PlanetMath: Hasse diagram

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Hasse diagram

(Definition)

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 and the edges correspond to the covering relation. More precisely an edge from $x\in A$ to $y\in A$ is present if

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

, then in

is drawn higher than

. 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

$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ & \{1,2,3\} & \ \{1,2\} \ar@{-... ...l] \ar@{-}[u] \ & \emptyset \ar@{-}[ul] \ar@{-}[u] \ar@{-}[ur] & } } \end{xy}$

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 $\{1,2,3\}$ .

"Hasse diagram" is owned by bbukh. [ full author list (2) | owner history (1) ]

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