Projection properties and reflexive binary relations

Given a structure \frak M\frak M for which constant mappings are endomorphisms, consider an inter n > 1; the structure is weakly n-projective if projections are the only retractions of \frak Mⁿ\frak {M^n} on its diagonal; it is strongly n-projective if projections are its only Hamming n-operations (i.e. n-ary operations regular with respect to each argument) which are identical on the diagonal. We establish general criteria of "projectivity" for reflexive binary relations. We obtain in particular, for each integer n, the existence of strongly n-projective (symmetric) graphs.