partial order

Partial order (Halbordnung)

The relation $R\subseteq A\times A$ is a partial order on $A$ if it is: reflexive, antisymmetric, transitive.

$(2^{\{1,2,3\}},\subseteq )$ … 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.

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EXAMPLE

Let $A=\mathbb{R},xRy⟺x\le y$ → total order
Let $A=\mathbb{R},xRy⟺x<y$ → partial order (not reflexive: $x\nless x$)

Let $A=\mathbb{N},xRy⟺x\mid y$ → partial order (e.g. $2\nmid 3,3\nmid 3$)

$A=2^M=\{x:x\subseteq M\}$, $YRY⟺X\subseteq Y$ → partial order, total order if $|M|\le 1$, since the empty set is comparable to the set with one element. But if there are two elements, we can put both into a set with a single element and then the two sets are not comparable.

$(\{R:R\subseteq A\times B\},\subseteq )$ → partial order (the set of all relations between two sets is partially ordered by inclusion)
$(\{R:R\subseteq A\times A\},\oplus )$ can be written as a pair $(A,R),R\subseteq A\times A$.

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