Hasse diagram

If $(A,\leq)$ is a finite poset, then it can be represented by a Hasse diagram, which is a graph whose vertices are elements of $A$ and the edges correspond to the covering relation. More precisely an edge from $x\in A$ to $y\in A$ is present if

• $x.

• There is no $z\in A$ such that $x and $z. (There are no in-between elements.)

If $x, then in $y$ is drawn higher than $x$. Because of that, the direction of the edges is never indicated in a Hasse diagram.

Example: If $A=\mathcal{P}(\{1,2,3\})$, the power set of $\{1,2,3\}$, and $\leq$ is the subset relation $\subseteq$, then Hasse diagram is

 $\xymatrix{&\{1,2,3\}&\\ \{1,2\}\ar@{-}[ur]&\{1,3\}\ar@{-}[u]&\{2,3\}\ar@{-}[ul]\\ \{1\}\ar@{-}[u]\ar@{-}[ur]&\{2\}\ar@{-}[ul]\ar@{-}[ur]&\{3\}\ar@{-}[ul]\ar@{-}% [u]\\ &\emptyset\ar@{-}[ul]\ar@{-}[u]\ar@{-}[ur]&}$

Even though $\{3\}<\{1,2,3\}$ (since $\{3\}\subset\{1,2,3\}$), there is no edge directly between them because there are inbetween elements: $\{2,3\}$ and $\{1,3\}$. However, there still remains an indirect path from $\{3\}$ to $\{1,2,3\}$.

Title Hasse diagram HasseDiagram 2013-03-22 12:15:23 2013-03-22 12:15:23 bbukh (348) bbukh (348) 18 bbukh (348) Definition msc 05C90 Poset PartialOrder