Automatic Semigroups¶
Semigroups defined by generators living in an ambient semigroup and represented by an automaton.
AUTHORS:
- Nicolas M. Thiéry 
- Aladin Virmaux 
- class sage.monoids.automatic_semigroup.AutomaticMonoid(generators, ambient, one, mul, category)[source]¶
- Bases: - AutomaticSemigroup- gens()[source]¶
- Return the family of monoid generators of - self.- EXAMPLES: - sage: from sage.monoids.automatic_semigroup import AutomaticSemigroup sage: R = IntegerModRing(28) sage: M = R.submonoid(Family({1: R(3), 2: R(5)})) sage: M.monoid_generators() Finite family {1: 3, 2: 5} - >>> from sage.all import * >>> from sage.monoids.automatic_semigroup import AutomaticSemigroup >>> R = IntegerModRing(Integer(28)) >>> M = R.submonoid(Family({Integer(1): R(Integer(3)), Integer(2): R(Integer(5))})) >>> M.monoid_generators() Finite family {1: 3, 2: 5} - Note that the monoid generators do not include the unit, unlike the semigroup generators: - sage: M.semigroup_generators() Family (1, 3, 5) - >>> from sage.all import * >>> M.semigroup_generators() Family (1, 3, 5) 
 - monoid_generators()[source]¶
- Return the family of monoid generators of - self.- EXAMPLES: - sage: from sage.monoids.automatic_semigroup import AutomaticSemigroup sage: R = IntegerModRing(28) sage: M = R.submonoid(Family({1: R(3), 2: R(5)})) sage: M.monoid_generators() Finite family {1: 3, 2: 5} - >>> from sage.all import * >>> from sage.monoids.automatic_semigroup import AutomaticSemigroup >>> R = IntegerModRing(Integer(28)) >>> M = R.submonoid(Family({Integer(1): R(Integer(3)), Integer(2): R(Integer(5))})) >>> M.monoid_generators() Finite family {1: 3, 2: 5} - Note that the monoid generators do not include the unit, unlike the semigroup generators: - sage: M.semigroup_generators() Family (1, 3, 5) - >>> from sage.all import * >>> M.semigroup_generators() Family (1, 3, 5) 
 - one()[source]¶
- Return the unit of - self.- EXAMPLES: - sage: from sage.monoids.automatic_semigroup import AutomaticSemigroup sage: R = IntegerModRing(21) sage: M = R.submonoid(()) sage: M.one() 1 sage: M.one().parent() is M True - >>> from sage.all import * >>> from sage.monoids.automatic_semigroup import AutomaticSemigroup >>> R = IntegerModRing(Integer(21)) >>> M = R.submonoid(()) >>> M.one() 1 >>> M.one().parent() is M True 
 - semigroup_generators()[source]¶
- Return the generators of - selfas a semigroup.- The generators of a monoid \(M\) as a semigroup are the generators of \(M\) as a monoid and the unit. - EXAMPLES: - sage: M = Monoids().free([1,2,3]) # needs sage.combinat sage: M.semigroup_generators() # needs sage.combinat Family (1, F[1], F[2], F[3]) - >>> from sage.all import * >>> M = Monoids().free([Integer(1),Integer(2),Integer(3)]) # needs sage.combinat >>> M.semigroup_generators() # needs sage.combinat Family (1, F[1], F[2], F[3]) 
 
- class sage.monoids.automatic_semigroup.AutomaticSemigroup(generators, ambient, one, mul, category)[source]¶
