presentation-examples: add abacus Jones

Author: james-d-mitchell

docs/source/data-structures/presentations/examples.rst

@@ -26,6 +26,7 @@ semigroup or monoid they define.
 .. autosummary::
     :signatures: short
 
+    abacus_jones_monoid
     alternating_group
     brauer_monoid
     chinese_monoid
@@ -75,6 +76,7 @@ documented in the :ref:`default-presentations` section.
 .. autosummary::
     :signatures: short
 
+    abacus_jones_monoid_AJP25
     alternating_group_Moo97
     brauer_monoid_KM07
     chinese_monoid_CEKNH01

docs/source/libsemigroups.bib

@@ -6,6 +6,16 @@
 
 %% Saved with string encoding Unicode (UTF-8) 
 
+@misc{Aicardi2025aa,
+  title={Framization and Deframization}, 
+  author={Francesca Aicardi and Jesús Juyumaya and Paolo Papi},
+  year={2025},
+  eprint={2405.10809},
+  archivePrefix={arXiv},
+  primaryClass={math.RA},
+  url={https://arxiv.org/abs/2405.10809}, 
+}
+
  @misc{Guennebaud2010aa,
 	author = {Ga\"{e}l Guennebaud and Beno\^{i}t Jacob and others},
 	title = {Eigen v3},

src/libsemigroups_pybind11/presentation/examples.py

@@ -15,6 +15,8 @@
 """
 
 from _libsemigroups_pybind11 import (  # pylint: disable=no-name-in-module
+    presentation_examples_abacus_jones_monoid as _abacus_jones_monoid,
+    presentation_examples_abacus_jones_monoid_AJP25 as _abacus_jones_monoid_AJP25,
     presentation_examples_alternating_group as _alternating_group,
     presentation_examples_alternating_group_Moo97 as _alternating_group_Moo97,
     presentation_examples_brauer_monoid as _brauer_monoid,
@@ -101,6 +103,8 @@
     wrap_cxx_free_fn as _wrap_cxx_free_fn,
 )
 
+abacus_jones_monoid = _wrap_cxx_free_fn(_abacus_jones_monoid)
+abacus_jones_monoid_AJP25 = _wrap_cxx_free_fn(_abacus_jones_monoid_AJP25)
 alternating_group = _wrap_cxx_free_fn(_alternating_group)
 alternating_group_Moo97 = _wrap_cxx_free_fn(_alternating_group_Moo97)
 brauer_monoid = _wrap_cxx_free_fn(_brauer_monoid)

src/presentation-examples.cpp

@@ -50,6 +50,7 @@ generators, as in Theorem 4.39 of :cite:`Gay1999aa`.
 
