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total_order [2013/09/04 17:04] nikolaj removed |
total_order [2014/03/21 11:11] (current) |
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| ===== Total order ===== | ===== Total order ===== | ||
| - | ==== Definition ==== | + | ==== Set ==== |
| - | | @#88DDEE: $X$ | | + | | @#55CCEE: context | @#55CCEE: $X$ | |
| - | | @#FFBB00: $ \le \in \text{TotalOrd}(X) $ | | + | | @#FFBB00: definiendum | @#FFBB00: $ \le\ \in\ \mathrm{it} $ | |
| - | | $ x\in \mathrm{dom}(\le) $ | | + | The relation $\le$ is an order relation if it's in the intersection of all total, all anti-symmetric and all transitive relation. Hence |
| - | | @#55EE55: $ x \le y\ \lor\ y\le x $ | | + | | @#55CCEE: context | @#55CCEE: $ \le\ \in\ \mathrm{Rel}(X) $ | |
| - | | @#55EE55: $ (x\le y) \land (y\le x) \implies (x=y) $ | | + | | $ x,y,z \in X $ | |
| - | | @#55EE55: $ (x \le y) \land (y \le z) \Leftrightarrow (x\le z) $ | | + | |
| - | Here we use infix notation: $x\le y\ \equiv\ \le(x,y)$. | + | | @#55EE55: postulate | @#55EE55: $ x \le y\ \lor\ y \le x $ | |
| + | | @#55EE55: postulate | @#55EE55: $ x\le y\ \land\ y\le x \implies (x=y) $ | | ||
| + | | @#55EE55: postulate | @#55EE55: $ x \le y\ \land\ y \le z \Leftrightarrow x\le z $ | | ||
| ==== Discussion ==== | ==== 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 === | === Reference === | ||
| - | Wikipedia: [[http://en.wikipedia.org/wiki/Order_relation|Order theory]] | + | Wikipedia: [[http://en.wikipedia.org/wiki/Total_order|Total order]] |
| - | ==== Context ==== | + | ==== Parents ==== |
| === Subset of === | === Subset of === | ||
| - | [[Total relation]] | + | [[Total relation]], [[Non-strict partial order]] |
