monoid.

This function returns a presentation for the hypoplactic monoid with n generators, as in Definition 4.2 of [Nov00]. This monoid is a quotient monoid of the plactic monoid, and this presentation includes the rules from plactic_monoid_Knu70.

Parameters:: n (int) – the number of generators.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 1.

presentation.examples.monogenic_semigroup(m: int, r: int) → Presentation

A presentation for a monogenic semigroup.

This function returns a presentation defining the monogenic semigroup defined by the presentation $\langle a \mid a^{m + r} = a^m \rangle$.

If m is 0, the presentation returned is a monoid presentation; otherwise, a semigroup presentation is returned.

Parameters:

m (int) – the index.
r (int) – the period.

Returns:

The specified presentation.

Return type:

Presentation

Raises:

LibsemigroupsError – if r == 0.

presentation.examples.motzkin_monoid(n: int) → Presentation

A presentation for the Motzkin monoid.

This function returns a monoid presentation defining the Motzkin monoid of degree n, as described in Theorem 4.1 of [PHL13], with the additional relations :math:` r_i t_i l_i = r_i ^ 2 ` added to fix a hole in Lemma 4.10 which rendered the presentation as stated in the paper incorrect.

Parameters:: n (int) – the degree.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 1.

Note

This function returns exactly the same presentation as motzkin_monoid_PHL13, and exists as a convenience function for when a presentation for the alternating group is required, but the specific presentation is not important.

presentation.examples.motzkin_monoid_PHL13(n: int) → Presentation

A presentation for the Motzkin monoid.

This function returns a monoid presentation defining the Motzkin monoid of degree n, as described in Theorem 4.1 of [PHL13], with the additional relations :math:` r_i t_i l_i = r_i^ 2 ` added to fix a hole in Lemma 4.10 which rendered the presentation as stated in the paper incorrect.

Parameters:: n (int) – the degree.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 1.

presentation.examples.not_renner_type_B_monoid(l: int, q: int) → Presentation

A presentation that incorrectly claims to be the Renner monoid of type B.

This functions returns a presentation that incorrectly claims to be the Renner monoid of type B with size l and Iwahori-Hecke deformation q.

When q == 0, this corresponds to Example 7.1.2 of [Gay18].

When q == 1, this corresponds to Section 3.2 of [God09].

Parameters:

l (int) – the size of the monoid.
q (int) – the Iwahori-Hecke deformation.

Returns:

The specified presentation

Return type:

Presentation

Raises:

LibsemigroupsError – if q != 0 or q != 1.

Note

This function returns exactly the same presentation as not_renner_type_B_monoid_Gay18, and exists as a convenience function for when a presentation for the alternating group is required, but the specific presentation is not important.

presentation.examples.not_renner_type_B_monoid_Gay18(l: int, q: int) → Presentation

A presentation that incorrectly claims to be the Renner monoid of type B.

This functions returns a presentation that incorrectly claims to be the Renner monoid of type B with size l and Iwahori-Hecke deformation q.

When q == 0, this corresponds to Example 7.1.2 of [Gay18].

When q == 1, this corresponds to Section 3.2 of [God09].

Parameters:

l (int) – the size of the monoid.
q (int) – the Iwahori-Hecke deformation.

Returns:

The specified presentation

Return type:

Presentation

Raises:

LibsemigroupsError – if q != 0 or q != 1.

presentation.examples.not_renner_type_D_monoid(l: int, q: int) → Presentation

A presentation that incorrectly claims to be the Renner monoid of type D.

This functions returns a presentation that incorrectly claims to be the Renner monoid of type D with size l and Iwahori-Hecke deformation q.

When q == 1, this corresponds to Section 3.3 of [God09].

Parameters:

l (int) – the size of the monoid.
q (int) – the Iwahori-Hecke deformation.

Returns:

The specified presentation

Return type:

Presentation

Raises:

LibsemigroupsError – if q != 0 or q != 1.

