Finite monoids¶
- class sage.categories.finite_monoids.FiniteMonoids(base_category)[source]¶
Bases:
CategoryWithAxiom_singletonThe category of finite (multiplicative)
monoids.A finite monoid is a
finite setsendowed with an associative unital binary operation \(*\).EXAMPLES:
sage: FiniteMonoids() Category of finite monoids sage: FiniteMonoids().super_categories() [Category of monoids, Category of finite semigroups]
>>> from sage.all import * >>> FiniteMonoids() Category of finite monoids >>> FiniteMonoids().super_categories() [Category of monoids, Category of finite semigroups]
FiniteMonoids() FiniteMonoids().super_categories()
- class ElementMethods[source]¶
Bases:
object- pseudo_order()[source]¶
Return the pair \([k, j]\) with \(k\) minimal and \(0\leq j <k\) such that
self^k == self^j.Note that \(j\) is uniquely determined.
EXAMPLES:
sage: M = FiniteMonoids().example(); M An example of a finite multiplicative monoid: the integers modulo 12 sage: x = M(2) sage: [ x^i for i in range(7) ] [1, 2, 4, 8, 4, 8, 4] sage: x.pseudo_order() [4, 2] sage: x = M(3) sage: [ x^i for i in range(7) ] [1, 3, 9, 3, 9, 3, 9] sage: x.pseudo_order() [3, 1] sage: x = M(4) sage: [ x^i for i in range(7) ] [1, 4, 4, 4, 4, 4, 4] sage: x.pseudo_order() [2, 1] sage: x = M(5) sage: [ x^i for i in range(7) ] [1, 5, 1, 5, 1, 5, 1] sage: x.pseudo_order() [2, 0]
>>> from sage.all import * >>> M = FiniteMonoids().example(); M An example of a finite multiplicative monoid: the integers modulo 12 >>> x = M(Integer(2)) >>> [ x**i for i in range(Integer(7)) ] [1, 2, 4, 8, 4, 8, 4] >>> x.pseudo_order() [4, 2] >>> x = M(Integer(3)) >>> [ x**i for i in range(Integer(7)) ] [1, 3, 9, 3, 9, 3, 9] >>> x.pseudo_order() [3, 1] >>> x = M(Integer(4)) >>> [ x**i for i in range(Integer(7)) ] [1, 4, 4, 4, 4, 4, 4] >>> x.pseudo_order() [2, 1] >>> x = M(Integer(5)) >>> [ x**i for i in range(Integer(7)) ] [1, 5, 1, 5, 1, 5, 1] >>> x.pseudo_order() [2, 0]
M = FiniteMonoids().example(); M x = M(2) [ x^i for i in range(7) ] x.pseudo_order() x = M(3) [ x^i for i in range(7) ] x.pseudo_order() x = M(4) [ x^i for i in range(7) ] x.pseudo_order() x = M(5) [ x^i for i in range(7) ] x.pseudo_order()
Todo
more appropriate name? see, for example, Jean-Eric Pin’s lecture notes on semigroups.
- class ParentMethods[source]¶
Bases:
object- nerve()[source]¶
The nerve (classifying space) of this monoid.
OUTPUT:
the nerve \(BG\) (if \(G\) denotes this monoid), as a simplicial set. The \(k\)-dimensional simplices of this object are indexed by products of \(k\) elements in the monoid:
\[a_1 * a_2 * \cdots * a_k\]The \(0\)-th face of this is obtained by deleting \(a_1\), and the \(k\)-th face is obtained by deleting \(a_k\). The other faces are obtained by multiplying elements: the 1st face is
\[(a1 * a_2) * \cdots * a_k\]and so on. See Wikipedia article Nerve_(category_theory), which describes the construction of the nerve as a simplicial set.
A simplex in this simplicial set will be degenerate if in the corresponding product of \(k\) elements, one of those elements is the identity. So we only need to keep track of the products of non-identity elements. Similarly, if a product \(a_{i-1} a_i\) is the identity element, then the corresponding face of the simplex will be a degenerate simplex.
