Semirigid equivalence relations - a new proof method

We show by a purely relational method that the joint-endomorphism of Zadori\´s three equivalence relations on a set A, |A|>2 is the clone consisting only of trivial functions, i.e., of the projections and constant functions. We use a so called "Wheatstone bridge" which is a device to yield an equivalence relation θ=W(α,β,γ) from a triple α,β,γ of equivalence relations such that if a function f:A→A preserves α,β,γ jointly, then it preserves θ. We also present a notion of compositions of two semirigid systems which preserve semirigidity. As an application of the composition we give three families of systems of five equivalence relations that are semirigid on the set A with |A|=4i, |A|=3i+1, or |A|=3i+2 for i≥1.