why does any partially ordered need to be antisymmetric?

why can't there be 2 elements in a poset with different values but order-wise have the same order priority? what's the motivation for this?

Does it even make sense to think of order without antisymmetry?

  • $\begingroup$ After a small Google search (you should try really!), this is called a pre-order, there is plenty of Wikipedia pages that talk about it! $\endgroup$ – eti902 May 9 '17 at 18:55
  • $\begingroup$ I know that pre-orders exist, your comment on my other question made me realize that my question was about symmetry (i realized that when you mentioned antisymmetry and i saw what really was not clicking for me). But my question is why to bother to divide preorders and partial orders at all, what's the motivation for adding the antisymmetry axiom, since "pre-order" makes me think that a pre-order is not quite an order. This sadly is not answered (or at least not in a way i understand) in wikipedia, i did search a bit in google but order theory seems to be obscure, i didnt find much. $\endgroup$ – Joaquin Brandan May 9 '17 at 19:01
  • $\begingroup$ @ÉtienneTétreault They are also called quasi-orders. I belive "quasi-order" was the original term and "pre-order" is a fashinable neologism, but I could be wrong. $\endgroup$ – bof May 9 '17 at 19:12
  • $\begingroup$ It may be similar to the motivation for the axiom $d(x,y)=0\implies x=y$ in metric spaces. Whatever that might be, I don't know the answer. $\endgroup$ – bof May 9 '17 at 19:16
  • $\begingroup$ I think that the reason in not a "deep" one. The original motivation for order is clearly the "natural" order of $\mathbb N$. Trying to define it absrtactly, we introduce the precise definitions of various properties: symmetry, reflexivity, transitivity, ... Thus, the next step is to find examples of the differetn "combinations" of them: pre-orders, partial order, ... $\endgroup$ – Mauro ALLEGRANZA May 10 '17 at 6:17

The answer seems to be that in any pre-order you can define an equivalence relation where two elements are equivalent if each is related to the other. When you mod out by the equivalence relation you get antisymmetry. In many applications the antisymmetry is important, but in some others not. That is why you have pre-orders and stronger orders. If the order has the stronger property it makes sense to acknowledge it.

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  • $\begingroup$ mh, there must be a historical reason then? at some point someone decided that asymmetry was very important, and it seems to be so important that most books about set theory i have come across dont mention pre-orders at all. It feels like there should be a concrete reason or bunch of reasons this is so important to acknowledge. Maybe there are important concepts in order theory that rely on antisymetry later on? $\endgroup$ – Joaquin Brandan May 9 '17 at 21:34
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    $\begingroup$ Of course. For example, in set theory, inclusion of sets gives a poset. If given two sets, each includes the other, then by antisymmetry the two sets are equal. This is a standard way of proving set equality. $\endgroup$ – Somos May 9 '17 at 21:43
  • $\begingroup$ mhh, it all kind of feels very arbitrary at this point, i will try to read about the development of set theory over time, with some luck i will be able to find why it's all the way it is. So far it seems it all started with number theory and developed from there, maybe i should start studying number theory and then read some of cantors papers or something. $\endgroup$ – Joaquin Brandan May 10 '17 at 2:39
  • $\begingroup$ It probably has something to do with the fact that comparison operation over $\mathbb{N} , \mathbb{R} $ are naturaly antisymetric. $\endgroup$ – Joaquin Brandan May 10 '17 at 2:41

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