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