- Bases: - UniqueRepresentation,- Parent- Semigroups defined by generators living in an ambient semigroup. - This implementation lazily constructs all the elements of the semigroup, and the right Cayley graph relations between them, and uses the latter as an automaton. - EXAMPLES: - sage: from sage.monoids.automatic_semigroup import AutomaticSemigroup sage: R = IntegerModRing(12) sage: M = AutomaticSemigroup(Family({1: R(3), 2: R(5)}), one=R.one()) sage: M in Monoids() True sage: M.one() 1 sage: M.one() in M True sage: g = M._generators; g Finite family {1: 3, 2: 5} sage: g[1]*g[2] 3 sage: M.some_elements() [1, 3, 5, 9] sage: M.list() [1, 3, 5, 9] sage: M.idempotents() [1, 9] - >>> from sage.all import * >>> from sage.monoids.automatic_semigroup import AutomaticSemigroup >>> R = IntegerModRing(Integer(12)) >>> M = AutomaticSemigroup(Family({Integer(1): R(Integer(3)), Integer(2): R(Integer(5))}), one=R.one()) >>> M in Monoids() True >>> M.one() 1 >>> M.one() in M True >>> g = M._generators; g Finite family {1: 3, 2: 5} >>> g[Integer(1)]*g[Integer(2)] 3 >>> M.some_elements() [1, 3, 5, 9] >>> M.list() [1, 3, 5, 9] >>> M.idempotents() [1, 9] - As can be seen above, elements are represented by default the corresponding element in the ambient monoid. One can also represent the elements by their reduced word: - sage: M.repr_element_method("reduced_word") sage: M.list() [[], [1], [2], [1, 1]] - >>> from sage.all import * >>> M.repr_element_method("reduced_word") >>> M.list() [[], [1], [2], [1, 1]] - In case the reduced word has not yet been calculated, the element will be represented by the corresponding element in the ambient monoid: - sage: R = IntegerModRing(13) sage: N = AutomaticSemigroup(Family({1: R(3), 2: R(5)}), one=R.one()) sage: N.repr_element_method("reduced_word") sage: n = N.an_element() sage: n [1] sage: n*n 9 - >>> from sage.all import * >>> R = IntegerModRing(Integer(13)) >>> N = AutomaticSemigroup(Family({Integer(1): R(Integer(3)), Integer(2): R(Integer(5))}), one=R.one()) >>> N.repr_element_method("reduced_word") >>> n = N.an_element() >>> n [1] >>> n*n 9 - Calling - construct(),- cardinality(), or- list(), or iterating through the monoid will trigger its full construction and, as a side effect, compute all the reduced words. The order of the elements, and the induced choice of reduced word is currently length-lexicographic (i.e. the chosen reduced word is of minimal length, and then minimal lexicographically w.r.t. the order of the indices of the generators):- sage: M.cardinality() 4 sage: M.list() [[], [1], [2], [1, 1]] sage: g = M._generators sage: g[1]*g[2] [1] sage: g[1].transition(1) [1, 1] sage: g[1] * g[1] [1, 1] sage: g[1] * g[1] * g[1] [1] sage: g[1].transition(2) [1] sage: g[1] * g[2] [1] sage: [ x.lift() for x in M.list() ] [1, 3, 5, 9] sage: # needs sage.graphs sage: G = M.cayley_graph(side='twosided'); G Looped multi-digraph on 4 vertices sage: G.edges(sort=True, key=str) [([1, 1], [1, 1], (2, 'left')), ([1, 1], [1, 1], (2, 'right')), ([1, 1], [1], (1, 'left')), ([1, 1], [1], (1, 'right')), ([1], [1, 1], (1, 'left')), ([1], [1, 1], (1, 'right')), ([1], [1], (2, 'left')), ([1], [1], (2, 'right')), ([2], [1], (1, 'left')), ([2], [1], (1, 'right')), ([2], [], (2, 'left')), ([2], [], (2, 'right')), ([], [1], (1, 'left')), ([], [1], (1, 'right')), ([], [2], (2, 'left')), ([], [2], (2, 'right'))] sage: list(map(sorted, M.j_classes())) [[[1], [1, 1]], [[], [2]]] sage: M.j_classes_of_idempotents() [[[1, 1]], [[]]] sage: M.j_transversal_of_idempotents() [[1, 1], []] sage: list(map(attrcall('pseudo_order'), M.list())) # needs sage.graphs [[1, 0], [3, 1], [2, 0], [2, 1]] - >>> from sage.all import * >>> M.cardinality() 4 >>> M.list() [[], [1], [2], [1, 1]] >>> g = M._generators >>> g[Integer(1)]*g[Integer(2)] [1] >>> g[Integer(1)].transition(Integer(1)) [1, 1] >>> g[Integer(1)] * g[Integer(1)] [1, 