Note

This function returns exactly the same presentation as not_renner_type_D_monoid_God09, and exists as a convenience function for when a presentation for the alternating group is required, but the specific presentation is not important.

presentation.examples.not_renner_type_D_monoid_God09(l: int, q: int) → Presentation

A presentation that incorrectly claims to be the Renner monoid of type D.

This functions returns a presentation that incorrectly claims to be the Renner monoid of type D with size l and Iwahori-Hecke deformation q.

When q == 1, this corresponds to Section 3.3 of [God09].

Parameters:

l (int) – the size of the monoid.
q (int) – the Iwahori-Hecke deformation.

Returns:

The specified presentation

Return type:

Presentation

Raises:

LibsemigroupsError – if q != 0 or q != 1.

presentation.examples.not_symmetric_group(n: int) → Presentation

A non-presentation for the symmetric group.

This function returns a monoid presentation which is claimed to define the symmetric group of degree n, but does not, as in Section 2.2 of [GKKL08].

Parameters:: n (int) – the claimed degree of the symmetric group.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 4.

Note

This function returns exactly the same presentation as not_symmetric_group_GKKL08, and exists as a convenience function for when a presentation for the alternating group is required, but the specific presentation is not important.

presentation.examples.not_symmetric_group_GKKL08(n: int) → Presentation

A non-presentation for the symmetric group.

This function returns a monoid presentation which is claimed to define the symmetric group of degree n, but does not, as in Section 2.2 of [GKKL08].

Parameters:: n (int) – the claimed degree of the symmetric group.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 4.

presentation.examples.order_preserving_cyclic_inverse_monoid(n: int) → Presentation

A presentation for the order preserving part of the cyclic inverse monoid.

This function returns a monoid presentation defining the order preserving part of the cyclic inverse monoid of degree n, as in Theorem 2.17 of [Fer22].

Parameters:: n (int) – the degree.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 3.

Note

This function returns exactly the same presentation as order_preserving_cyclic_inverse_monoid_Fer22, and exists as a convenience function for when a presentation for the alternating group is required, but the specific presentation is not important.

presentation.examples.order_preserving_cyclic_inverse_monoid_Fer22(n: int) → Presentation

A presentation for the order preserving part of the cyclic inverse monoid.

This function returns a monoid presentation defining the order preserving part of the cyclic inverse monoid of degree n, as in Theorem 2.17 of [Fer22].

Parameters:: n (int) – the degree.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 3.

presentation.examples.order_preserving_monoid(n: int) → Presentation

A presentation for the monoid of order preserving mappings.

This function returns a monoid presentation defining the monoid of order preserving transformations of degree n, as described in Section 2 of [AR00]. This presentation has $2n - 2$ generators and $n^2$ relations.

Parameters:: n (int) – the degree.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 3.

Note

This function returns exactly the same presentation as order_preserving_monoid_AR00, and exists as a convenience function for when a presentation for the alternating group is required, but the specific presentation is not important.

presentation.examples.order_preserving_monoid_AR00(n: int) → Presentation

A presentation for the monoid of order preserving mappings.

This function returns a monoid presentation defining the monoid of order preserving transformations of degree n, as described in Section 2 of [AR00]. This presentation has $2n - 2$ generators and $n^2$ relations.

Parameters:: n (int) – the degree.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 3.

presentation.examples.orientation_preserving_monoid(n: int) → Presentation

A presentation for the monoid of orientation preserving mappings.

This function returns a monoid presentation defining the monoid of orientation preserving mappings on a finite chain of order n, as described in [AR00].

Parameters:: n (int) – the order of the chain.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 3.

Note

This function returns exactly the same presentation as orientation_preserving_monoid_AR00, and exists as a convenience function for when a presentation for the alternating group is required, but the specific presentation is not important.

presentation.examples.orientation_preserving_monoid_AR00(n: int) → Presentation

A presentation for the monoid of orientation preserving mappings.

This function returns a monoid presentation defining the monoid of orientation preserving mappings on a finite chain of order n, as described in [AR00].

