Non-associative algebra

Medial magma

In abstract algebra, a medial magma or medial groupoid is a magma or groupoid (that is, a set with a binary operation) which satisfies the identity , or more simply for all x, y, u and v, using the convention that juxtaposition denotes the same operation but has higher precedence. This identity has been variously called medial, abelian, alternation, transposition, interchange, bi-commutative, bisymmetric, surcommutative, etc. Any commutative semigroup is a medial magma, and a medial magma has an identity element if and only if it is a commutative monoid. The "only if" direction is the Eckmann–Hilton argument. Another class of semigroups forming medial magmas are normal bands. Medial magmas need not be associative: for any nontrivial abelian group with operation + and integers m ≠ n, the new binary operation defined by yields a medial magma which in general is neither associative nor commutative. Using the categorical definition of product, for a magma M, one may define the Cartesian square magma M × M with the operation (x, y) ∙ (u, v) = (x ∙ u, y ∙ v) . The binary operation ∙ of M, considered as a mapping from M × M to M, maps (x, y) to x ∙ y, (u, v) to u ∙ v, and (x ∙ u, y ∙ v)  to (x ∙ u) ∙ (y ∙ v) .Hence, a magma M is medial if and only if its binary operation is a magma homomorphism from M × M to M. This can easily be expressed in terms of a commutative diagram, and thus leads to the notion of a medial magma object in a category with a Cartesian product. (See the discussion in auto magma object.) If f and g are endomorphisms of a medial magma, then the mapping  f∙g defined by pointwise multiplication is itself an endomorphism. It follows that the set End(M) of all endomorphisms of a medial magma M is itself a medial magma. (Wikipedia).

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Related pages

Algebraic structure | Category of medial magmas | Quasigroup | Identity element | Isotopy of loops | Product (category theory) | Semigroup | Arity | Homomorphism | Commutative diagram | Set (mathematics) | Integer | Identity (mathematics) | Eckmann–Hilton argument | Category theory | Abstract algebra | Cartesian closed category | Magma (algebra) | Binary operation | Endomorphism | Abelian group | Monoid