1] >>> g[Integer(1)] * g[Integer(1)] * g[Integer(1)] [1] >>> g[Integer(1)].transition(Integer(2)) [1] >>> g[Integer(1)] * g[Integer(2)] [1] >>> [ x.lift() for x in M.list() ] [1, 3, 5, 9] >>> # needs sage.graphs >>> G = M.cayley_graph(side='twosided'); G Looped multi-digraph on 4 vertices >>> G.edges(sort=True, key=str) [([1, 1], [1, 1], (2, 'left')), ([1, 1], [1, 1], (2, 'right')), ([1, 1], [1], (1, 'left')), ([1, 1], [1], (1, 'right')), ([1], [1, 1], (1, 'left')), ([1], [1, 1], (1, 'right')), ([1], [1], (2, 'left')), ([1], [1], (2, 'right')), ([2], [1], (1, 'left')), ([2], [1], (1, 'right')), ([2], [], (2, 'left')), ([2], [], (2, 'right')), ([], [1], (1, 'left')), ([], [1], (1, 'right')), ([], [2], (2, 'left')), ([], [2], (2, 'right'))] >>> list(map(sorted, M.j_classes())) [[[1], [1, 1]], [[], [2]]] >>> M.j_classes_of_idempotents() [[[1, 1]], [[]]] >>> M.j_transversal_of_idempotents() [[1, 1], []] >>> list(map(attrcall('pseudo_order'), M.list())) # needs sage.graphs [[1, 0], [3, 1], [2, 0], [2, 1]] - We can also use it to get submonoids from groups. We check that in the symmetric group, a transposition and a long cycle generate the whole group: - sage: # needs sage.groups sage: G5 = SymmetricGroup(5) sage: N = AutomaticSemigroup(Family({1: G5([2,1,3,4,5]), 2: G5([2,3,4,5,1])}), ....: one=G5.one()) sage: N.repr_element_method("reduced_word") sage: N.cardinality() == G5.cardinality() True sage: N.retract(G5((1,4,3,5,2))) [1, 2, 1, 2, 2, 1, 2, 1, 2, 2] sage: N.from_reduced_word([1, 2, 1, 2, 2, 1, 2, 1, 2, 2]).lift() (1,4,3,5,2) - >>> from sage.all import * >>> # needs sage.groups >>> G5 = SymmetricGroup(Integer(5)) >>> N = AutomaticSemigroup(Family({Integer(1): G5([Integer(2),Integer(1),Integer(3),Integer(4),Integer(5)]), Integer(2): G5([Integer(2),Integer(3),Integer(4),Integer(5),Integer(1)])}), ... one=G5.one()) >>> N.repr_element_method("reduced_word") >>> N.cardinality() == G5.cardinality() True >>> N.retract(G5((Integer(1),Integer(4),Integer(3),Integer(5),Integer(2)))) [1, 2, 1, 2, 2, 1, 2, 1, 2, 2] >>> N.from_reduced_word([Integer(1), Integer(2), Integer(1), Integer(2), Integer(2), Integer(1), Integer(2), Integer(1), Integer(2), Integer(2)]).lift() (1,4,3,5,2) - We can also create a semigroup of matrices, where we define the multiplication as matrix multiplication: - sage: # needs sage.modules sage: M1 = matrix([[0,0,1],[1,0,0],[0,1,0]]) sage: M2 = matrix([[0,0,0],[1,1,0],[0,0,1]]) sage: M1.set_immutable() sage: M2.set_immutable() sage: def prod_m(x, y): ....: z=x*y ....: z.set_immutable() ....: return z sage: Mon = AutomaticSemigroup([M1,M2], mul=prod_m, ....: category=Monoids().Finite().Subobjects()) sage: Mon.cardinality() 24 sage: C = Mon.cayley_graph() # needs sage.graphs sage: C.is_directed_acyclic() # needs sage.graphs False - >>> from sage.all import * >>> # needs sage.modules >>> M1 = matrix([[Integer(0),Integer(0),Integer(1)],[Integer(1),Integer(0),Integer(0)],[Integer(0),Integer(1),Integer(0)]]) >>> M2 = matrix([[Integer(0),Integer(0),Integer(0)],[Integer(1),Integer(1),Integer(0)],[Integer(0),Integer(0),Integer(1)]]) >>> M1.set_immutable() >>> M2.set_immutable() >>> def prod_m(x, y): ... z=x*y ... z.set_immutable() ... return z >>> Mon = AutomaticSemigroup([M1,M2], mul=prod_m, ... category=Monoids().Finite().Subobjects()) >>> Mon.cardinality() 24 >>> C = Mon.cayley_graph() # needs sage.graphs >>> C.is_directed_acyclic() # needs