# motivation behind antisymmetry axiom on partially ordered sets

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?

• 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! – eti902 May 9 '17 at 18:55
• 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. – Joaquin Brandan May 9 '17 at 19:01
• @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. – bof May 9 '17 at 19:12
• 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. – bof May 9 '17 at 19:16
• 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, ... – Mauro ALLEGRANZA May 10 '17 at 6:17

• It probably has something to do with the fact that comparison operation over $\mathbb{N} , \mathbb{R}$ are naturaly antisymetric. – Joaquin Brandan May 10 '17 at 2:41