Differences
This shows you the differences between two versions of the page.
quasigroup [2013/08/06 21:39] nikolaj created |
quasigroup [2014/03/21 11:11] |
||
---|---|---|---|
Line 1: | Line 1: | ||
- | ===== Quasigroup ===== | ||
- | ==== Definition ==== | ||
- | | @#88DDEE: $X$ | | ||
- | | @#55EE55: $ \langle X,* \rangle \in \text{Quasigroup}(X)$ | | ||
- | |||
- | | @#88DDEE: $*\in \mathrm{magma}(X)$ | | ||
- | |||
- | | @#DDDDDD: $a,b,x,y\in X$ | | ||
- | |||
- | | @#55EE55: $ \forall a.\ \forall b.\ \exists x.\ a*x=b $ | | ||
- | | @#55EE55: $ \forall a.\ \forall b.\ \exists y.\ y*a=b $ | | ||
- | |||
- | Here we used infix notation for "$*$". | ||
- | |||
- | ==== Ramifications ==== | ||
- | === Discussion === | ||
- | |||
- | The binary operation is often called //multiplication//. | ||
- | |||
- | The axioms $*\in \mathrm{binaryOp}(X)$ above means that a monoid is closed with respect to the multiplication. | ||
- | |||
- | One generally calls $X$ the quasigroup, i.e. the set where the operation "$*$" is defined on. | ||
- | |||
- | ==== Reference ==== | ||
- | Wikipedia: [[http://en.wikipedia.org/wiki/Quasigroup|Quasigroup]] | ||
- | ==== Context ==== | ||
- | === Subset of === | ||
- | [[Magma]] |