EXAMPLES:
The nerve (classifying space) of the cyclic group of order 2 is infinite-dimensional real projective space.
sage: Sigma2 = groups.permutation.Cyclic(2) # needs sage.groups sage: BSigma2 = Sigma2.nerve() # needs sage.graphs sage.groups sage: BSigma2.cohomology(4, base_ring=GF(2)) # needs sage.graphs sage.groups sage.modules Vector space of dimension 1 over Finite Field of size 2
>>> from sage.all import * >>> Sigma2 = groups.permutation.Cyclic(Integer(2)) # needs sage.groups >>> BSigma2 = Sigma2.nerve() # needs sage.graphs sage.groups >>> BSigma2.cohomology(Integer(4), base_ring=GF(Integer(2))) # needs sage.graphs sage.groups sage.modules Vector space of dimension 1 over Finite Field of size 2
Sigma2 = groups.permutation.Cyclic(2) # needs sage.groups BSigma2 = Sigma2.nerve() # needs sage.graphs sage.groups BSigma2.cohomology(4, base_ring=GF(2)) # needs sage.graphs sage.groups sage.modules
The \(k\)-simplices of the nerve are named after the chains of \(k\) non-unit elements to be multiplied. The group \(\Sigma_2\) has two elements, written
()(the identity element) and(1,2)in Sage. So the 1-cells and 2-cells in \(B\Sigma_2\) are:sage: BSigma2.n_cells(1) # needs sage.graphs sage.groups [(1,2)] sage: BSigma2.n_cells(2) # needs sage.graphs sage.groups [(1,2) * (1,2)]
>>> from sage.all import * >>> BSigma2.n_cells(Integer(1)) # needs sage.graphs sage.groups [(1,2)] >>> BSigma2.n_cells(Integer(2)) # needs sage.graphs sage.groups [(1,2) * (1,2)]
BSigma2.n_cells(1) # needs sage.graphs sage.groups BSigma2.n_cells(2) # needs sage.graphs sage.groups
Another construction of the group, with different names for its elements:
sage: # needs sage.groups sage.rings.number_field sage: C2 = groups.misc.MultiplicativeAbelian([2]) sage: BC2 = C2.nerve() sage: BC2.n_cells(0) [1] sage: BC2.n_cells(1) [f] sage: BC2.n_cells(2) [f * f]
>>> from sage.all import * >>> # needs sage.groups sage.rings.number_field >>> C2 = groups.misc.MultiplicativeAbelian([Integer(2)]) >>> BC2 = C2.nerve() >>> BC2.n_cells(Integer(0)) [1] >>> BC2.n_cells(Integer(1)) [f] >>> BC2.n_cells(Integer(2)) [f * f]
# needs sage.groups sage.rings.number_field C2 = groups.misc.MultiplicativeAbelian([2]) BC2 = C2.nerve() BC2.n_cells(0) BC2.n_cells(1) BC2.n_cells(2)
With mod \(p\) coefficients, \(B \Sigma_p\) should have its first nonvanishing homology group in dimension \(p\):
sage: Sigma3 = groups.permutation.Symmetric(3) # needs sage.groups sage: BSigma3 = Sigma3.nerve() # needs sage.graphs sage.groups sage: BSigma3.homology(range(4), base_ring=GF(3)) # needs sage.graphs sage.groups {0: Vector space of dimension 0 over Finite Field of size 3, 1: Vector space of dimension 0 over Finite Field of size 3, 2: Vector space of dimension 0 over Finite Field of size 3, 3: Vector space of dimension 1 over Finite Field of size 3}
>>> from sage.all import * >>> Sigma3 = groups.permutation.Symmetric(Integer(3)) # needs sage.groups >>> BSigma3 = Sigma3.nerve() # needs sage.graphs sage.groups >>> BSigma3.homology(range(Integer(4)), base_ring=GF(Integer(3))) # needs sage.graphs sage.groups {0: Vector space of dimension 0 over Finite Field of size 3, 1: Vector space of dimension 0 over Finite Field of size 3, 2: Vector space of dimension 0 over Finite Field of size 3, 3: Vector space of dimension 1 over Finite Field of size 3}
