partial order

Definition

A relation $\leq$ is a partial order on a set $S$ if it satisfies:

• reflexivity: $a \leq a$ for all $a \in S$
• antisymmetry: $a \leq b$ and $b \leq a$ implies $a = b$
• transitivity: $a \leq b$ and $b \leq c$ implies $a \leq c$

A partially ordered set (poset) is an ordered pair $P = (X, \leq)$ consisting of a set $X$ (ground set of $P$) and partial order $\leq$ on $X$.

Notes

• total order has additional property that for any $a \neq b$, either $a < b$ or $b < a$

References

1. https://mathworld.wolfram.com/PartialOrder.html
2. https://ncatlab.org/nlab/show/partial+order
3. https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/A_Spiral_Workbook_for_Discrete_Mathematics_(Kwong)/07%3A_Relations/7.04%3A_Partial_and_Total_Ordering