Parameters:: n (int) – the order of the chain.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 3.

presentation.examples.orientation_preserving_reversing_monoid(n: int) → Presentation

A presentation for the monoid of orientation preserving or reversing mappings.

This function returns a monoid presentation defining the monoid of orientation preserving or reversing mappings on a finite chain of order n, as described in [AR00].

Parameters:: n (int) – the order of the chain.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 3.

Note

This function returns exactly the same presentation as orientation_preserving_reversing_monoid_AR00, and exists as a convenience function for when a presentation for the alternating group is required, but the specific presentation is not important.

presentation.examples.orientation_preserving_reversing_monoid_AR00(n: int) → Presentation

A presentation for the monoid of orientation preserving or reversing mappings.

This function returns a monoid presentation defining the monoid of orientation preserving or reversing mappings on a finite chain of order n, as described in [AR00].

Parameters:: n (int) – the order of the chain.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 3.

presentation.examples.partial_brauer_monoid(n: int) → Presentation

A presentation for the partial Brauer monoid.

This function returns a monoid presentation defining the partial Brauer monoid of degree n, as described in Theorem 5.1 of [KM07].

Parameters:: n (int) – the degree.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 1.

Note

This function returns exactly the same presentation as partial_brauer_monoid_KM07, and exists as a convenience function for when a presentation for the alternating group is required, but the specific presentation is not important.

presentation.examples.partial_brauer_monoid_KM07(n: int) → Presentation

A presentation for the partial Brauer monoid.

This function returns a monoid presentation defining the partial Brauer monoid of degree n, as described in Theorem 5.1 of [KM07].

Parameters:: n (int) – the degree.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 1.

presentation.examples.partial_isometries_cycle_graph_monoid(n: int) → Presentation

A presentation for the monoid of partial isometries of a cycle graph.

This function returns a monoid presentation defining the monoid of partial isometries of an $n$-cycle graph, as in Theorem 2.8 of [FP22]

Parameters:: n (int) – the number of vertices of the cycle graph.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 3.

Note

This function returns exactly the same presentation as partial_isometries_cycle_graph_monoid_FP22, and exists as a convenience function for when a presentation for the alternating group is required, but the specific presentation is not important.

presentation.examples.partial_isometries_cycle_graph_monoid_FP22(n: int) → Presentation

A presentation for the monoid of partial isometries of a cycle graph.

This function returns a monoid presentation defining the monoid of partial isometries of an $n$ -cycle graph, as in Theorem 2.8 of [FP22]

Parameters:: n (int) – the number of vertices of the cycle graph.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 3.

presentation.examples.partial_transformation_monoid(n: int) → Presentation

A presentation for the partial transformation monoid.

This function returns a monoid presentation defining the partial transformation monoid of degree n, as in Theorem 1.6 of [MW24].

Parameters:: n (int) – the degree of the partial transformation monoid.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 2.

See also

For a specific presentation of the full transformation monoid, see one of the following functions:

partial_transformation_monoid_Shu60;

partial_transformation_monoid_MW24.

presentation.examples.partial_transformation_monoid_MW24(n: int) → Presentation

A presentation for the partial transformation monoid.

This function returns a monoid presentation defining the partial transformation monoid of degree n, as in Theorem 1.6 of [MW24].

Parameters:: n (int) – the degree of the partial transformation monoid.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 2.

presentation.examples.partial_transformation_monoid_Shu60(n: int) → Presentation

A presentation for the partial transformation monoid.

This function returns a monoid presentation defining the partial transformation monoid of degree n due to Shutov [Shu60], as in Theorem 9.4.1 of [GM09].

Parameters:: n (int) – the degree of the partial transformation monoid.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 4.

presentation.examples.partition_monoid(n: int) → Presentation

A presentation for the partition monoid.

This function returns a monoid presentation defining the partition monoid of degree n, as in Theorem 41 of [Eas11].

Parameters:: n (int) – the degree.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 4.

See also

For a specific presentation of the symmetric group, see one of the following functions:

partition_monoid_Eas11;

partition_monoid_HR05.

presentation.examples.partition_monoid_Eas11(n: int) → Presentation

A presentation for the partition monoid.

