> For the complete documentation index, see [llms.txt](https://mandober.gitbook.io/math-debrief/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://mandober.gitbook.io/math-debrief/900-index/enums/enum-algebras/magma.md).

# magma

## Magma

<https://en.wikipedia.org/wiki/Magma_(algebra>)

### Types of magma

Magmas are not often studied as such; instead there are several different kinds of magma, depending on what axioms the operation is required to satisfy. Commonly studied types of magma include:

Quasigroup A magma where division is always possible

Loop A quasigroup with an identity element

Semigroup A magma where the operation is associative

Inverse semigroup A semigroup with inverse.

Semilattice A semigroup where the operation is commutative and idempotent

Monoid A semigroup with an identity element

Group A monoid with inverse elements, or equivalently, an associative loop, or a non-empty associative quasigroup

Abelian group A group where the operation is commutative Note that each of divisibility and invertibility imply the cancellation property.

## Classification of magmas by properties

* Medial
* Left semimedial
* Right semimedial
* Semimedial
* Trimedial
* Entropic
* Left distributive
* Right distributive
* Autodistributive
* Commutative
* Idempotent
* Unipotent
* Zeropotent
* Alternative
* Power-associative
* Flexible
* A semigroup, or associative
* A left unar
* A right unar
* Semigroup with zero multiplication, or null semigroup
* Unital
* Left-cancellative
* Right-cancellative
* Cancellative
* A semigroup with left zeros
* A semigroup with right zeros

A magma `(S, •)`, with `x,y,u,z ∈ S` is called:

* **Medial**            if `xy • uz ≡ xu • yz`
* **Left semimedial**   if `xx • yz ≡ xy • xz`
* **Right semimedial**  if `yz • xx ≡ yx • zx`
* **Semimedial** if it is left and right semimedial
* **Trimedial**  if any triple of (not necessarily distinct) elements generates a medial submagma
* **Entropic** if it is a homomorphic image of a medial cancellation magma
* **Left distributive** if `x • yz ≡ xy • xz`
* **Right distributive** if `yz • x ≡ yx • zx`
* **Autodistributive** if it is left and right distributive
* **Commutative** if `xy ≡ yx`
* **Idempotent**  if `xx ≡ x`
* **Unipotent**   if `xx ≡ yy`
* **Zeropotent**  if `xx • y` ≡ `xx` ≡ `y • xx`
* **Alternative** if `xx • y` ≡ `x • xy` and `x • yy` ≡ `xy • y`
* **Power-associative** if a submagma generated by any element is associative
* **Flexible** if `xy • x` ≡ `x • yx`
* **associative** (or a semigroup) if `x • yz` ≡ `xy • z`
* **left unar**  if `xy` ≡ `xz`
* **right unar** if `yx` ≡ `zx`
* **null semigroup** *semigroup with zero multiplication* if `xy ≡ uv`
* **Unital** if it has an identity element
* **Left-cancellative**  if `∀xyz. xy = xz => y = z`
* **Right-cancellative** if `∀xyz. yx = zx => y = z`
* **Cancellative**       if right-cancellative and left-cancellative
* *semigroup with left zeros*  if it is a semigroup and `∀x. x ≡ xy`
* *semigroup with right zeros* if it is a semigroup and `∀x. x ≡ yx`