The Hasse diagram

Observation — Hasse diagram

To visualise a finite partial order one draws the Hasse diagram: each element is a point and edges are drawn between $a$ and $b$ only if $a\prec b$ and there exists no $c$ such that $a\prec c\prec b$ (covering relations). Below, the diagram of the divisors of $12$ with the relation “divides”:

The divisors of $12$ ordered by divisibility: $1$ at the bottom, $12$ at the top.

The $1$ sits at the bottom (it divides everything), the $12$ at the top (a multiple of everything). The branches that separate — for example $2$ and $3$ above $1$ — declare incomparability: the relation is partial.

Links

Topics: Set theory
Concepts: Hasse diagram · Partial order
People: Hasse