(≤k)-reconstructible binary relations

A relation R is (≤k)-reconstructible (k a positive integer) if it is isomorphic with any relation S on the same vertex set with the property that the relations induced by R and S on any set of at most k vertices are isomorphic; it is (≤k)-self dual if every restriction to at most k vertices is self dual, i.e.  isomorphic to its dual relation (the relation obtained by reversing its arcs). In particular, relying on the description of (≤k)-self dual binary relations, we characterize, for each k≥4, all (≤k)-reconstructible binary relations: A binary relation is (≤k)-reconstructible if and only if its modules that are chains are finite and its (≤k)-self dual modules are self dual.