sage.graphs False - Let us construct and play with the 0-Hecke Monoid: - sage: # needs sage.graphs sage.modules sage: W = WeylGroup(['A',4]); W.rename('W') sage: ambient_monoid = FiniteSetMaps(W, action='right') sage: pi = W.simple_projections(length_increasing=True).map(ambient_monoid) sage: M = AutomaticSemigroup(pi, one=ambient_monoid.one()); M A submonoid of (Maps from W to itself) with 4 generators sage: M.repr_element_method("reduced_word") sage: sorted(M._elements_set, key=str) [[1], [2], [3], [4], []] sage: M.construct(n=10) sage: sorted(M._elements_set, key=str) [[1, 2], [1, 3], [1, 4], [1], [2, 1], [2, 3], [2], [3], [4], []] sage: elt = M.from_reduced_word([3,1,2,4,2]) sage: M.construct(up_to=elt) sage: len(M._elements_set) 36 sage: M.cardinality() 120 - >>> from sage.all import * >>> # needs sage.graphs sage.modules >>> W = WeylGroup(['A',Integer(4)]); W.rename('W') >>> ambient_monoid = FiniteSetMaps(W, action='right') >>> pi = W.simple_projections(length_increasing=True).map(ambient_monoid) >>> M = AutomaticSemigroup(pi, one=ambient_monoid.one()); M A submonoid of (Maps from W to itself) with 4 generators >>> M.repr_element_method("reduced_word") >>> sorted(M._elements_set, key=str) [[1], [2], [3], [4], []] >>> M.construct(n=Integer(10)) >>> sorted(M._elements_set, key=str) [[1, 2], [1, 3], [1, 4], [1], [2, 1], [2, 3], [2], [3], [4], []] >>> elt = M.from_reduced_word([Integer(3),Integer(1),Integer(2),Integer(4),Integer(2)]) >>> M.construct(up_to=elt) >>> len(M._elements_set) 36 >>> M.cardinality() 120 - We check that the 0-Hecke monoid is \(J\)-trivial and contains \(2^4\) idempotents: - sage: len(M.idempotents()) # needs sage.graphs sage.modules 16 sage: all(len(j) == 1 for j in M.j_classes()) # needs sage.graphs sage.modules True - >>> from sage.all import * >>> len(M.idempotents()) # needs sage.graphs sage.modules 16 >>> all(len(j) == Integer(1) for j in M.j_classes()) # needs sage.graphs sage.modules True - Note - Unlike what the name of the class may suggest, this currently implements only a subclass of automatic semigroups; essentially the finite ones. See Wikipedia article Automatic_semigroup. - Warning - AutomaticSemigroupis designed primarily for finite semigroups. This property is not checked automatically (this would be too costly, if not undecidable). Use with care for an infinite semigroup, as certain features may require constructing all of it:- sage: M = AutomaticSemigroup([2], category = Monoids().Subobjects()); M A submonoid of (Integer Ring) with 1 generators sage: M.retract(2) 2 sage: M.retract(3) # not tested: runs forever trying to find 3 - >>> from sage.all import * >>> M = AutomaticSemigroup([Integer(2)], category = Monoids().Subobjects()); M A submonoid of (Integer Ring) with 1 generators >>> M.retract(Integer(2)) 2 >>> M.retract(Integer(3)) # not tested: runs forever trying to find 3 - class Element(ambient_element, parent)[source]¶
- Bases: - ElementWrapper- lift()[source]¶
