nLab super commutative monoid

Contents

Context

Algebra

higher algebra

universal algebra

supersymmetry

Applications

Monoid theory

monoid theory in algebra:

Contents

Idea

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

Definition

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

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

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

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