What properties must a relation have for it to be called a ''partial ordering''?

Answer 1

For a relation, $, to be called a partial ordering on a set, S, the following three properties must be met:

1) If T is any subset of S, then T $ T.

2) If T and U are any two subsets of S that meet the condition T $ U as well as the condition U $ T, then T = U.

3) If T, U, and V are any three subsets of S that meet the condition T $ U as well as the condition U $ V, then T $ V.

For the relation, $, to be called a total ordering on the set, S, the following statement must hold in addition to the previous three:

If T and U are any two subsets of S, then either T $ U or U $ T.

This final property is called totality.

For an example of a partial ordering relation, see the related link on "less than or equal to."

Also, see the corresponding related link for the definition of "relation."