Semirigid Systems of Equivalence Relations

A system M of equivalence relations on a set E is semirigid if only the projections and constant functions preserve all members of M. We construct semirigid systems of three equivalence relations. Our construction leads to the examples given by Zadori in 1983 and to many others and also extends to some infinite cardinalities. As a consequence, we show that for each cardinal κ, κ ∉ {2, 4}, κ ≤ 2^(N0) there exists a semirigid system of three equivalences on a set of cardinality κ.