Consistency of Natural Relations on Sets

The natural relations for sets are those deenable in terms of the emptiness of the subsets corresponding to Boolean combinations of the sets. For pairs of sets, there are just ve natural relations of interest, namely, strict inclusion in each direction, disjointness, intersection with the universe being covered, or not. Let N denote f1; 2; : : :; ng and ? N 2 denote f(i; j) j i; j 2 N and i < jg. A function on ? N 2 speciies one of these relations for each pair of indices. Then 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 speciied by hold between each associated pair of the sets. Firstly, it is proved that if is consistent on all subsets of N of size three then is consistent on N. Secondly, explicit conditions that make 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 2. A part of the work was done while the second author was visiting Brown University. The third author gratefully acknowledges support from the British Council while visiting Tohoku University, and from the EC under ESPRIT LTR Project no. 20244 | ALCOM-IT.