Medial_magma
In abstract algebra, a medial magma or medial groupoid is a magma or groupoid (that is, a set with a binary operation) that satisfies the identity
- (x • y) • (u • v) = (x • u) • (y • v),
or more simply,
- xy • uv = xu • yv
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, entropic, etc.[1]
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.[2] Medial magmas need not be associative: for any nontrivial abelian group with operation + and integers m ≠ n, the new binary operation defined by x • y = mx + ny yields a medial magma that 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
- (f • g)(x) = f(x) • g(x)
is itself an endomorphism. It follows that the set End(M) of all endomorphisms of a medial magma M is itself a medial magma.