Medial magma
This article defines a property that can be evaluated for a magma, and is invariant under isomorphisms of magmas.
View other such properties
Definition
A magma is termed a medial magma if it satisfies the following identity:
.
This identity is termed the medial identity.
If a medial magma has a two-sided neutral element (i.e., identity element) then it must be an abelian monoid. This is a special case of the general result known as Eckmann-Hilton duality, which says that if two binary operations commute and have a common neutral element, they must coincide and both be commutative.
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Abelian semigroup | ||||
| Medial quasigroup |