Editing Medial magma (section)

{{for|the triple product|Median algebra}}

In [[abstract algebra]], a '''medial magma''' or '''medial groupoid''' is a [[Magma (algebra)|magma]] or [[Magma (algebra)#History_and_terminology|groupoid]] (that is, a [[Set (mathematics)|set]] with a [[binary operation]]) that satisfies the [[identity (mathematics)|identity]]
: {{math|1=(''x'' • ''y'') • (''u'' • ''v'') = (''x'' • ''u'') • (''y'' • ''v'')}},
or more simply,
: {{math|1=''xy'' • ''uv'' = ''xu'' • ''yv''}}
for all {{math|''x''}}, {{math|''y''}}, {{math|''u''}} and {{math|''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'', [[#Generalizations|''entropic'']], etc.{{sfn|ps=|Ježek|Kepka|1983}}

Any [[Semigroup|commutative semigroup]] is a medial magma, and a medial magma has an [[identity element]] if and only if it is a [[Monoid#Commutative_monoid|commutative]] [[monoid]]. The "only if" direction is the [[Eckmann–Hilton argument]]. Another class of semigroups forming medial magmas are [[Band (mathematics)|normal bands]].{{sfn|ps=|Yamada|1971}} Medial magmas need not be associative: for any nontrivial [[abelian group]] with operation {{math|+}} and [[integer]]s {{math|''m'' ≠ ''n''}},  the new binary operation defined by {{math|1=''x'' • ''y'' = ''mx'' + ''ny''}} yields a medial magma that in general is neither associative nor commutative.

Using the [[category theory|categorical]] definition of [[product (category theory)|product]], for a magma {{math|''M''}}, one may define the [[Cartesian square]] magma&nbsp;{{math|''M'' × ''M''}} with the operation
: {{math|1=(''x'', ''y'') • (''u'', ''v'') = (''x'' • ''u'', ''y'' • ''v'')}}.
The binary operation {{math|•}} of&nbsp;{{math|''M''}}, considered as a mapping from {{math|''M'' × ''M''}} to {{math|''M''}}, maps {{math|(''x'', ''y'')}} to {{math|''x'' • ''y''}}, {{math|(''u'', ''v'')}} to {{math|''u'' • ''v''}}, and {{math|(''x'' • ''u'', ''y'' • ''v'') }} to {{math|(''x'' • ''u'') • (''y'' • ''v'') }}.
Hence, a magma&nbsp;{{math|''M''}} is medial if and only if its binary operation is a magma [[homomorphism]] from&nbsp;{{math|''M'' × ''M''}} to&nbsp;{{math|''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 [[Cartesian closed category|category with a Cartesian product]]. (See the discussion in auto magma object.)

If {{math|''f''}} and {{math|''g''}} are [[endomorphism]]s of a medial magma, then the mapping {{math|''f'' • ''g''}} defined by pointwise multiplication 
: {{math|1=(''f'' • ''g'')(''x'') = ''f''(''x'') • ''g''(''x'')}}
is itself an endomorphism. It follows that the set {{math|End(''M'')}} of all endomorphisms of a medial magma {{math|''M''}} is itself a medial magma.