Sigma3 = groups.permutation.Symmetric(3) # needs sage.groups BSigma3 = Sigma3.nerve() # needs sage.graphs sage.groups BSigma3.homology(range(4), base_ring=GF(3)) # needs sage.graphs sage.groups
Note that we can construct the \(n\)-skeleton for \(B\Sigma_2\) for relatively large values of \(n\), while for \(B\Sigma_3\), the complexes get large pretty quickly:
sage: # needs sage.graphs sage.groups sage: Sigma2.nerve().n_skeleton(14) Simplicial set with 15 non-degenerate simplices sage: BSigma3 = Sigma3.nerve() sage: BSigma3.n_skeleton(3) Simplicial set with 156 non-degenerate simplices sage: BSigma3.n_skeleton(4) Simplicial set with 781 non-degenerate simplices
>>> from sage.all import * >>> # needs sage.graphs sage.groups >>> Sigma2.nerve().n_skeleton(Integer(14)) Simplicial set with 15 non-degenerate simplices >>> BSigma3 = Sigma3.nerve() >>> BSigma3.n_skeleton(Integer(3)) Simplicial set with 156 non-degenerate simplices >>> BSigma3.n_skeleton(Integer(4)) Simplicial set with 781 non-degenerate simplices
# needs sage.graphs sage.groups Sigma2.nerve().n_skeleton(14) BSigma3 = Sigma3.nerve() BSigma3.n_skeleton(3) BSigma3.n_skeleton(4)
Finally, note that the classifying space of the order \(p\) cyclic group is smaller than that of the symmetric group on \(p\) letters, and its first homology group appears earlier:
sage: # needs sage.graphs sage.groups sage.rings.number_field sage: C3 = groups.misc.MultiplicativeAbelian([3]) sage: list(C3) [1, f, f^2] sage: BC3 = C3.nerve() sage: BC3.n_cells(1) [f, f^2] sage: BC3.n_cells(2) [f * f, f * f^2, f^2 * f, f^2 * f^2] sage: len(BSigma3.n_cells(2)) 25 sage: len(BC3.n_cells(3)) 8 sage: len(BSigma3.n_cells(3)) 125 sage: BC3.homology(range(4), base_ring=GF(3)) {0: Vector space of dimension 0 over Finite Field of size 3, 1: Vector space of dimension 1 over Finite Field of size 3, 2: Vector space of dimension 1 over Finite Field of size 3, 3: Vector space of dimension 1 over Finite Field of size 3} sage: BC5 = groups.permutation.Cyclic(5).nerve() sage: BC5.homology(range(4), base_ring=GF(5)) {0: Vector space of dimension 0 over Finite Field of size 5, 1: Vector space of dimension 1 over Finite Field of size 5, 2: Vector space of dimension 1 over Finite Field of size 5, 3: Vector space of dimension 1 over Finite Field of size 5}
>>> from sage.all import * >>> # needs sage.graphs sage.groups sage.rings.number_field >>> C3 = groups.misc.MultiplicativeAbelian([Integer(3)]) >>> list(C3) [1, f, f^2] >>> BC3 = C3.nerve() >>> BC3.n_cells(Integer(1)) [f, f^2] >>> BC3.n_cells(Integer(2)) [f * f, f * f^2, f^2 * f, f^2 * f^2] >>> len(BSigma3.n_cells(Integer(2))) 25 >>> len(BC3.n_cells(Integer(3))) 8 >>> len(BSigma3.n_cells(Integer(3))) 125 >>> BC3.homology(range(Integer(4)), base_ring=GF(Integer(3))) {0: Vector space of dimension 0 over Finite Field of size 3, 1: Vector space of dimension 1 over Finite Field of size 3, 2: