Let 
be a partially ordered set. Then an element 
is said to be maximal if there is no 
such that 
and 
. Equivalently, an element 
is maximal if 
for any 
implies 
.
Note that the definition for a maximal element above is true for any two elements of a partially ordered set that are comparable. However, it may be the case that two elements of a given partial ordering are not comparable.
