Definition:Hasse Diagram

Definition
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.

Also known as
Some sources refer to this as a nodal diagram.

Also presented as
Some sources draw arrows on their edges, so as to make $G$ a directed graph, but this is usually considered unnecessary.

Some sources do not label the vertices, on the grounds that the structure of the Hasse diagram is the important part.