# Differences

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 non-strict_partial_order [2013/09/04 17:27]nikolaj created non-strict_partial_order [2014/03/21 11:11] (current) 2013/09/04 17:43 nikolaj 2013/09/04 17:27 nikolaj created Next revision Previous revision 2013/09/04 17:43 nikolaj 2013/09/04 17:27 nikolaj created Line 1: Line 1: ===== Non-strict partial order ===== ===== Non-strict partial order ===== - ==== Definition ​==== + ==== Set ==== - | @#88DDEE: $X$ | + | @#55CCEE: context ​    | @#55CCEE: $X$ | - | @#FFBB00: $\le\ \in\ \mathrm{it}$ | + | @#FFBB00: definiendum ​| @#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 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: $\le\ \in\ \mathrm{Rel}(X)$ | + | @#55CCEE: context ​    | @#55CCEE: $\le\ \in\ \mathrm{Rel}(X)$ | | $x,y,z \in X$ | | $x,y,z \in X$ | - | @#55EE55: $x \le x$ | + | @#55EE55: postulate ​  | @#55EE55: $x \le x$ | - | @#55EE55: $x\le y\ \land\ y\le x \implies (x=y)$ | + | @#55EE55: postulate ​  | @#55EE55: $x\le y\ \land\ y\le x \implies (x=y)$ | - | @#55EE55: $x \le y\ \land\ y \le z \Leftrightarrow x\le z$ | + | @#55EE55: postulate ​  | @#55EE55: $x \le y\ \land\ y \le z \Leftrightarrow x\le z$ | Here we use infix notation: $x\le y\ \equiv\ \le(x,y)$. Here we use infix notation: $x\le y\ \equiv\ \le(x,y)$. Line 19: Line 19: === Reference === === Reference === Wikipedia: [[http://​en.wikipedia.org/​wiki/​Order_relation|Order theory]], [[http://​en.wikipedia.org/​wiki/​Poset|Poset]] Wikipedia: [[http://​en.wikipedia.org/​wiki/​Order_relation|Order theory]], [[http://​en.wikipedia.org/​wiki/​Poset|Poset]] - ==== Context ​==== + ==== Parents ​==== === Subset of === === Subset of === [[Reflexive relation]], [[Anti-symmetric relation]], [[Transitive relation]] [[Reflexive relation]], [[Anti-symmetric relation]], [[Transitive relation]] + === Equivalent to === + [[Poset]]