ProofWiki:Sandbox

Magma

An algebraic structure $\left({S, \circ}\right)$ with:

Semigroup

A magma $\left({S, \circ}\right)$ with:

Commutative semigroup

A semigroup $\left({S, \circ}\right)$ with:

Monoid

A semigroup $\left({S, \circ}\right)$ with:

Commutative monoid

A monoid $\left({G, \circ}\right)$ with:

Group

A monoid $\left({G, \circ}\right)$ with:

Abelian group

A group $\left({G, \circ}\right)$ with:

Ringoid

An algebraic structure $\left({R, +, \times}\right)$ where:

Semiring

A ringoid $\left({R, +, \times}\right)$ where:

$\left({R, +}\right)$ and $\left({R, \times}\right)$ are semigroups

Additive semiring

A semiring $\left({R, +, \times}\right)$ where:

$\left({R, +}\right)$ is a commutative semigroup

Rig

An additive semiring $\left({R, +, \times}\right)$ where:

$\left({R, +}\right)$ is a commutative monoid

Ring

A Rig $\left({R, +, \times}\right)$ where:

$\left({R, +}\right)$ is an Abelian group

Commutative ring

A ring $\left({R, +, \times}\right)$ where:

$\left({R, \times}\right)$ is an commutative semigroup

Ring with unity

A ring $\left({R, +, \times}\right)$ where:

$\left({R, \times}\right)$ is a monoid

Commutative and unitary ring

A ring with unity $\left({R, +, \times}\right)$ where:

$\left({R \setminus \left\{{0}\right\}, \times}\right)$ is a commutative monoid

Division ring

A ring with unity $\left({R, +, \times}\right)$ where:

$\left({R \setminus \left\{{0}\right\}, \times}\right)$ is a group

Integral domain

A commutative and unitary ring $\left({R, +, \times}\right)$ where:

Field

A division ring $\left({F, +, \times}\right)$ where:

$\left({F \setminus \left\{{0}\right\}, \times}\right)$ is an Abelian group

Vector space

$\left({G, +_G, \circ}\right)_K$ where:

$\left({G, +_G}\right)$ is an Abelian group

$\left({K, +_K, \times_K}\right)$ is a field