- Lift the element - selfinto its ambient semigroup.- EXAMPLES: - sage: from sage.monoids.automatic_semigroup import AutomaticSemigroup sage: R = IntegerModRing(18) sage: M = AutomaticSemigroup(Family({1: R(3), 2: R(5)})) sage: M.repr_element_method("reduced_word") sage: m = M.an_element(); m [1] sage: type(m) <class 'sage.monoids.automatic_semigroup.AutomaticSemigroup_with_category.element_class'> sage: m.lift() 3 sage: type(m.lift()) <class 'sage.rings.finite_rings.integer_mod.IntegerMod_int'> - >>> from sage.all import * >>> from sage.monoids.automatic_semigroup import AutomaticSemigroup >>> R = IntegerModRing(Integer(18)) >>> M = AutomaticSemigroup(Family({Integer(1): R(Integer(3)), Integer(2): R(Integer(5))})) >>> M.repr_element_method("reduced_word") >>> m = M.an_element(); m [1] >>> type(m) <class 'sage.monoids.automatic_semigroup.AutomaticSemigroup_with_category.element_class'> >>> m.lift() 3 >>> type(m.lift()) <class 'sage.rings.finite_rings.integer_mod.IntegerMod_int'> 
 - reduced_word()[source]¶
- Return the length-lexicographic shortest word of - self.- OUTPUT: list of indexes of the generators - Obtaining the reduced word requires having constructed the Cayley graph of the semigroup up to - self. If this is not the case, an error is raised.- EXAMPLES: - sage: from sage.monoids.automatic_semigroup import AutomaticSemigroup sage: R = IntegerModRing(15) sage: M = AutomaticSemigroup(Family({1: R(3), 2: R(5)}), one=R.one()) sage: M.construct() sage: for m in M: print((m, m.reduced_word())) (1, []) (3, [1]) (5, [2]) (9, [1, 1]) (0, [1, 2]) (10, [2, 2]) (12, [1, 1, 1]) (6, [1, 1, 1, 1]) - >>> from sage.all import * >>> from sage.monoids.automatic_semigroup import AutomaticSemigroup >>> R = IntegerModRing(Integer(15)) >>> M = AutomaticSemigroup(Family({Integer(1): R(Integer(3)), Integer(2): R(Integer(5))}), one=R.one()) >>> M.construct() >>> for m in M: print((m, m.reduced_word())) (1, []) (3, [1]) (5, [2]) (9, [1, 1]) (0, [1, 2]) (10, [2, 2]) (12, [1, 1, 1]) (6, [1, 1, 1, 1]) 
 - transition(i)[source]¶
- The multiplication on the right by a generator. - INPUT: - i– an element from the indexing set of the generators
 - This method computes - self * self._generators[i].- EXAMPLES: - sage: from sage.monoids.automatic_semigroup import AutomaticSemigroup sage: R = IntegerModRing(17) sage: M = AutomaticSemigroup(Family({1: R(3), 2: R(5)}), one=R.one()) sage: M.repr_element_method("reduced_word") sage: M.construct() sage: a = M.an_element() sage: a.transition(1) [1, 1] sage: a.transition(2) [1, 2] - >>> from sage.all import * >>> from sage.monoids.automatic_semigroup import AutomaticSemigroup >>> R = IntegerModRing(Integer(17)) >>> M = AutomaticSemigroup(Family({Integer(1): R(Integer(3)), Integer(2): R(Integer(5))}), one=R.one()) >>> M.repr_element_method("reduced_word") >>> M.construct() >>> a = M.an_element() >>> a.transition(Integer(1)) [1, 1] >>> a.transition(Integer(2)) [1, 2] 
 
 - ambient()[source]¶
- Return the ambient semigroup of - self.- EXAMPLES: - sage: from sage.monoids.automatic_semigroup import AutomaticSemigroup sage: R = IntegerModRing(12) sage: M = AutomaticSemigroup(Family({1: R(3), 2: R(5)}), one=R.one()) sage: M.ambient() Ring of integers modulo 12 sage: # needs sage.modules sage: M1 = matrix([[0,0,1],[1,0,0],[0,1,0]]) sage: M2 = matrix([[0,0,0],[1,1,0],[0,0,1]]) sage: M1.set_immutable() sage: M2.set_immutable() sage: def prod_m(x, y): ....: z=x*y ....: z.set_immutable() ....: return z sage: Mon = AutomaticSemigroup([M1,M2], mul=prod_m) sage: Mon.ambient() Full MatrixSpace of 3 by 3 dense matrices over Integer Ring - >>> from sage.all import * >>> from sage.monoids.automatic_semigroup import AutomaticSemigroup >>> R = IntegerModRing(Integer(12)) >>> M = AutomaticSemigroup(Family({Integer(1): R(Integer(3)), Integer(2): R(Integer(5))}), one=R.one()) >>> M.ambient() Ring