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Name for a poset where incomparability is an equivalence relation

0

$begingroup$

Say I have a partial order $leq$ on a set $S$. Let me write $a sim b$ if $a$ and $b$ are incomparable under this order.

Is there a name for the following restriction on the partial order?

$sim$ is an equivalence relation.

If this holds, then we can use the equivalence relation to partition $S$ into totally ordered subsets where $S_i preceq S_j$ if and only if $a leq b$ and $b notleq a$ for all $a in S_i$ and $b in S_j$. Each each $S_i$ contains only elements that are not comparable under the original partial order. So it seems like a nice property.

order-theory equivalence-relations

share|cite|improve this question

asked Mar 18 at 19:38

usulusul

1,7101422

$endgroup$

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  $begingroup$
  Since $anotsim a,$ it ($sim$) cannot be an equivalence relation, unless the poset is empty.
  $endgroup$
  – Thomas Andrews
  Mar 18 at 19:42
  
  

  
  
  
  


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  1
  

  
  
  
  
  
  

  
  $begingroup$
  Even after correcting for Thomas' comment, the notion you'll get will just be equivalent to the poset being a disjoint union of linear orders. Such posets are not terribly interesting and so do not warrant a name. Certainly, most posets in nature are not of that type.
  $endgroup$
  – Ittay Weiss
  Mar 18 at 19:54
  

  
  
  
  


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  $begingroup$
  Yes, I forgot the condition "or a=b". @IttayWeiss, if you can convert that comment into an answer, I can accept it.
  $endgroup$
  – usul
  Mar 18 at 21:29
  

  
  
  
  

add a comment | 

0

$begingroup$

Say I have a partial order $leq$ on a set $S$. Let me write $a sim b$ if $a$ and $b$ are incomparable under this order.

Is there a name for the following restriction on the partial order?

$sim$ is an equivalence relation.

If this holds, then we can use the equivalence relation to partition $S$ into totally ordered subsets where $S_i preceq S_j$ if and only if $a leq b$ and $b notleq a$ for all $a in S_i$ and $b in S_j$. Each each $S_i$ contains only elements that are not comparable under the original partial order. So it seems like a nice property.

order-theory equivalence-relations

share|cite|improve this question

asked Mar 18 at 19:38

usulusul

1,7101422

$endgroup$

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  Since $anotsim a,$ it ($sim$) cannot be an equivalence relation, unless the poset is empty.
  $endgroup$
  – Thomas Andrews
  Mar 18 at 19:42
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Even after correcting for Thomas' comment, the notion you'll get will just be equivalent to the poset being a disjoint union of linear orders. Such posets are not terribly interesting and so do not warrant a name. Certainly, most posets in nature are not of that type.
  $endgroup$
  – Ittay Weiss
  Mar 18 at 19:54
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Yes, I forgot the condition "or a=b". @IttayWeiss, if you can convert that comment into an answer, I can accept it.
  $endgroup$
  – usul
  Mar 18 at 21:29
  

  
  
  
  

add a comment | 

$begingroup$

Say I have a partial order $leq$ on a set $S$. Let me write $a sim b$ if $a$ and $b$ are incomparable under this order.

Is there a name for the following restriction on the partial order?

$sim$ is an equivalence relation.

If this holds, then we can use the equivalence relation to partition $S$ into totally ordered subsets where $S_i preceq S_j$ if and only if $a leq b$ and $b notleq a$ for all $a in S_i$ and $b in S_j$. Each each $S_i$ contains only elements that are not comparable under the original partial order. So it seems like a nice property.

order-theory equivalence-relations

share|cite|improve this question

asked Mar 18 at 19:38

usulusul

1,7101422

$endgroup$

share|cite|improve this question

asked Mar 18 at 19:38

usulusul

1,7101422

share|cite|improve this question

asked Mar 18 at 19:38

usulusul

1,7101422

asked Mar 18 at 19:38

usulusul

1,7101422

asked Mar 18 at 19:38

usulusul

1,7101422