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total_order [2013/09/04 17:04] nikolaj removed |
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| - | ===== Total order ===== | ||
| - | ==== Definition ==== | ||
| - | | @#88DDEE: $X$ | | ||
| - | | @#FFBB00: $ \le \in \text{TotalOrd}(X) $ | | ||
| - | |||
| - | | $ x\in \mathrm{dom}(\le) $ | | ||
| - | |||
| - | | @#55EE55: $ x \le y\ \lor\ y\le x $ | | ||
| - | | @#55EE55: $ (x\le y) \land (y\le x) \implies (x=y) $ | | ||
| - | | @#55EE55: $ (x \le y) \land (y \le z) \Leftrightarrow (x\le z) $ | | ||
| - | |||
| - | Here we use infix notation: $x\le y\ \equiv\ \le(x,y)$. | ||
| - | |||
| - | ==== Discussion ==== | ||
| - | The relation $\le$ is an order relation if it's in the intersection of all reflexive, all anti-symmetric and all transitive relation. Hence | ||
| - | |||
| - | The first axiom $ x \le y\ \lor\ y\le x $ is called //totality// and implies $ x \le x $. Therefore a linear order is a partial order, although the converse is not true in general. | ||
| - | === Reference === | ||
| - | Wikipedia: [[http://en.wikipedia.org/wiki/Order_relation|Order theory]] | ||
| - | ==== Context ==== | ||
| - | === Subset of === | ||
| - | [[Total relation]] | ||
