\begin{definition}[Definition:Hasse Diagram]
Let $\struct {S, \preceq}$ be an ordered set.
A '''Hasse diagram''' is a method of representing $\struct {S, \preceq}$ as a graph $G$, in which:
:$(1):\quad$ The vertices of $G$ represent the elements of $S$
:$(2):\quad$ The edges of $G$ represent the elements of $\preceq$
:$(3):\quad$ If $x, y \in S: x \preceq y$ then the edge representing $x \preceq y$ is drawn so that $x$ is lower down the page than $y$.
::::That is, the edge ascends (usually obliquely) from $x$ to $y$
:$(4):\quad$ If $x \preceq y$ and $y \preceq z$, then as an ordering is transitive it follows that $x \preceq z$.
::::But in a '''Hasse diagram''', the relation $x \preceq z$ is not shown.
::::Transitivity is implicitly expressed by the fact that $z$ is higher up than $x$, and can be reached by tracing a path from $x$ to $z$ completely through ascending edges.
\end{definition}