examples: add example of non-iso. fp monoids

examples/not_renner_B_5_1.py

@@ -0,0 +1,38 @@
+"""
+In this file we give the "proof" that the monoid defined by the presentation
+not_renner_type_B_monoid(5, 1) is not isomorphic to the monoid defined by
+renner_type_B_monoid(5, 1).
+"""
+
+from libsemigroups_pybind11 import Sims2, Presentation, WordGraph
+from libsemigroups_pybind11 import word_graph
+from libsemigroups_pybind11.presentation import examples
+
+
+def check_incompat(wg: WordGraph, p: Presentation) -> bool:
+    for i in range(0, len(p.rules), 2):
+        if not word_graph.is_compatible(wg, 0, 4096, p.rules[i], p.rules[i + 1]):
+            return True
+    return False
+
+
+def witness(wg: WordGraph, p: Presentation) -> tuple[list[int], list[int]]:
+    for i in range(0, len(p.rules), 2):
+        if not word_graph.is_compatible(wg, 0, 1, p.rules[i], p.rules[i + 1]):
+            return (p.rules[i], p.rules[i + 1])
+
+
+p = examples.not_renner_type_B_monoid(5, 1)
+q = examples.renner_type_B_monoid(5, 1)
+
+# We iterate 2-sided congruences of the monoid defined by "p" with the aim of
+# finding one which is not compatible with the relations in "q"
+twosided_congs = Sims2(p)
+wg = twosided_congs.find_if(4096, lambda wg: check_incompat(wg, q))
+word_graph.number_of_nodes_reachable_from(wg, 0)  # 2177
+wg.induced_subgraph(0, 2177)
+witness(wg, q)  # ([5, 0, 1, 2, 3, 4, 0, 1, 2, 3, 0, 1, 2, 0, 1, 0, 5], [10])
+# meaning that the paths with source 0 labelled by these two words should lead to the same
+# node but do not.
+word_graph.follow_path(wg, 0, [5, 0, 1, 2, 3, 4, 0, 1, 2, 3, 0, 1, 2, 0, 1, 0, 5])  # 6
+word_graph.follow_path(wg, 0, [10])  # 7