 :raises LibsemigroupsError:  if ``l < 2``.
 )pbdoc");
+
     m.def("presentation_examples_dual_symmetric_inverse_monoid_EEF07",
           &examples::dual_symmetric_inverse_monoid_EEF07,
           py::arg("n"),
@@ -1701,6 +1702,7 @@ includes the rules from :any:`plactic_monoid_Knu70`.
       alternating group is required, but the specific presentation
       is not important.
 )pbdoc");
+
     m.def("presentation_examples_sigma_plactic_monoid",
           &examples::sigma_plactic_monoid,
           py::arg("sigma"),
@@ -1724,6 +1726,7 @@ and is given in :any:`stylic_monoid`.
 :returns: The specified presentation.
 :rtype: Presentation
 )pbdoc");
+
     m.def("presentation_examples_zero_rook_monoid",
           &examples::zero_rook_monoid,
           py::arg("n"),
@@ -1849,6 +1852,7 @@ When ``q == 1``, this corresponds to Theorem 8.4.43 of :cite:`Gay2018aa`.
       alternating group is required, but the specific presentation
       is not important.
 )pbdoc");
+
     m.def("presentation_examples_not_renner_type_D_monoid",
           &examples::not_renner_type_D_monoid,
           py::arg("l"),
@@ -1879,6 +1883,68 @@ When ``q == 1``, this corresponds to Section 3.3 of :cite:`Godelle2009aa`.
       and exists as a convenience function for when a presentation for the
       alternating group is required, but the specific presentation
       is not important.
+)pbdoc");
+
+    m.def("presentation_examples_abacus_jones_monoid",
+          &presentation::examples::abacus_jones_monoid,
+          py::arg("n"),
+          py::arg("d"),
+          R"pbdoc(
+:sig=(n: int, d: int) -> Presentation:
+
+A presentation for the abacus Jones monoid.
+
+This function returns a monoid presentation defining the abacus Jones monoid of
+degree *n*; as defined in Proposition 3.2 of :cite:`Aicardi2025aa`. The
+abacus Jones monoid is formed by the diagrams of the Jones monoid
+(aka the Temperley-Lieb monoid) whose arcs have at most ``d − 1``
+beads sliding on each arc. This presentation has :math:`2n - 1`
+generators, :math:`5n ^ 2 - 5n + 2 + 2(n - 1)(d - 1)` relations, and
+defines a monoid of size :math:`C_n\ d ^n` where :math:`C_n` is the
+:math:`n`-th Catalan number :math:`\frac{1}{n + 1}\binom{2n}{n}`.
+
+:param n: the degree.
+:type n: int
+
+:param d: one more than the maximum number of beads on each arc of the abacus.
+:type d: int
+
+:returns: The specified presentation.
+:rtype: Presentation
+
+:raises LibsemigroupsError:  if ``n < 3`` or if ``d = 0``.
+
+.. seealso::  :any:`abacus_jones_monoid_AJP25`.
+)pbdoc");
+
+    m.def("presentation_examples_abacus_jones_monoid_AJP25",
+          &presentation::examples::abacus_jones_monoid_AJP25,
+          py::arg("n"),
+          py::arg("d"),
+          R"pbdoc(
+:sig=(n: int, d: int) -> Presentation:
+
+A presentation for the abacus Jones monoid.
+
+This function returns a monoid presentation defining the abacus Jones monoid of
+degree *n*; as defined in Proposition 3.2 of :cite:`Aicardi2025aa`. The
+abacus Jones monoid is formed by the diagrams of the Jones monoid
+(aka the Temperley-Lieb monoid) whose arcs have at most ``d − 1``
+beads sliding on each arc. This presentation has :math:`2n - 1`
+generators, :math:`5n ^ 2 - 5n + 2 + 2(n - 1)(d - 1)` relations, and
+defines a monoid of size :math:`C_n\ d ^n` where :math:`C_n` is the
+:math:`n`-th Catalan number :math:`\frac{1}{n + 1}\binom{2n}{n}`.
+
+:param n: the degree.
+:type n: int
+
+:param d: one more than the maximum number of beads on each arc of the abacus.
+:type d: int
+
+:returns: The specified presentation.
+:rtype: Presentation
+
+:raises LibsemigroupsError:  if ``n < 3`` or if ``d = 0``.
 )pbdoc");
   }  // init_presentation_examples
 

tests/test_presentation_examples.py

@@ -391,3 +391,25 @@ def test_orientation_preserving_reversing_monoid_AR00():
     p = examples.orientation_preserving_reversing_monoid_AR00(5)
     tc = ToddCoxeter(congruence_kind.twosided, p)
     assert tc.number_of_classes() == 1015
+
+
+def test_abacus_jones():
+    ReportGuard(False)
+    with pytest.raises(LibsemigroupsError):
+        examples.abacus_jones_monoid(0, 1)
+    with pytest.raises(LibsemigroupsError):
+        examples.abacus_jones_monoid(1, 1)
+    with pytest.raises(LibsemigroupsError):
+        examples.abacus_jones_monoid(2, 1)
+    with pytest.raises(LibsemigroupsError):
+        examples.abacus_jones_monoid(3, 0)
+
+    p = examples.abacus_jones_monoid(3, 2)
+    tc = ToddCoxeter(congruence_kind.twosided, p)
+    assert tc.number_of_classes() == 40
+    assert tc.number_of_classes() == 5 * 2**3
+
+    p = examples.abacus_jones_monoid_AJP25(3, 2)
+    tc = ToddCoxeter(congruence_kind.twosided, p)
+    assert tc.number_of_classes() == 40
+    assert tc.number_of_classes() == 5 * 2**3