of integers modulo 12 >>> # needs sage.modules >>> M1 = matrix([[Integer(0),Integer(0),Integer(1)],[Integer(1),Integer(0),Integer(0)],[Integer(0),Integer(1),Integer(0)]]) >>> M2 = matrix([[Integer(0),Integer(0),Integer(0)],[Integer(1),Integer(1),Integer(0)],[Integer(0),Integer(0),Integer(1)]]) >>> M1.set_immutable() >>> M2.set_immutable() >>> def prod_m(x, y): ... z=x*y ... z.set_immutable() ... return z >>> Mon = AutomaticSemigroup([M1,M2], mul=prod_m) >>> Mon.ambient() Full MatrixSpace of 3 by 3 dense matrices over Integer Ring 
 - an_element()[source]¶
- Return the first given generator of - self.- EXAMPLES: - sage: from sage.monoids.automatic_semigroup import AutomaticSemigroup sage: R = IntegerModRing(16) sage: M = AutomaticSemigroup(Family({1: R(3), 2: R(5)}), one=R.one()) sage: M.an_element() 3 - >>> from sage.all import * >>> from sage.monoids.automatic_semigroup import AutomaticSemigroup >>> R = IntegerModRing(Integer(16)) >>> M = AutomaticSemigroup(Family({Integer(1): R(Integer(3)), Integer(2): R(Integer(5))}), one=R.one()) >>> M.an_element() 3 
 - cardinality()[source]¶
- Return the cardinality of - self.- EXAMPLES: - sage: from sage.monoids.automatic_semigroup import AutomaticSemigroup sage: R = IntegerModRing(12) sage: M = AutomaticSemigroup(Family({1: R(3), 2: R(5)}), one=R.one()) sage: M.cardinality() 4 - >>> from sage.all import * >>> from sage.monoids.automatic_semigroup import AutomaticSemigroup >>> R = IntegerModRing(Integer(12)) >>> M = AutomaticSemigroup(Family({Integer(1): R(Integer(3)), Integer(2): R(Integer(5))}), one=R.one()) >>> M.cardinality() 4 
 - construct(up_to=None, n=None)[source]¶
- Construct the elements of - self.- INPUT: - up_to– an element of- selfor of the ambient semigroup
- n– integer or- None(default:- None)
 - This construct all the elements of this semigroup, their reduced words, and the right Cayley graph. If \(n\) is specified, only the \(n\) first elements of the semigroup are constructed. If - elementis specified, only the elements up to- ambient_elementare constructed.- EXAMPLES: - sage: # needs sage.groups sage.modules sage: from sage.monoids.automatic_semigroup import AutomaticSemigroup sage: W = WeylGroup(['A',3]); W.rename('W') sage: ambient_monoid = FiniteSetMaps(W, action='right') sage: pi = W.simple_projections(length_increasing=True).map(ambient_monoid) sage: M = AutomaticSemigroup(pi, one=ambient_monoid.one()); M A submonoid of (Maps from W to itself) with 3 generators sage: M.repr_element_method("reduced_word") sage: sorted(M._elements_set, key=str) [[1], [2], [3], []] sage: elt = M.from_reduced_word([2,3,1,2]) sage: M.construct(up_to=elt) sage: len(M._elements_set) 19 sage: M.cardinality() 24 - >>> from sage.all import * >>> # needs sage.groups sage.modules >>> from sage.monoids.automatic_semigroup import AutomaticSemigroup >>> W = WeylGroup(['A',Integer(3)]); W.rename('W') >>> ambient_monoid = FiniteSetMaps(W, action='right') >>> pi = W.simple_projections(length_increasing=True).map(ambient_monoid) >>> M = AutomaticSemigroup(pi, one=ambient_monoid.one()); M A submonoid of (Maps from W to itself) with 3 generators >>> M.repr_element_method("reduced_word") >>> sorted(M._elements_set, key=str) [[1], [2], [3], []] >>> elt = M.from_reduced_word([Integer(2),Integer(3),Integer(1),Integer(2)]) >>> M.construct(up_to=elt) >>> len(M._elements_set) 19 >>> M.cardinality() 24 