This function returns a monoid presentation defining the partition monoid of degree n, as in Theorem 41 of [Eas11].

Parameters:: n (int) – the degree.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 4.

presentation.examples.partition_monoid_HR05(n: int) → Presentation

A presentation for the partition monoid.

This function returns a monoid presentation defining the partition monoid of degree n, as in [HR05].

Parameters:: n (int) – the degree.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 1.

presentation.examples.plactic_monoid(n: int) → Presentation

A presentation for the plactic monoid.

This function returns a monoid presentation defining the plactic monoid with n generators, as in see Theorem 6 of [Knu70].

Parameters:: n (int) – the number of generators.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 2.

Note

This function returns exactly the same presentation as plactic_monoid_Knu70, and exists as a convenience function for when a presentation for the alternating group is required, but the specific presentation is not important.

presentation.examples.plactic_monoid_Knu70(n: int) → Presentation

A presentation for the plactic monoid.

This function returns a monoid presentation defining the plactic monoid with n generators, as in see Theorem 6 of [Knu70].

Parameters:: n (int) – the number of generators.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 2.

presentation.examples.rectangular_band(m: int, n: int) → Presentation

A presentation for a rectangular band.

This function returns a semigroup presentation defining the m by n rectangular band, as given in Proposition 4.2 of [ACOConnorR00].

Parameters:

m (int) – the number of rows.
n (int) – the number of columns.

Returns:

The specified presentation.

Return type:

Presentation

Raises:

LibsemigroupsError – if m == 0.
LibsemigroupsError – if n == 0.

Note

This function returns exactly the same presentation as rectangular_band_ACOR00, and exists as a convenience function for when a presentation for the alternating group is required, but the specific presentation is not important.

presentation.examples.rectangular_band_ACOR00(m: int, n: int) → Presentation

A presentation for a rectangular band.

This function returns a semigroup presentation defining the m by n rectangular band, as given in Proposition 4.2 of [ACOConnorR00].

Parameters:

m (int) – the number of rows.
n (int) – the number of columns.

Returns:

The specified presentation.

Return type:

Presentation

Raises:

LibsemigroupsError – if m == 0.
LibsemigroupsError – if n == 0.

presentation.examples.renner_type_B_monoid(l: int, q: int) → Presentation

A presentation for the Renner monoid of type B.

This functions returns a presentation for the Renner monoid of type B with size l and Iwahori-Hecke deformation q.

When q == 0, this corresponds to Definition 8.4.1 and Example 8.4.2 of [Gay18].

When q == 1, this corresponds to Theorem 8.4.19 of [Gay18].

Parameters:

l (int) – the size of the monoid.
q (int) – the Iwahori-Hecke deformation.

Returns:

The specified presentation

Return type:

Presentation

Raises:

LibsemigroupsError – if q != 0 or q != 1.

Note

This function returns exactly the same presentation as renner_type_B_monoid_Gay18, and exists as a convenience function for when a presentation for the alternating group is required, but the specific presentation is not important.

presentation.examples.renner_type_B_monoid_Gay18(l: int, q: int) → Presentation

A presentation for the Renner monoid of type B.

This functions returns a presentation for the Renner monoid of type B with size l and Iwahori-Hecke deformation q.

When q == 0, this corresponds to Definition 8.4.1 and Example 8.4.2 of [Gay18].

When q == 1, this corresponds to Theorem 8.4.19 of [Gay18].

Parameters:

l (int) – the size of the monoid.
q (int) – the Iwahori-Hecke deformation.

Returns:

The specified presentation

Return type:

Presentation

Raises:

LibsemigroupsError – if q != 0 or q != 1.

presentation.examples.renner_type_D_monoid(l: int, q: int) → Presentation

A presentation for the Renner monoid of type D.

This functions returns a presentation for the Renner monoid of type D with size l and Iwahori-Hecke deformation q.

When q == 0, this corresponds Definition 8.4.22 of [Gay18].

When q == 1, this corresponds to Theorem 8.4.43 of [Gay18].

Parameters:

l (int) – the size of the monoid.
q (int) – the Iwahori-Hecke deformation.

Returns:

The specified presentation

Return type:

Presentation

Raises:

LibsemigroupsError – if q != 0 or q != 1.

Note

This function returns exactly the same presentation as renner_type_D_monoid_Gay18, and exists as a convenience function for when a presentation for the alternating group is required, but the specific presentation is not important.

presentation.examples.renner_type_D_monoid_Gay18(l: int, q: int) → Presentation

A presentation for the Renner monoid of type D.

This functions returns a presentation for the Renner monoid of type D with size l and Iwahori-Hecke deformation q.

When q == 0, this corresponds Definition 8.4.22 of [Gay18].

When q == 1, this corresponds to Theorem 8.4.43 of [Gay18].

Parameters:

l (int) – the size of the monoid.
q (int) – the Iwahori-Hecke deformation.

Returns:

The specified presentation

Return type:

Presentation

Raises:

LibsemigroupsError – if q != 0 or q != 1.

presentation.examples.sigma_plactic_monoid(sigma: list[int]) → Presentation

A presentation for the $\sigma$-plactic monoid.

This function returns a presentation for the $\sigma$-plactic monoid with sigma.size() generators, where the image of $\sigma$ is given by the values in sigma.

The $\sigma$-plactic monoid is the quotient of the plactic monoid by the least congruence containing the relation $a^{\sigma(a)} = a$ for each $a$ in the alphabet. When $\sigma(a) = 2$ for all $a$, the resultant $\sigma$-plactic monoid is known as the stylic monoid, and is given in stylic_monoid.

Parameters:: sigma (list[int]) – a list representing the image of $\sigma$.
Returns:: The specified presentation.
Return type:: Presentation

presentation.examples.sigma_plactic_monoid_AHMNT24(sigma: list[int]) → Presentation

A presentation for the $\sigma$-plactic monoid.

This function returns a presentation for the $\sigma$-plactic monoid with sigma.size() generators as in Section 3.1 [AHM+24]. The image of $\sigma$ is given by the values in sigma.

The $\sigma$-plactic monoid is the quotient of the plactic monoid by the least congruence containing the relation $a^{\sigma(a)} = a$ for each $a$ in the alphabet. When $\sigma(a) = 2$ for all $a$, the resultant $\sigma$-plactic monoid is known as the stylic monoid, and is given in stylic_monoid.

Parameters:: sigma (list[int]) – a list representing the image of $sigma$.
Returns:: The specified presentation
Return type:: Presentation
Raises:: LibsemigroupsError – if len(sigma) < 1.

presentation.examples.singular_brauer_monoid(n: int) → Presentation

A presentation for the singular part of the Brauer monoid.

This function returns a monoid presentation for the singular part of the Brauer monoid of degree n, as in Theorem 5 of [MM07].

Parameters:: n (int) – the degree.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 3.

Note

This function returns exactly the same presentation as singular_brauer_monoid_MM07, and exists as a convenience function for when a presentation for the alternating group is required, but the specific presentation is not important.

presentation.examples.singular_brauer_monoid_MM07(n: int) → Presentation

A presentation for the singular part of the Brauer monoid.

This function returns a monoid presentation for the singular part of the Brauer monoid of degree n, as in Theorem 5 of [MM07].

Parameters:: n (int) – the degree.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 3.

presentation.examples.special_linear_group_2(q: int) → Presentation

A presentation for the special linear group $\mathrm{SL}(2, q)$.

This function returns a presentation for the special linear group $\mathrm{SL}(2, q)$ (also written $\mathrm{SL(2, \mathbb{Z}_q)}$), where q is an odd prime, as in Theorem 4 of [CR80].

Parameters:: q (int) – the order of the finite field over which the special linear group is constructed. This should be an odd prime for the returned presentation to define claimed group.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if q < 3.

