nLab super commutative monoid




Super-Algebra and Super-Geometry

Monoid theory



A /2\mathbb{Z}/2\mathbb{Z}-graded commutative monoid.


An super commutative monoid is a commutative monoid MM with decomposition functions 𝒟 0:MM\mathcal{D}_0:M \to M and 𝒟 1:MM\mathcal{D}_1:M \to M, such that

  • for all a:Ma:M, a=𝒟 0(a)+𝒟 1(a)a = \mathcal{D}_0(a) + \mathcal{D}_1(a)
  • 𝒟 0(0)=0\mathcal{D}_0(0) = 0
  • for all a:Ma:M, and b:Mb:M, 𝒟 0(a+b)=𝒟 0(a)+𝒟 0(b)\mathcal{D}_0(a + b) = \mathcal{D}_0(a) + \mathcal{D}_0(b)
  • 𝒟 1(0)=0\mathcal{D}_1(0) = 0
  • for all a:Ma:M, and b:Mb:M, 𝒟 1(a+b)=𝒟 1(a)+𝒟 1(b)\mathcal{D}_1(a + b) = \mathcal{D}_1(a) + \mathcal{D}_1(b)
  • for all a:Ma:M, 𝒟 0(𝒟 0(a))=𝒟 0(a)\mathcal{D}_0(\mathcal{D}_0(a)) = \mathcal{D}_0(a)
  • for all a:Ma:M, 𝒟 0(𝒟 1(a))=0\mathcal{D}_0(\mathcal{D}_1(a)) = 0
  • for all a:Ma:M, 𝒟 1(𝒟 0(a))=0\mathcal{D}_1(\mathcal{D}_0(a)) = 0
  • for all a:Ma:M, 𝒟 1(𝒟 1(a))=𝒟 1(a)\mathcal{D}_1(\mathcal{D}_1(a)) = \mathcal{D}_1(a)

As a result, the image of the two decompostion functions im(𝒟 0)\im(\mathcal{D}_0) and im(𝒟 1)\im(\mathcal{D}_1) are commutative monoids and there exists a monoid isomorphism i:Vim(𝒟 0)im(𝒟 1)i:V \cong \im(\mathcal{D}_0) \otimes \im(\mathcal{D}_1), where ABA \otimes B is the tensor product of commutative monoids.

The elements of im(𝒟 0)\im(\mathcal{D}_0) are called even elements or bosonic elements, and the elements of im(𝒟 1)\im(\mathcal{D}_1) are called odd elements or fermionic elements.

See also

Created on May 11, 2022 at 01:32:51. See the history of this page for a list of all contributions to it.