Add a presentation for the Grammic monoid

docs/libsemigroups.bib

@@ -5,6 +5,18 @@
 
 %% Saved with string encoding Unicode (UTF-8) 
 
+@article{Luo2026aa,
+	author = {Luo, Yan Feng and Xie, Jia Jia and Zhang, Wen Ting},
+	title = {A presentation of the grammic monoid},
+	journal = {Semigroup Forum},
+	volume = {113},
+	number = {2},
+	pages = {575--593},
+	year = {2026},
+	doi = {10.1007/s00233-026-10671-1},
+	url = {https://doi.org/10.1007/s00233-026-10671-1}
+}
+
 @misc{Hivert2026aa,
 	author = {Hivert, Florent and Scott, Jeanne},
 	title = {Diagrammatic {Okada} monoid and cellularity of the {Okada} algebra},

include/libsemigroups/presentation-examples.hpp

@@ -55,6 +55,7 @@ namespace libsemigroups {
     Presentation<word_type> dual_symmetric_inverse_monoid(size_t n);
     Presentation<word_type> fibonacci_semigroup(size_t r, size_t n);
     Presentation<word_type> full_transformation_monoid(size_t n);
+    Presentation<word_type> grammic_monoid(size_t n);
     Presentation<word_type> hypo_plactic_monoid(size_t n);
     Presentation<word_type> monogenic_semigroup(size_t m, size_t r);
     Presentation<word_type> motzkin_monoid(size_t n);
@@ -336,6 +337,22 @@ namespace libsemigroups {
     [[nodiscard]] Presentation<word_type>
     full_transformation_monoid_MW24_b(size_t n);
 
+    //! \brief A presentation for the grammic monoid.
+    //!
+    //! This function returns a monoid presentation defining the grammic monoid
+    //! with \p n generators, as in Theorem 3.17 of \cite Luo2026aa.
+    //!
+    //! This presentation consists of the Knuth relations from
+    //! \ref plactic_monoid_Knu70 together with \f$dbac = badc\f$ for all
+    //! \f$0 \leq a < b \leq c < d < n\f$.
+    //!
+    //! \param n the number of generators.
+    //!
+    //! \returns A value of type `Presentation<word_type>`.
+    //!
+    //! \throws LibsemigroupsException if `n < 1`.
+    [[nodiscard]] Presentation<word_type> grammic_monoid_LXZ26(size_t n);
+
     //! \brief A presentation for the hypoplactic monoid.
     //!
     //! This function returns a presentation for the hypoplactic monoid with
@@ -1019,6 +1036,20 @@ namespace libsemigroups {
     //! * `full_transformation_monoid_MW24_b`.
     [[nodiscard]] Presentation<word_type> full_transformation_monoid(size_t n);
 
+    //! \copydoc grammic_monoid_LXZ26
+    //!
+    //! \note
+    //! This function performs exactly the same as `grammic_monoid_LXZ26`,
+    //! and exists as a convenience function for when a presentation for the
+    //! grammic monoid is required, but the relations of the presentation are
+    //! not important.
+    //!
+    //! \sa
+    //! `grammic_monoid_LXZ26`.
+    [[nodiscard]] inline Presentation<word_type> grammic_monoid(size_t n) {
+      return grammic_monoid_LXZ26(n);
+    }
+
     //! \copydoc hypo_plactic_monoid_Nov00
     //!
     //! \note

src/presentation-examples.cpp

@@ -456,6 +456,25 @@ namespace libsemigroups {
       return p;
     }
 
+    // Theorem 3.17 of https://doi.org/10.1007/s00233-026-10671-1.
+    Presentation<word_type> grammic_monoid_LXZ26(size_t n) {
+      auto p = plactic_monoid_Knu70(n);
+      p.contains_empty_word(true);
+      // Use dbac = badc as in the abstract; the Knuth relations identify
+      // badc and bdac, the right-hand side used in Equation (3.3).
+      for (size_t d = 2; d < n; ++d) {
+        for (size_t c = 1; c < d; ++c) {
+          for (size_t b = 1; b <= c; ++b) {
+            for (size_t a = 0; a < b; ++a) {
+              presentation::add_rule_no_checks(
+                  p, word_type({d, b, a, c}), word_type({b, a, d, c}));
+            }
+          }
+        }
+      }
+      return p;
+    }
+
     Presentation<word_type> catalan_monoid_Sol96(size_t n) {
       if (n < 1) {
         LIBSEMIGROUPS_EXCEPTION(