Note

This function returns exactly the same presentation as special_linear_group_2_CR80, and exists as a convenience function for when a presentation for the alternating group is required, but the specific presentation is not important.

presentation.examples.special_linear_group_2_CR80(q: int) → Presentation

A presentation for the special linear group $\mathrm{SL}(2, q)$.

This function returns a presentation for the special linear group $\mathrm{SL}(2, q)$ (also written $\mathrm{SL(2, \mathbb{Z}_q)}$), where q is an odd prime, as in Theorem 4 of [CR80].

Parameters:: q (int) – the order of the finite field over which the special linear group is constructed. This should be an odd prime for the returned presentation to define claimed group.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if q < 3.

presentation.examples.stellar_monoid(l: int) → Presentation

A presentation for the stellar monoid.

This function returns a monoid presentation defining the stellar monoid with l generators, as in Theorem 4.39 of [GH19].

Parameters:: l (int) – the number of generators.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if l < 2.

Note

This function returns exactly the same presentation as stellar_monoid_GH19, and exists as a convenience function for when a presentation for the alternating group is required, but the specific presentation is not important.

presentation.examples.stellar_monoid_GH19(l: int) → Presentation

A presentation for the stellar monoid.

This function returns a monoid presentation defining the stellar monoid with l generators, as in Theorem 4.39 of [GH19].

Parameters:: l (int) – the number of generators.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if l < 2.

presentation.examples.stylic_monoid(n: int) → Presentation

A presentation for the stylic monoid.

This function returns a monoid presentation defining the stylic monoid with n generators, as in Theorem 8.1 of [AR22].

Parameters:: n (int) – the number of generators.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 2.

Note

This function returns exactly the same presentation as stylic_monoid_AR22, and exists as a convenience function for when a presentation for the alternating group is required, but the specific presentation is not important.

presentation.examples.stylic_monoid_AR22(n: int) → Presentation

A presentation for the stylic monoid.

This function returns a monoid presentation defining the stylic monoid with n generators, as in Theorem 8.1 of [AR22].

Parameters:: n (int) – the number of generators.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 2.

presentation.examples.symmetric_group(n: int) → Presentation

A presentation for the symmetric group.

This function returns a monoid presentation for the symmetric group of degree n, as on page 169 of [Car56]. This presentation has $n - 1$ generators and $(n - 1)^2$ relations.

Parameters:: n (int) – the degree of the symmetric group.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 2.

See also

For a specific presentation of the symmetric group, see one of the following functions:

symmetric_group_Bur12;

symmetric_group_Car56;

symmetric_group_Moo97_a;

symmetric_group_Moo97_b.

presentation.examples.symmetric_group_Bur12(n: int) → Presentation

A presentation for the symmetric group.

This function returns a monoid presentation for the symmetric group of degree n, as in p.464 of [Bur12]. This presentation has $n - 1$ generators and $n^3 - 5n^2 + 9n - 5$ relations.

Parameters:: n (int) – the degree of the symmetric group.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 2.

presentation.examples.symmetric_group_Car56(n: int) → Presentation

A presentation for the symmetric group.

This function returns a monoid presentation for the symmetric group of degree n, as on page 169 of [Car56]. This presentation has $n - 1$ generators and $(n - 1)^2$ relations.

Parameters:: n (int) – the degree of the symmetric group.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 2.

presentation.examples.symmetric_group_Moo97_a(n: int) → Presentation

A presentation for the symmetric group.

This function returns a monoid presentation for the symmetric group of degree n, as in Theorem A of [Moo97]. This presentation has $n - 1$ generators and $\frac{1}{2}n(n - 1)$ relations.

Parameters:: n (int) – the degree of the symmetric group.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 2.

presentation.examples.symmetric_group_Moo97_b(n: int) → Presentation

A presentation for the symmetric group.

This function returns a monoid presentation for the symmetric group of degree n, as in in Theorem A’ of [Moo97]. This presentation has $2$ generators and $n + 1$ relations for $n \geq 4$. If $n<4$ then there are $4$ relations.

