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non-strict_partial_order [2013/09/04 17:27] nikolaj created |
non-strict_partial_order [2014/03/21 11:11] |
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- | ===== Non-strict partial order ===== | ||
- | ==== Definition ==== | ||
- | | @#88DDEE: $X$ | | ||
- | | @#FFBB00: $ \le\ \in\ \mathrm{it} $ | | ||
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- | The relation $\le$ is an order relation if it's in the intersection of all reflexive, all anti-symmetric and all transitive relation. Hence | ||
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- | | @#88DDEE: $ \le\ \in\ \mathrm{Rel}(X) $ | | ||
- | | $ x,y,z \in X $ | | ||
- | |||
- | | @#55EE55: $ x \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 $ | | ||
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- | Here we use infix notation: $x\le y\ \equiv\ \le(x,y)$. | ||
- | |||
- | ==== Discussion ==== | ||
- | === Reference === | ||
- | Wikipedia: [[http://en.wikipedia.org/wiki/Order_relation|Order theory]], [[http://en.wikipedia.org/wiki/Poset|Poset]] | ||
- | ==== Context ==== | ||
- | === Subset of === | ||
- | [[Reflexive relation]], [[Anti-symmetric relation]], [[Transitive relation]] |