I first quote a definition from Clone theory in Universal Algebra: A binary relation $\rho$ on a set U is strongly rigid if every universal algebra on U such that $\rho$ is a subuniverse of its square is trivial, i.e., the clone preserving ρ has only projections. It is known that there are only two strongly rigid relations on a 3-element domain. I found that

1) On a 4-element domain {0,1,2,3}, the following binary relation is possibly a strongly rigid relation: {(0,1), (0,2), (0,3), (1,0), (1,2), (2,0), (2,1), (2,3), (3,1), (3,2)}

2) On a 5-element domain {0,1,2,3,4}, the following relation is possibly a strongly rigid relation: {(0,2), (0,3), (0,4), (1,0), (1,3), (1,4), (2,0), (2,1), (2,4), (3,0), (3,1), (3,2), (3,4), (4,0), (4,1), (4,2), (4,3)}.

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Could you let me know if there is an easy way to prove the above statements?
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This could be an algebraic proof, or by using a computer program.