Category:Hasse Diagrams

From ProofWiki
Jump to: navigation, search

This category contains results about Hasse Diagrams.
Definitions specific to this category can be found in Definitions/Hasse Diagrams.

Let $\left({S, \preceq}\right)$ be an ordered set.

A Hasse diagram is a method of representing $\left({S, \preceq}\right)$ 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.

This category currently contains no pages or media.