Finite Monoids

class sage.categories.finite_monoids.FiniteMonoids(base_category)

Bases: sage.categories.category_with_axiom.CategoryWithAxiom_singleton

The category of finite (multiplicative) monoids.

A finite monoid is a finite sets endowed with an associative unital binary operation \(*\).

EXAMPLES:

sage: FiniteMonoids()
Category of finite monoids
sage: FiniteMonoids().super_categories()
[Category of monoids, Category of finite semigroups]
class ElementMethods
pseudo_order()

Returns 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]

TODO: more appropriate name? see, for example, Jean-Eric Pin’s lecture notes on semigroups.

class ParentMethods
nerve()

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 0th 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)
sage: BSigma2 = Sigma2.nerve()
sage: BSigma2.cohomology(4, base_ring=GF(2))
Vector space of dimension 1 over Finite Field of size 2

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)
[(1,2)]
sage: BSigma2.n_cells(2)
[(1,2) * (1,2)]

Another construction of the group, with different names for its elements:

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]

With mod \(p\) coefficients, \(B \Sigma_p\) should have its first nonvanishing homology group in dimension \(p\):

sage: Sigma3 = groups.permutation.Symmetric(3)
sage: BSigma3 = Sigma3.nerve()
sage: BSigma3.homology(range(4), base_ring=GF(3))
{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}

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: 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

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: 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(5), 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,
 4: Vector space of dimension 1 over Finite Field of size 3}

sage: BC5 = groups.permutation.Cyclic(5).nerve()
sage: BC5.homology(range(5), 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,
4: Vector space of dimension 1 over Finite Field of size 5}
rhodes_radical_congruence(base_ring=None)

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()
[(0, 6), (2, 8), (4, 10)]
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])]

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()
[]
sage: DihedralGroup(10).rhodes_radical_congruence()
[]

REFERENCES: