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

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

If $x<y$ , 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
Canonical name	HasseDiagram
Date of creation	2013-03-22 12:15:23
Last modified on	2013-03-22 12:15:23
Owner	bbukh (348)
Last modified by	bbukh (348)
Numerical id	18
Author	bbukh (348)
Entry type	Definition
Classification	msc 05C90
Related topic	Poset
Related topic	PartialOrder