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In abstract algebra, a medial magma, or medial groupoid, is a set with a binary operation which satisfies the identity :, or more simply, using the convention that juxtaposition denotes the same operation but has higher precedence. A magma or groupoid is an algebraic structure that generalizes a group. This identity has been variously called ''medial'', ''abelian'', ''alternation'', ''transposition'', ''interchange'', ''bi-commutative'', ''bisymmetric'', ''surcommutative'', ''entropic'' etc.〔(Historical comments ) J.Jezek and T.Kepka: Medial groupoids Rozpravy CSAV, Rada mat. a prir. ved 93/2 (1983), 93 pp〕 Any commutative semigroup is a medial magma, and a medial magma has an identity element if and only if it is a commutative monoid. Another class of semigroups forming medial magmas are the normal bands.〔.〕 Medial magmas need not be associative: for any nontrivial abelian group and integers , replacing the group operation with the binary operation yields a medial magma which in general is neither associative nor commutative. Using the categorial definition of the product, one may define the Cartesian square magma with the operation : . The binary operation of , considered as a function on , maps to , to , and to . Hence, a magma is medial if and only if its binary operation is a magma homomorphism from to . 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 and are endomorphisms of a medial magma, then the mapping defined by pointwise multiplication : is itself an endomorphism. == Bruck–Murdoch–Toyoda theorem == The Bruck–Murdoch-Toyoda theorem provides the following characterization of medial quasigroups. Given an abelian group and two commuting automorphisms φ and ψ of , define an operation on by : where some fixed element of . It is not hard to prove that forms a medial quasigroup under this operation. The Bruck–Toyoda theorem states that every medial quasigroup is of this form, i.e. is isomorphic to a quasigroup defined from an abelian group in this way. In particular, every medial quasigroup is isotopic to an abelian group. The result was obtained independently in 1941 by D.C. Murdoch and K. Toyoda. It was then rediscovered by Bruck in 1944. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「medial magma」の詳細全文を読む スポンサード リンク
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