Hasse diagram HasseDiagram 2002-02-02 03:10:14 2004-04-11 15:26:58 Definition \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{xypic} If $(A,\leq)$ is a finite poset, then it can be represented by a \emph{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 \begin{itemize} \item $x < y$. \item There is no $z \in A$ such that $x < z$ and $z < y$. (There are no in-between elements.) \end{itemize} 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. \emph{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 $\{1,2,3\}$.