Semirigid Equivalence Relations on a Finite Set

A system R of equivalence relations on a set A (with at least 3 elements) is semirigid ;/ only the trivial opera tions (that is the projections and constant functions) preserve all members of R. To a system R of equivalence relations we associate a graph Gr. We observe that ifR is semirigid then the graph Gr is 2-connected. We show that the converse holds if all the members of R are atoms of the lattice E of equivalence relations on A. We present a notion of graphical composition of semirigid systems and show that it preserves semirigidity.