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- This article is about medial in mathematics. For other uses, see medial (disambiguation).

- $(x\; cdot\; y)\; cdot\; (u\; cdot\; z)\; =\; (x\; cdot\; u)\; cdot\; (y\; cdot\; z)$, or more simply, $xycdot\; uz\; =\; xucdot\; yz$

using the convention that juxtaposition has higher precedence. This identity has been variously called medial, abelian, alternation, transposition, bi-commutative, bisymmetric, surcommutative, entropic, 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. An elementary example of a nonassociative medial quasigroup can be constructed as follows: take an abelian group except the group of order 2 (written additively) and define a new operation by x * y = (− x) + (− y).

A magma M is medial if and only if its binary operation is a homomorphism from the Cartesian square M x 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

- $(fcdot\; g)(x)\; =\; f(x)cdot\; g(x)$

is itself an endomorphism.

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Last updated on Saturday February 16, 2008 at 19:34:45 PST (GMT -0800)

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This article is licensed under the GNU Free Documentation License.

Last updated on Saturday February 16, 2008 at 19:34:45 PST (GMT -0800)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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