Status: #good
Tags: #definition

partially ordered set

A partially ordered set, or poset, is a set $S$ along with binary relations $\ge$ on $S$ satisfying the following properties:

• (reflexivity) $x\ge x$
• (transitivity) $x\ge y,y\ge z⟹x\ge z$
• (antisymmetry) $x\ge y,y\ge x⟹x=y$

Notationally, we use $a<b$ to mean $a\le b$ and $a\ne b$, and we'll flip the arrows.

If there is a unique minimal element (and note that this might not exist), we denote it $\hat{0}$.

Examples

• $[n]=\{1,\dots ,n\}$ under the usual $\le$
• Boolean lattice