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loop [2013/08/06 21:42] nikolaj created |
loop [2014/03/21 11:11] |
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| - | ===== Loop ===== | ||
| - | ==== Definition ==== | ||
| - | | @#88DDEE: $X$ | | ||
| - | | @#55EE55: $ \langle X,* \rangle \in \text{Loop}(X)$ | | ||
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| - | | @#88DDEE: $\langle X,* \rangle \in \mathrm{Quasigroup}(X)$ | | ||
| - | | @#DDDDDD: $a,a^{-1}\in X$ | | ||
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| - | | @#55EE55: $\forall a.\ \exists a^{-1}.\ (a*a^{-1}=a^{-1}*a=e)$ | | ||
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| - | Here we used infix notation for "$*$". | ||
| - | |||
| - | ==== Ramifications ==== | ||
| - | === Discussion === | ||
| - | |||
| - | The binary operation is often called //multiplication//. | ||
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| - | The axioms $*\in \mathrm{binaryOp}(X)$ above means that a monoid is closed with respect to the multiplication. | ||
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| - | One generally calls $X$ the loop, i.e. the set where the operation "$*$" is defined on. | ||
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| - | ==== Reference ==== | ||
| - | Wikipedia: [[http://en.wikipedia.org/wiki/Quasigroup|Quasigroup]] | ||
| - | ==== Context ==== | ||
| - | === Subset of === | ||
| - | [[Quasigroup]] | ||
