An order is a relation R that satisfies transitivity. i.e., such that if aRb and bRc, then aRc.
A strict partial order is a relation < on S that satisfies irreflexivity(\(\forall x\in S \lnot x<x\)) and transitivity(\(\forall x,y,z \in S, x<y\land y<z \rightarrow x<z\))
We can define a new relation \(\leq\) based on any relation <: \( x\leq y\leftrightarrow x<y \lor x=y\)
If < is a strict partial order, the defined \(\leq\) is called non-strict partial order.
From the properties of strict partial order <, we can derive the following properties of the non-strict partial \(\leq\):
- relexivity: \(\forall x \in S, x\leq x\)
- transitivity: \(\forall x,y,z \in S, x\leq y\land y\leq z \rightarrow x\leq z\)
- antisymmetry: \(\forall x,y \in S, x\leq y\land y\leq x \rightarrow x=y\)
Both strict partial order and non-strict partial order are partial order.
A strict total order is a strict partial order < that also satisfies \(\forall x,y \in S x<y\lor x=y\lor y<x\).
A non-strict total order is a non-strict partial order \(\leq\) that also satisfies \(\forall x,y \in S x\leq y\lor x=y\lor y\leq x\).
Both strict total order and non-strict total order are total order.
References:
https://en.wikipedia.org/wiki/Total_order
https://en.wikipedia.org/wiki/Partially_ordered_set
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