def __init__(self, automaton, alphabet, category, side):
"""
sage: automaton = DiGraph( [ (1, 1, "b"), (1, 2, "a"),
... (2, 2, "a"), (2, 2, "c"), (2, 3, "b"),
... (3, 2, "a"), (3, 3, "b"), (3, 3, "c") ] )
sage: from sage.monoids.transition_monoid import TransitionMonoidOfDeterministicAutomaton
sage: M = TransitionMonoidOfDeterministicAutomaton(automaton)
sage: M.cardinality()
7
sage: M.list()
[[], ['a'], ['b'], ['c'], ['a', 'b'], ['b', 'c'], ['c', 'a']]
sage: TestSuite(M).run()
Since ``automaton`` is mutable, we made a private copy of it::
sage: M.automaton() == automaton
True
sage: M.automaton() is automaton
False
"""
# Should check that automaton is deterministic
self._automaton = automaton
# We would want to have FiniteSetPartialMaps
ambient = FiniteSetMaps( list(automaton.vertices())+[None], action=side)
def generator(a):
return ambient( functools.partial(automaton.transition, a = a) )
generators = Family( alphabet, generator )
AutomaticMonoid.__init__(self, generators, ambient, ambient.one(), operator.mul, category)
def __init__(self, n=3, action="left"):
r"""
EXAMPLES::
sage: from sage_semigroups.categories.examples.finite_h_trivial_monoids import MonoidOfOrderPreservingMaps
sage: p = MonoidOfOrderPreservingMaps(4)
sage: TestSuite(p).run()
"""
assert n in NN
domain = IntegerRange(Integer(1), Integer(n + 1))
index_set = IntegerRange(Integer(1), Integer(n))
ambient_monoid = FiniteSetMaps(domain, action=action)
def pi(i):
return ambient_monoid.from_dict(
dict([(k, k) for k in domain if k != i + 1] + [(i + 1, i)]))
def opi(i):
return ambient_monoid.from_dict(
dict([(k, k) for k in domain if k != i] + [(i, i + 1)]))
piopi = Family(
dict([[i, pi(i)] for i in index_set] + [[-i, opi(i)]
for i in index_set]))
AutomaticMonoid.__init__(
self,
piopi,
ambient_monoid,
one=ambient_monoid.one(),
mul=operator.mul,
category=Monoids().HTrivial().Finite().FinitelyGenerated()
& Monoids().Transformation().Subobjects())
def __init__(self, n):
ambient_monoid = FiniteSetMaps(range(-n, 0) + range(1, n + 1),
action="right")
pi = Family(
range(1, n), lambda j: ambient_monoid.from_dict(
dict([(i, i) for i in range(1, n + 1) if i != j + 1] + [(
j + 1, j)] + [(i, i) for i in range(-n, 0)
if i != -j] + [(-j, -j - 1)])))
category = Monoids().JTrivial().Finite() & Monoids().Transformation(
).Subobjects()
AutomaticMonoid.__init__(self,
pi,
ambient_monoid,
one=ambient_monoid.one(),
mul=operator.mul,
category=category)
def __init__(self, W):
self.lattice = W.domain()
self.W = W
from sage.sets.finite_set_maps import FiniteSetMaps
ambient_monoid = FiniteSetMaps(self.W, action="right")
pi = self.W.simple_projections(
length_increasing=True).map(ambient_monoid)
category = Monoids().JTrivial().Finite() & Monoids().Transformation(
).Subobjects()
AutomaticMonoid.__init__(self,
pi,
ambient_monoid,
one=ambient_monoid.one(),
mul=operator.mul,
category=category)
self.domain = self.W
self.codomain = self.W
def __init__(self, n, action="right"):
ambient_monoid = FiniteSetMaps(range(1, n + 1), action=action)
def pii(i):
return ambient_monoid.from_dict(
{j: j - 1 if j == i + 1 else j
for j in range(1, n + 1)})
pi = Family(range(1, n), pii)
category = Monoids().JTrivial().Finite() & Monoids().Transformation(
).Subobjects()
AutomaticMonoid.__init__(self,
pi,
ambient_monoid,
one=ambient_monoid.one(),
mul=operator.mul,
category=category)
def __init__(self, poset):
self.poset = poset
support = poset.list()
ambient_monoid = FiniteSetMaps(support, action="right")
def genij(ij):
(i, j) = ij
return ambient_monoid.from_dict(
{k: i if k == j else k
for k in support})
index = map(tuple,
poset.cover_relations()) # index elems must be hashable
self.pi = Family(index, genij)
category = Monoids().JTrivial().Finite() & Monoids().Transformation(
).Subobjects()
AutomaticMonoid.__init__(self,
self.pi,
ambient_monoid,
one=ambient_monoid.one(),
mul=operator.mul,
category=category)
def __init__(self, n = 3, action="left"):
r"""
EXAMPLES::
sage: from sage_semigroups.categories.examples.finite_h_trivial_monoids import MonoidOfOrderPreservingMaps
sage: p = MonoidOfOrderPreservingMaps(4)
sage: TestSuite(p).run()
"""
assert n in NN
domain = IntegerRange(Integer(1),Integer(n+1))
index_set = IntegerRange(Integer(1),Integer(n))
ambient_monoid = FiniteSetMaps(domain, action=action)
def pi(i):
return ambient_monoid.from_dict(dict([(k, k) for k in domain if k != i+1]+[(i+1,i)]))
def opi(i):
return ambient_monoid.from_dict(dict([(k, k) for k in domain if k != i ]+[(i,i+1)]))
piopi = Family(dict([ [ i, pi(i) ] for i in index_set] +
[ [-i, opi(i) ] for i in index_set]))
AutomaticMonoid.__init__(self, piopi,
ambient_monoid, one=ambient_monoid.one(), mul=operator.mul,
category=Monoids().HTrivial().Finite().FinitelyGenerated() &
Monoids().Transformation().Subobjects()
)