partially ordered set no relation infimum
a
        ≤
        x
      
    
    {\displaystyle a\leq x}
   for all 
  
    
      
        x
        ∈
        S
        .
      
    
    {\displaystyle x\in S.}
  A lower bound 
  
    
      
        a
      
    
    {\displaystyle a}
   of 
  
    
      
        S
      
    
    {\displaystyle S}
   is called an infimum (or greatest lower bound, or meet) of 
  
    
      
        S
      
    
    {\displaystyle S}
   iffor all lower bounds 
  
    
      
        y
      
    
    {\displaystyle y}
   of 
  
    
      
        S
      
    
    {\displaystyle S}
   in 
  
    
      
        P
        ,
      
    
    {\displaystyle P,}
   
  
    
      
        y
        ≤
        a
      
    
    {\displaystyle y\leq a}
   (
  
    
      
        a
      
    
    {\displaystyle a}
   is larger than any other lower bound).Similarly, an upper bound of a subset 
  
    
      
        S
      
    
    {\displaystyle S}
   of a partially ordered set 
  
    
      
        (
        P
        ,
        ≤
        )
      
    
    {\displaystyle (P,\leq )}
   is an element 
  
    
      
        b
      
    
    {\displaystyle b}
   of 
  
    
      
        P
      
    
    {\displaystyle P}
   such that
no relation