 - from_reduced_word(l)[source]¶
- Return the element of - selfobtained from the reduced word- l.- INPUT: - l– list of indices of the generators
 - Note - We do not save the given reduced word - las an attribute of the element, as some elements above in the branches may have not been explored by the iterator yet.- EXAMPLES: - sage: # needs sage.groups sage: from sage.monoids.automatic_semigroup import AutomaticSemigroup sage: G4 = SymmetricGroup(4) sage: M = AutomaticSemigroup(Family({1:G4((1,2)), 2:G4((1,2,3,4))}), ....: one=G4.one()) sage: M.from_reduced_word([2, 1, 2, 2, 1]).lift() (1,3) sage: M.from_reduced_word([2, 1, 2, 2, 1]) == M.retract(G4((3,1))) True - >>> from sage.all import * >>> # needs sage.groups >>> from sage.monoids.automatic_semigroup import AutomaticSemigroup >>> G4 = SymmetricGroup(Integer(4)) >>> M = AutomaticSemigroup(Family({Integer(1):G4((Integer(1),Integer(2))), Integer(2):G4((Integer(1),Integer(2),Integer(3),Integer(4)))}), ... one=G4.one()) >>> M.from_reduced_word([Integer(2), Integer(1), Integer(2), Integer(2), Integer(1)]).lift() (1,3) >>> M.from_reduced_word([Integer(2), Integer(1), Integer(2), Integer(2), Integer(1)]) == M.retract(G4((Integer(3),Integer(1)))) True 
 - gens()[source]¶
- Return the family of generators of - self.- EXAMPLES: - sage: from sage.monoids.automatic_semigroup import AutomaticSemigroup sage: R = IntegerModRing(28) sage: M = AutomaticSemigroup(Family({1: R(3), 2: R(5)})) sage: M.semigroup_generators() Finite family {1: 3, 2: 5} - >>> from sage.all import * >>> from sage.monoids.automatic_semigroup import AutomaticSemigroup >>> R = IntegerModRing(Integer(28)) >>> M = AutomaticSemigroup(Family({Integer(1): R(Integer(3)), Integer(2): R(Integer(5))})) >>> M.semigroup_generators() Finite family {1: 3, 2: 5} 
 - lift(x)[source]¶
- Lift an element of - selfinto its ambient space.- EXAMPLES: - sage: from sage.monoids.automatic_semigroup import AutomaticSemigroup sage: R = IntegerModRing(15) sage: M = AutomaticSemigroup(Family({1: R(3), 2: R(5)}), one=R.one()) sage: a = M.an_element() sage: a.lift() in R True sage: a.lift() 3 sage: [m.lift() for m in M] [1, 3, 5, 9, 0, 10, 12, 6] - >>> from sage.all import * >>> from sage.monoids.automatic_semigroup import AutomaticSemigroup >>> R = IntegerModRing(Integer(15)) >>> M = AutomaticSemigroup(Family({Integer(1): R(Integer(3)), Integer(2): R(Integer(5))}), one=R.one()) >>> a = M.an_element() >>> a.lift() in R True >>> a.lift() 3 >>> [m.lift() for m in M] [1, 3, 5, 9, 0, 10, 12, 6] 
 - list()[source]¶
- Return the list of elements of - self.- EXAMPLES: - sage: from sage.monoids.automatic_semigroup import AutomaticSemigroup sage: R = IntegerModRing(12) sage: M = AutomaticSemigroup(Family({1: R(3), 2: R(5)}), one=R.one()) sage: M.repr_element_method("reduced_word") sage: M.list() [[], [1], [2], [1, 1]] - >>> from sage.all import * >>> from sage.monoids.automatic_semigroup import AutomaticSemigroup >>> R = IntegerModRing(Integer(12)) >>> M = AutomaticSemigroup(Family({Integer(1): R(Integer(3)), Integer(2): R(Integer(5))}), one=R.one()) >>> M.repr_element_method("reduced_word") >>> M.list() [[], [1], [2], [1, 1]] 
 - product(x, y)[source]¶