tests/test-presentation-examples-2.cpp

@@ -320,4 +320,71 @@ namespace libsemigroups {
     REQUIRE(kb.number_of_classes() == POSITIVE_INFINITY);
   }
 
+  LIBSEMIGROUPS_TEST_CASE("Example",
+                          "110",
+                          "grammic_monoid arguments and small ranks",
+                          "[quick][presentation-examples]") {
+    REQUIRE_THROWS_AS(grammic_monoid(0), LibsemigroupsException);
+    REQUIRE_THROWS_AS(grammic_monoid_LXZ26(0), LibsemigroupsException);
+
+    auto p = grammic_monoid(1);
+    REQUIRE(p.alphabet() == 0_w);
+    REQUIRE(p.rules.empty());
+    REQUIRE(p.contains_empty_word());
+    REQUIRE(grammic_monoid(2) == plactic_monoid(2));
+
+    for (size_t n = 1; n <= 6; ++n) {
+      p = grammic_monoid(n);
+      REQUIRE(p == grammic_monoid_LXZ26(n));
+      REQUIRE(p.alphabet().size() == n);
+      REQUIRE(p.contains_empty_word());
+      REQUIRE_NOTHROW(p.throw_if_bad_alphabet_or_rules());
+    }
+  }
+
+  LIBSEMIGROUPS_TEST_CASE("Example",
+                          "111",
+                          "grammic_monoid(3) is a proper plactic quotient",
+                          "[quick][presentation-examples]") {
+    auto rg = ReportGuard(false);
+    auto p  = grammic_monoid(3);
+    REQUIRE(p.rules.size() == 18);
+    KnuthBendix kb(congruence_kind::twosided, p);
+    REQUIRE(is_obviously_infinite(kb));
+    REQUIRE(kb.number_of_classes() == POSITIVE_INFINITY);
+    // The b = c case: cbab = bacb, equivalently cbab = bcab.
+    REQUIRE(knuth_bendix::contains(kb, 2101_w, 1021_w));
+    REQUIRE(knuth_bendix::contains(kb, 2101_w, 1201_w));
+    REQUIRE(!knuth_bendix::contains(kb, 01_w, 10_w));
+    REQUIRE(!knuth_bendix::contains(kb, 0_w, 00_w));
+    REQUIRE(knuth_bendix::normal_forms(kb).min(4).max(4).count() == 38);
+
+    KnuthBendix plactic(congruence_kind::twosided, plactic_monoid(3));
+    REQUIRE(!knuth_bendix::contains(plactic, 2101_w, 1021_w));
+  }
+
+  LIBSEMIGROUPS_TEST_CASE("Example",
+                          "112",
+                          "grammic_monoid relations for larger ranks",
+                          "[quick][presentation-examples]") {
+    auto p        = grammic_monoid(4);
+    auto expected = plactic_monoid(4).rules;
+    // All five instances of dbac = badc with a < b <= c < d.
+    expected.insert(expected.end(),
+                    {2101_w,
+                     1021_w,
+                     3101_w,
+                     1031_w,
+                     3102_w,
+                     1032_w,
+                     3202_w,
+                     2032_w,
+                     3212_w,
+                     2132_w});
+    REQUIRE(p.rules == expected);
+    // There are binomial(n + 1, 4) extra relations.
+    REQUIRE(grammic_monoid(5).rules.size() == 110);
+    REQUIRE(grammic_monoid(6).rules.size() == 210);
+  }
+
 }  // namespace libsemigroups