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Wikipedia states:

There is a 1-to-1 correspondence between preorders and pairs (partition, partial order).

I tried to calculate all preorder relations on a set with cardnilaity $n=2,3,4$ and I could not do that for the case $n=4$, also my strategy for calculating them is not easier than Wiki's .

Can someone explain why the theorem is true?

Also Wikipedia states:

The number of preorders is the sum of the number of partial orders on every partition.

In my opinion there exist just one partial order on every partiton, although it's not true ( maybe we should count the number of cells in every partition).

Foe example consider $A=\left\{1,2\right\}$, then I listed the partial order relations on this set: $$\left\{\left(1,1\right)\left(2,2\right)\right\}$$ $$\left\{\left(1,1\right)\left(2,2\right)\left(1,2\right)\right\}$$ $$\left\{\left(1,1\right)\left(2,2\right)\left(2,1\right)\right\}$$

The corresponding partitions are: $$\left\{\left\{1\right\},\left\{2\right\}\right\}$$ $$\left\{\left\{1,2\right\},\left\{2\right\}\right\}$$ $$\left\{\left\{1\right\},\left\{2,1\right\}\right\}$$

But the secod and third case are not partition, so where I'm wrong (Also I guess I've missed one of the partial order relations)

For example:

for $n = 3$:

$1$ partition of $3$, giving 1 preorder

$3$ partitions of $2 + 1$, giving $3 × 3 = 9$ preorders

$1$ partition of $1 + 1 + 1$, giving $19$ preorders

I.e., together, $29$ preorders.

I cannot understand what has been done there and cannot understand the link between partitions and preorders.

Why the example is true?

I trieds

  • You seem to have taken that first claim from the Wikipedia article on preorders. Did you read the three paragraphs leading up to that claim? They explain it quite thoroughly. – Arthur Feb 03 '20 at 15:26
  • @Arthur, yes but could not understand –  Feb 03 '20 at 15:27
  • Cool, so you have read it. Then, what parts of it could you not understand? – Arthur Feb 03 '20 at 15:27
  • why the number of preorders is related to partitions? –  Feb 03 '20 at 15:28
  • The wikipedia article explains that quite thoroughly. As it stands, anyone reading your question as it stands would just reproduce the wikipedia argument and think they have answered your question. Please, update your post with explicit references to the Wikipedia argument and what parts you do and do not get so that we can actually help you understand. – Arthur Feb 03 '20 at 15:29
  • @Arthur,I've added some notes –  Feb 03 '20 at 15:45

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