Hasse Diagrams for Partially Ordered Sets 📊

We introduce Hasse diagrams for representing partially ordered sets. Recall a partially ordered set consists of a set A with a partial order R. To be a partial order, R must be transitive, antisymmetric, and reflexive. A common example is the divides relation. To create the Hasse diagram for a partially ordered set (or PO set), we represent each element in the set with a vertex, then when vertex x relates to vertex y, we draw an arrow from x to y. Then remove the arrows that exist as a result of transitivity, then redraw what remains so that all arrows are oriented in the same direction (up, down, left, or right), now the arrows can be removed as it is understood that each line indicates relation in the same direction. We'll see three examples, including a Hasse diagram for a total order. #DiscreteMath

We'll also discuss minimal and maximal elements of partially ordered sets, and see how to find minimal and maximal elements in Hasse diagrams.

Antisymmetric Relations: https://youtu.be/fu9j4cPhaLo
Partially Ordered Sets: https://youtu.be/XJQqDDTNvJA

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