Vector space of dimension 1 over Finite Field of size 3, 3: Vector space of dimension 1 over Finite Field of size 3} >>> BC5 = groups.permutation.Cyclic(Integer(5)).nerve() >>> BC5.homology(range(Integer(4)), base_ring=GF(Integer(5))) {0: Vector space of dimension 0 over Finite Field of size 5, 1: Vector space of dimension 1 over Finite Field of size 5, 2: Vector space of dimension 1 over Finite Field of size 5, 3: Vector space of dimension 1 over Finite Field of size 5}
# needs sage.graphs sage.groups sage.rings.number_field C3 = groups.misc.MultiplicativeAbelian([3]) list(C3) BC3 = C3.nerve() BC3.n_cells(1) BC3.n_cells(2) len(BSigma3.n_cells(2)) len(BC3.n_cells(3)) len(BSigma3.n_cells(3)) BC3.homology(range(4), base_ring=GF(3)) BC5 = groups.permutation.Cyclic(5).nerve() BC5.homology(range(4), base_ring=GF(5))
- rhodes_radical_congruence(base_ring=None)[source]¶
Return the Rhodes radical congruence of the semigroup.
The Rhodes radical congruence is the congruence induced on S by the map \(S \rightarrow kS \rightarrow kS / rad kS\) with k a field.
INPUT:
base_ring– (default: \(\QQ\)) a field
OUTPUT:
A list of couples (m, n) with \(m \neq n\) in the lexicographic order for the enumeration of the monoid
self.
EXAMPLES:
sage: M = Monoids().Finite().example() sage: M.rhodes_radical_congruence() # needs sage.modules [(0, 6), (2, 8), (4, 10)] sage: # needs sage.combinat sage.groups sage.modules sage: from sage.monoids.hecke_monoid import HeckeMonoid sage: H3 = HeckeMonoid(SymmetricGroup(3)) sage: H3.repr_element_method(style='reduced') sage: H3.rhodes_radical_congruence() [([1, 2], [2, 1]), ([1, 2], [1, 2, 1]), ([2, 1], [1, 2, 1])]
>>> from sage.all import * >>> M = Monoids().Finite().example() >>> M.rhodes_radical_congruence() # needs sage.modules [(0, 6), (2, 8), (4, 10)] >>> # needs sage.combinat sage.groups sage.modules >>> from sage.monoids.hecke_monoid import HeckeMonoid >>> H3 = HeckeMonoid(SymmetricGroup(Integer(3))) >>> H3.repr_element_method(style='reduced') >>> H3.rhodes_radical_congruence() [([1, 2], [2, 1]), ([1, 2], [1, 2, 1]), ([2, 1], [1, 2, 1])]
M = Monoids().Finite().example() M.rhodes_radical_congruence() # needs sage.modules # needs sage.combinat sage.groups sage.modules from sage.monoids.hecke_monoid import HeckeMonoid H3 = HeckeMonoid(SymmetricGroup(3)) H3.repr_element_method(style='reduced') H3.rhodes_radical_congruence()
By Maschke’s theorem, every group algebra over \(\QQ\) is semisimple hence the Rhodes radical of a group must be trivial:
sage: SymmetricGroup(3).rhodes_radical_congruence() # needs sage.combinat sage.groups sage.modules [] sage: DihedralGroup(10).rhodes_radical_congruence() # needs sage.groups sage.modules []
>>> from sage.all import * >>> SymmetricGroup(Integer(3)).rhodes_radical_congruence() # needs sage.combinat sage.groups sage.modules [] >>> DihedralGroup(Integer(10)).rhodes_radical_congruence() # needs sage.groups sage.modules []
SymmetricGroup(3).rhodes_radical_congruence() # needs sage.combinat sage.groups sage.modules DihedralGroup(10).rhodes_radical_congruence() # needs sage.groups sage.modules
REFERENCES: