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A partially ordered set has the following properties:

a ≤ a (reflexivity);

if a ≤ b and b ≤ a, then a = b (antisymmetry);

if a ≤ b and b ≤ c, then a ≤ c (transitivity);

I think that the easiest way is to visualize it with a Hasse-diagram:enter image description here

So I obtained the solution 16 partially ordered sets have a set of 4 elements have.

My second question: How many total order sets(which has the following properties: reflexivity,antisymmetry,transitivity,trichotomous) does a 4 element set have?

Zauberkerl
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    The answer to the latter is: $4!=24$ distinct orderings, but only one such ordering up to isomorphism. The latter distinction is something you must take into account for partial orders, too, since (for example) all $24$ total orders are of "type P." – Cameron Buie Apr 25 '16 at 17:58
  • I still don't see exactly why is the latter $4!$, where do you use the fact that this is trichotomous? Is my answer to the first question correct? is that just the power set? – Zauberkerl Apr 25 '16 at 18:23
  • A reflexive relation on a non-empty set cannot be trichotomous. I think you mean "total," instead. Totality is used to prove that every total order on a four element set is of "type P." As for why there are $24$ such: How many options are there for the least element? Once that's chosen, how many options are there for the second-least element? (etc.) – Cameron Buie Apr 25 '16 at 19:46
  • As a result, your answer is not correct, though it seems you've correctly determined all the possible Hasse diagrams (I haven't had the chance to sit down and work it all out), which would mean you've determined all partial orders on a four-element set up to isomorphism. – Cameron Buie Apr 25 '16 at 19:53
  • I mean "totally ordered set", \ http://mathworld.wolfram.com/TotallyOrderedSet.html in this way. with the properties: reflexivity,antisymmetry,transitivity,comparability (trichotomy law) – Zauberkerl Apr 25 '16 at 19:56
  • I got the result to the first question that there are 219 possible variations. – Zauberkerl Apr 27 '16 at 07:28

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