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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
- $x < y$
- 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 $\{1,2,3\}$
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