def __init__(self, Q):
"""
Initialize ``self``.
INPUT:
- a :class:`~sage.graphs.digraph.DiGraph`.
EXAMPLES:
Note that usually a path semigroup is created using
:meth:`sage.graphs.digraph.DiGraph.path_semigroup`. Here, we
demonstrate the direct construction::
sage: Q = DiGraph({1:{2:['a','b'], 3:['c']}, 2:{3:['d']}}, immutable=True)
sage: from sage.quivers.path_semigroup import PathSemigroup
sage: P = PathSemigroup(Q)
sage: P is DiGraph({1:{2:['a','b'], 3:['c']}, 2:{3:['d']}}).path_semigroup() # indirect doctest
True
sage: P
Partial semigroup formed by the directed paths of Multi-digraph on 3 vertices
While hardly of any use, it is possible to construct the path
semigroup of an empty quiver (it is, of course, empty)::
sage: D = DiGraph({})
sage: A = D.path_semigroup(); A
Partial semigroup formed by the directed paths of Digraph on 0 vertices
sage: A.list()
[]
"""
#########
## Verify that the graph labels are acceptable for this implementation ##
# Check that edges are labelled with nonempty strings and don't begin
# with 'e_' or contain '*'
for x in Q.edge_labels():
if not isinstance(x, str) or x == '':
raise ValueError(
"edge labels of the digraph must be nonempty strings")
if x[0:2] == 'e_':
raise ValueError(
"edge labels of the digraph must not begin with 'e_'")
if x.find('*') != -1:
raise ValueError(
"edge labels of the digraph must not contain '*'")
# Check validity of input: vertices have to be labelled 1,2,3,... and
# edge labels must be unique
for v in Q:
if v not in ZZ:
raise ValueError(
"vertices of the digraph must be labelled by integers")
if len(set(Q.edge_labels())) != Q.num_edges():
raise ValueError("edge labels of the digraph must be unique")
nvert = len(Q.vertices())
## Determine the category which this (partial) semigroup belongs to
if Q.is_directed_acyclic() and not Q.has_loops():
cat = FiniteEnumeratedSets()
else:
cat = InfiniteEnumeratedSets()
names = ['e_{0}'.format(v)
for v in Q.vertices()] + [e[2] for e in Q.edges()]
self._quiver = Q
if nvert == 1:
cat = cat.join([cat, Monoids()])
else:
# TODO: Make this the category of "partial semigroups"
cat = cat.join([cat, Semigroups()])
Parent.__init__(self, names=names, category=cat)
def __init__(self, Q):
"""
Initialize ``self``.
INPUT:
- a :class:`~sage.graphs.digraph.DiGraph`.
EXAMPLES:
Note that usually a path semigroup is created using
:meth:`sage.graphs.digraph.DiGraph.path_semigroup`. Here, we
demonstrate the direct construction::
sage: Q = DiGraph({1:{2:['a','b'], 3:['c']}, 2:{3:['d']}}, immutable=True)
sage: from sage.quivers.path_semigroup import PathSemigroup
sage: P = PathSemigroup(Q)
sage: P is DiGraph({1:{2:['a','b'], 3:['c']}, 2:{3:['d']}}).path_semigroup() # indirect doctest
True
sage: P
Partial semigroup formed by the directed paths of Multi-digraph on 3 vertices
While hardly of any use, it is possible to construct the path
semigroup of an empty quiver (it is, of course, empty)::
sage: D = DiGraph({})
sage: A = D.path_semigroup(); A
Partial semigroup formed by the directed paths of Digraph on 0 vertices
sage: A.list()
[]
.. TODO::
When the graph has more than one edge, the proper category would be
a "partial semigroup" or a "semigroupoid" but definitely not a
semigroup!
