hasse-diagram

Hasse-Diagram

A hasse diagram is like a relation graph for partial orders, so we omit the reflexive edges for simplicty. Every edge signifies a relation.
There is a hierarchy: Lower elements are subsets of the upper ones, so we can omit the arrows.
Every set connected via multiple relations could be made in a single step (transitive), but we also omit those arrows.

$(2^{\{1,2,3\}},\subseteq )$ … a partial order.

$\mathcal{P}(\{1,2,3\})$ = power set = All possible subsets of $\{1,2,3\}$

It’s not a total order since for example $\{1\}$ and $\{2\}$ are incomparable (neither is a subset of the other).
For $(2^{\{1,2,3\}},\supseteq )$, we mirror the diagram along the horizontal axis.

The natural numbers $\mathbb{N}$ are a linear order.

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