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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} $ | 
- 
-The relation $\le$ is an order relation if it's in the intersection of all reflexive, all anti-symmetric and all transitive relation. Hence  
- 
-| @#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 $ | 
- 
-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]] 
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