stl ordering - strict weak ordering

Question

Why does STL work with a comparison function that is strict weak ordering? Why can\'t it be partial ordering?

Answer 1

Simply, a strict weak ordering is defined as an ordering that defines a (computable) equivalence relation. The equivalence classes are ordered by the strict weak ordering: a strict weak ordering is a strict ordering on equivalence classes.

A partial ordering (that is not a strict weak ordering) does not define an equivalence relation, so any specification using the concept of "equivalent elements" is meaningless with a partial ordering that is not a strict weak ordering. All STL associative containers use this concept at some point, so all these specifications are meaningless with a partial ordering that is not a strict weak ordering.

Because a partial ordering (that is not a strict weak ordering) does not necessarily define any strict ordering, you cannot "sort elements" in the common sense according to partial ordering (all you can do is a "topological sort" which has weaker properties).

Given

• a mathematical set S
• a partial ordering < over S
• a value x in S

you can define a partition of S (every element of S is either in L(x), I(x) or G(x)):

L(x) = { y in S | y

A sequence is sorted according to < iff for every x in the sequence, elements of L(x) appear first in the sequence, followed by elements of I(x), followed by elements of G(x).

A sequence is topologically sorted iff for every element y that appears after another element x in the sequence, y is not less than x. It is a weaker constraint than being sorted.

It is trivial to prove that every element of L(x) is less than any element of G(x). There is no general relation between elements of L(x) and elements of I(x), or between elements of I(x) and elements of G(x). However, if < is a strict weak ordering, than every element of L(x) is less than any element of I(x), and than any element of I(x) is less than any element of G(x).

If < is a strict weak ordering, and x then any element of L(x) U I(x) is less then any element I(y) U G(y): any element not greater than x is less than any element not less that y. This does not necessarily hold for a partial ordering.