The natural relations for sets are those definable in
terms of the emptiness of the subsets
corresponding to Boolean combinations of the sets. For pairs of sets, there
are just
five natural relations of interest, namely, strict inclusion in each direction,
disjointness,
intersection with the universe being covered, or not. Let N denote
{1, 2, …, n} and (N2) denote
{(i, j)[mid ]i, j∈N and
i<j}. A function μ on (N2)
specifies one of these relations for each pair of indices. Then μ is
said to be
consistent on M⊆N if and only if there exists
a collection of sets corresponding to indices in M such that the
relations specified by μ
hold between each associated pair of the sets. Firstly, it is proved that
if μ is consistent
on all subsets of N of size three then μ is consistent on N.
Secondly, explicit conditions
that make μ consistent on a subset of size three are given as generalized
transitivity laws.
Finally, it is shown that the result concerning binary natural relations
can be generalized
to r-ary natural relations for arbitrary r[ges ]2.