Category:Definitions/Hasse Diagrams

This category contains definitions related to Hasse Diagrams.
Related results can be found in Category:Hasse Diagrams.

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.