def test_FreeGroup__init__():
x, y, z = map(Symbol, "xyz")
assert len(FreeGroup("x, y, z").generators) == 3
assert len(FreeGroup(x).generators) == 1
assert len(FreeGroup(("x", "y", "z"))) == 3
assert len(FreeGroup((x, y, z)).generators) == 3
def simplify_presentation(*args, **kwargs):
'''
For an instance of `FpGroup`, return a simplified isomorphic copy of
the group (e.g. remove redundant generators or relators). Alternatively,
a list of generators and relators can be passed in which case the
simplified lists will be returned.
By default, the generators of the group are unchanged. If you would
like to remove redundant generators, set the keyword argument
`change_gens = True`.
'''
change_gens = kwargs.get("change_gens", False)
if len(args) == 1:
if not isinstance(args[0], FpGroup):
raise TypeError("The argument must be an instance of FpGroup")
G = args[0]
gens, rels = simplify_presentation(G.generators,
G.relators,
change_gens=change_gens)
if gens:
return FpGroup(gens[0].group, rels)
return FpGroup(FreeGroup([]), [])
elif len(args) == 2:
gens, rels = args[0][:], args[1][:]
if not gens:
return gens, rels
identity = gens[0].group.identity
else:
if len(args) == 0:
m = "Not enough arguments"
else:
m = "Too many arguments"
raise RuntimeError(m)
prev_gens = []
prev_rels = []
while not set(prev_rels) == set(rels):
prev_rels = rels
while change_gens and not set(prev_gens) == set(gens):
prev_gens = gens
gens, rels = elimination_technique_1(gens, rels, identity)
rels = _simplify_relators(rels, identity)
if change_gens:
syms = [g.array_form[0][0] for g in gens]
F = free_group(syms)[0]
identity = F.identity
gens = F.generators
subs = dict(zip(syms, gens))
for j, r in enumerate(rels):
a = r.array_form
rel = identity
for sym, p in a:
rel = rel * subs[sym]**p
rels[j] = rel
return gens, rels
def test_FreeGroup__getitem__():
assert F[0:] == FreeGroup("x, y, z")
assert F[1:] == FreeGroup("y, z")
assert F[2:] == FreeGroup("z")
def test_simplify_presentation():
# ref #16083
G = simplify_presentation(FpGroup(FreeGroup([]), []))
assert not G.generators
assert not G.relators
import networkx as nx
from graphs.oriented_labelled_graph_with_basepoint import OrientedLabelledGraphWithBasePoint
from graphs.graph_from_free_group_factory import GraphFromFreeGroupFactory, CoreGraph
from sympy.combinatorics.free_groups import FreeGroup, FreeGroupElement
from itertools import product, combinations
from typing import Iterable
from utilities.general_utilities import memorize
n = 3
generators_string = ",".join(["a%d" % d for d in range(1, n+1)])
F = FreeGroup(generators_string)
a1, a2, a3 = F.generators
free_group_graph = GraphFromFreeGroupFactory.get_subgroup_core(F.generators)
def get_edge_crosses(graph, element):
path = graph.get_membership_path(element)
edge_crosses = {}
for x in path:
u, v, n, s = x
if s.ext_rep[1] < 0:
u2 = u
u = v
v = u2
generator = s if s.ext_rep[1] > 0 else s.inverse()
edge_id = (u, v, n, generator)
if edge_id not in edge_crosses:
edge_crosses[edge_id] = 0