Parameters:: n (int) – the degree of the symmetric group.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 2.

presentation.examples.symmetric_inverse_monoid(n: int) → Presentation

A presentation for the partial transformation monoid.

This function returns a monoid presentation defining the partial transformation monoid of degree n, as in Theorem 1.6 of [MW24].

Parameters:: n (int) – the degree of the partial transformation monoid.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 2.

See also

For a specific presentation of the full transformation monoid, see one of the following functions:

symmetric_inverse_monoid_Sol04;

symmetric_inverse_monoid_Shu60;

symmetric_inverse_monoid_MW24.

presentation.examples.symmetric_inverse_monoid_MW24(n: int) → Presentation

A presentation for the symmetric inverse monoid.

This function returns a monoid presentation defining the partial transformation monoid of degree n, as in Theorem 1.4 of [MW24].

Parameters:: n (int) – the degree of the symmetric inverse monoid.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 4.

presentation.examples.symmetric_inverse_monoid_Shu60(n: int) → Presentation

A presentation for the symmetric inverse monoid.

This function returns a monoid presentation defining the symmetric inverse monoid of degree n due to Shutov [Shu60], as in Theorem 9.2.2 of [GM09].

Parameters:: n (int) – the degree of the symmetric inverse monoid.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 4.

presentation.examples.symmetric_inverse_monoid_Sol04(n: int) → Presentation

A presentation for the symmetric inverse monoid.

This function returns a monoid presentation defining the symmetric inverse monoid of degree n, as in Example 7.1.2 of [Gay18].

Parameters:: n (int) – the degree of the symmetric inverse monoid.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 2.

presentation.examples.temperley_lieb_monoid(n: int) → Presentation

A presentation for the Temperley-Lieb monoid.

This function returns a monoid presentation defining the Temperley-Lieb monoid with n generators, as described in Theorem 2.2 of [Eas21].

Parameters:: n (int) – the number of generators.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 3.

Note

This function returns exactly the same presentation as temperley_lieb_monoid_Eas21, and exists as a convenience function for when a presentation for the alternating group is required, but the specific presentation is not important.

presentation.examples.temperley_lieb_monoid_Eas21(n: int) → Presentation

A presentation for the Temperley-Lieb monoid.

This function returns a monoid presentation defining the Temperley-Lieb monoid with n generators, as described in Theorem 2.2 of [Eas21].

Parameters:: n (int) – the number of generators.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 3.

presentation.examples.uniform_block_bijection_monoid(n: int) → Presentation

A presentation for the uniform block bijection monoid.

This function returns a monoid presentation defining the uniform block bijection monoid of degree n, as in [Fit03].

Parameters:: n (int) – the degree.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 3.

Note

This function returns exactly the same presentation as uniform_block_bijection_monoid_Fit03, and exists as a convenience function for when a presentation for the alternating group is required, but the specific presentation is not important.

presentation.examples.uniform_block_bijection_monoid_Fit03(n: int) → Presentation

A presentation for the uniform block bijection monoid.

This function returns a monoid presentation defining the uniform block bijection monoid of degree n, as in [Fit03].

Parameters:: n (int) – the degree.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 3.

presentation.examples.zero_rook_monoid(n: int) → Presentation

A presentation for the $0$-rook monoid.

This function returns a presentation for the $0$-rook monoid of degree n, as in Definition 4.1.1 in [Gay18]

Parameters:: n (int) – the degree.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 2.

Note

This function returns exactly the same presentation as zero_rook_monoid_Gay18, and exists as a convenience function for when a presentation for the alternating group is required, but the specific presentation is not important.

presentation.examples.zero_rook_monoid_Gay18(n: int) → Presentation

A presentation for the $0$-rook monoid.

This function returns a presentation for the $0$ -rook monoid of degree n, as in Definition 4.1.1 in [Gay18].

Parameters:: n (int) – the degree.
Returns:: The specified presentation.
Return type:: Presentation
Raises:: LibsemigroupsError – if n < 2.