- Return the product of two elements in - self. It is done by retracting the multiplication in the ambient semigroup.- EXAMPLES: - sage: from sage.monoids.automatic_semigroup import AutomaticSemigroup sage: R = IntegerModRing(12) sage: M = AutomaticSemigroup(Family({1: R(3), 2: R(5)}), one=R.one()) sage: a = M[1] sage: b = M[2] sage: a*b [1] - >>> from sage.all import * >>> from sage.monoids.automatic_semigroup import AutomaticSemigroup >>> R = IntegerModRing(Integer(12)) >>> M = AutomaticSemigroup(Family({Integer(1): R(Integer(3)), Integer(2): R(Integer(5))}), one=R.one()) >>> a = M[Integer(1)] >>> b = M[Integer(2)] >>> a*b [1] 
 - repr_element_method(style='ambient')[source]¶
- Set the representation of the elements of the monoid. - INPUT: - style– “ambient” or “reduced_word”
 - EXAMPLES: - sage: from sage.monoids.automatic_semigroup import AutomaticSemigroup sage: R = IntegerModRing(17) sage: M = AutomaticSemigroup(Family({1: R(3), 2: R(5)}), one=R.one()) sage: M.list() [1, 3, 5, 9, 15, 8, 10, 11, 7, 6, 13, 16, 4, 14, 12, 2] sage: M.repr_element_method("reduced_word") sage: M.list() [[], [1], [2], [1, 1], [1, 2], [2, 2], [1, 1, 1], [1, 1, 2], [1, 2, 2], [2, 2, 2], [1, 1, 1, 1], [1, 1, 1, 2], [1, 1, 2, 2], [1, 1, 1, 1, 2], [1, 1, 1, 2, 2], [1, 1, 1, 1, 2, 2]] - >>> from sage.all import * >>> from sage.monoids.automatic_semigroup import AutomaticSemigroup >>> R = IntegerModRing(Integer(17)) >>> M = AutomaticSemigroup(Family({Integer(1): R(Integer(3)), Integer(2): R(Integer(5))}), one=R.one()) >>> M.list() [1, 3, 5, 9, 15, 8, 10, 11, 7, 6, 13, 16, 4, 14, 12, 2] >>> M.repr_element_method("reduced_word") >>> M.list() [[], [1], [2], [1, 1], [1, 2], [2, 2], [1, 1, 1], [1, 1, 2], [1, 2, 2], [2, 2, 2], [1, 1, 1, 1], [1, 1, 1, 2], [1, 1, 2, 2], [1, 1, 1, 1, 2], [1, 1, 1, 2, 2], [1, 1, 1, 1, 2, 2]] 
 - retract(ambient_element, check=True)[source]¶
- Retract an element of the ambient semigroup into - self.- EXAMPLES: - sage: # needs sage.groups sage: from sage.monoids.automatic_semigroup import AutomaticSemigroup sage: S5 = SymmetricGroup(5); S5.rename('S5') sage: M = AutomaticSemigroup(Family({1:S5((1,2)), 2:S5((1,2,3,4))}), ....: one=S5.one()) sage: m = M.retract(S5((3,1))); m (1,3) sage: m.parent() is M True sage: M.retract(S5((4,5)), check=False) (4,5) sage: M.retract(S5((4,5))) Traceback (most recent call last): ... ValueError: (4,5) not in A subgroup of (S5) with 2 generators - >>> from sage.all import * >>> # needs sage.groups >>> from sage.monoids.automatic_semigroup import AutomaticSemigroup >>> S5 = SymmetricGroup(Integer(5)); S5.rename('S5') >>> M = AutomaticSemigroup(Family({Integer(1):S5((Integer(1),Integer(2))), Integer(2):S5((Integer(1),Integer(2),Integer(3),Integer(4)))}), ... one=S5.one()) >>> m = M.retract(S5((Integer(3),Integer(1)))); m (1,3) >>> m.parent() is M True >>> M.retract(S5((Integer(4),Integer(5))), check=False) (4,5) >>> M.retract(S5((Integer(4),Integer(5)))) Traceback (most recent call last): ... ValueError: (4,5) not in A subgroup of (S5) with 2 generators 
 - semigroup_generators()[source]¶
- Return the family of generators of - self.- EXAMPLES: - sage: from sage.monoids.automatic_semigroup import AutomaticSemigroup sage: R = IntegerModRing(28) sage: M = AutomaticSemigroup(Family({1: R(3), 2: R(5)})) sage: M.semigroup_generators() Finite family {1: 3, 2: 5} - >>> from sage.all import * >>> from sage.monoids.automatic_semigroup import AutomaticSemigroup >>> R = IntegerModRing(Integer(28)) >>> M = AutomaticSemigroup(Family({Integer(1): R(Integer(3)), Integer(2): R(Integer(5))})) >>> M.semigroup_generators() Finite family {1: 3, 2: 5}