"""
#########
## Verify that the graph labels are acceptable for this implementation ##
# Check that edges are labelled with nonempty strings and don't begin
# with 'e_' or contain '*'
labels = Q.edge_labels()
if len(set(labels)) != len(labels):
raise ValueError("edge labels of the digraph must be unique")
for x in labels:
if not isinstance(x, str) or x == '':
raise ValueError("edge labels of the digraph must be nonempty strings")
if x[0:2] == 'e_':
raise ValueError("edge labels of the digraph must not begin with 'e_'")
if x.find('*') != -1:
raise ValueError("edge labels of the digraph must not contain '*'")
# Check validity of input: vertices have to be labelled 1,2,3,... and
# edge labels must be unique
for v in Q:
if not isinstance(v, integer_types + (Integer,)):
raise ValueError("vertices of the digraph must be labelled by integers")
## Determine the category which this (partial) semigroup belongs to
if Q.is_directed_acyclic():
cat = FiniteEnumeratedSets()
else:
cat = InfiniteEnumeratedSets()
self._sorted_edges = tuple(sorted(Q.edges(), key=lambda x:x[2]))
self._labels = tuple([x[2] for x in self._sorted_edges])
self._label_index = {s[2]: i for i,s in enumerate(self._sorted_edges)}
self._nb_arrows = max(len(self._sorted_edges), 1)
names = ['e_{0}'.format(v) for v in Q.vertices()] + list(self._labels)
self._quiver = Q
if Q.num_verts() == 1:
cat = cat.join([cat,Monoids()])
else:
cat = cat.join([cat,Semigroups()])
Parent.__init__(self, names=names, category=cat)
def __init__(self, Q):
"""
Initialize ``self``.
INPUT:
- a :class:`~sage.graphs.digraph.DiGraph`.
EXAMPLES:
Note that usually a path semigroup is created using
:meth:`sage.graphs.digraph.DiGraph.path_semigroup`. Here, we
demonstrate the direct construction::
sage: Q = DiGraph({1:{2:['a','b'], 3:['c']}, 2:{3:['d']}}, immutable=True)
sage: from sage.quivers.path_semigroup import PathSemigroup
sage: P = PathSemigroup(Q)
sage: P is DiGraph({1:{2:['a','b'], 3:['c']}, 2:{3:['d']}}).path_semigroup() # indirect doctest
True
sage: P
Partial semigroup formed by the directed paths of Multi-digraph on 3 vertices
While hardly of any use, it is possible to construct the path
semigroup of an empty quiver (it is, of course, empty)::
sage: D = DiGraph({})
sage: A = D.path_semigroup(); A
Partial semigroup formed by the directed paths of Digraph on 0 vertices
sage: A.list()
[]
"""
#########
## Verify that the graph labels are acceptable for this implementation ##
# Check that edges are labelled with nonempty strings and don't begin
# with 'e_' or contain '*'
for x in Q.edge_labels():
if not isinstance(x, str) or x == '':
raise ValueError("edge labels of the digraph must be nonempty strings")
if x[0:2] == 'e_':
raise ValueError("edge labels of the digraph must not begin with 'e_'")
if x.find('*') != -1:
raise ValueError("edge labels of the digraph must not contain '*'")
# Check validity of input: vertices have to be labelled 1,2,3,... and
# edge labels must be unique
for v in Q:
if v not in ZZ:
raise ValueError("vertices of the digraph must be labelled by integers")
if len(set(Q.edge_labels())) != Q.num_edges():
raise ValueError("edge labels of the digraph must be unique")
nvert = len(Q.vertices())
## Determine the category which this (partial) semigroup belongs to
if Q.is_directed_acyclic() and not Q.has_loops():
cat = FiniteEnumeratedSets()
else:
cat = InfiniteEnumeratedSets()
names = ['e_{0}'.format(v) for v in Q.vertices()] + [e[2] for e in Q.edges()]
self._quiver = Q
if nvert == 1:
cat = cat.join([cat,Monoids()])
else:
# TODO: Make this the category of "partial semigroups"
cat = cat.join([cat,Semigroups()])
Parent.__init__(self, names=names, category=cat)
