A system M of equivalence relations on a set E is semirigid if only the identity and constant functions preserve all members of M. We construct semirigid systems of three equivalence relations. Our construction leads to the examples given by Zádori in 1983 and to many others and also extends to some infinite cardinalities. As a consequence, we show that on every set of at most continuum cardinality distinct from 2 and 4 there exists a semirigid system of three equivalence relations.

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Abst r actLet 5 be a fixed order of height at least 2 on a set A (i.e. contains a chain a < b < c). We show that all the isotone clones preserving orders on A isomorphic to 5 intersect in the clone lib. of trivial functions (i.e. all the projections and all the constant operations on A ) . We further show that for A finite with at least 8 elements and for any 6-element set there exist 2 orders on A such that every joint endomorphism is trivial (i.e. idA or constants). The same is true for intersections of isotone clones. This ields that with the above restrictions there are 4 maximar isotone clones intersecting in I(a. Separate considerations are given on the intersections of maximal isotone clones for IAl = 3 and 4.