# Explanation: Hasse Diagram

(related to Chapter: Order Relations)

A Hasse diagram is a method named after Helmut Hasse (1898 - 1979), who used it to visualize a poset $(V,\preceq )$, which has only a finite number1 of elements. A Hasse diagram of a set $V$ can be drawn by applying the following rules:

• Draw the elements of $V$ as points in the plane and label them.
• If $a\preceq b$ for some two elements $a$ and $b$, then draw $b$ "above" $a$ and connect both by an arrow or line $ab$.
• If $a\preceq c$ because $a\preceq b\preceq c$ for some $b$, avoid drawing an additional arrow $ac$, because there are already arrows $ab$ and $bc.$

As an example, consider the poset of the divisors of $24,$ i.e. the set $D:=\{1,2,3,4,6,8,12,24\}$ with the order $\prec:=\{(a,b)\in D\times D: a\mid b\}$. There are elements, which are incomparable (e.g. $3$ and $8$ since $3\not\mid 8$). Otherewise, there are many transitivities, e.g. $1 \mid 2$, $2\mid 4$, $4\mid 8$, and $8\mid 24$). To see the Hasse diagram, click on the Evaluate button below: §§§1

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#### Footnotes

1. We will define later on what "finite" means in a strict mathematical manner. For the time being, it is sufficient to rely on the meaning of this word in the English language.