MathDB

Let f be a ternary operation on a set of at least four elements for which (1)

f ( x , x , y ) \equiv f ( x , y , x ) \equiv f( x , y , y ) \equiv x

(2)

f ( x , y , z ) = f ( y , z , x ) = f ( y , x , z ) \in \{ x , y , z \}

for pairwise distinct x,y,z. Prove that f is a nontrivial composition of g such that g is not a composition of f. (The n-variable operation g is trivial if

g(x_1, ..., x_n) \equiv x_i

for some i (

1 \leq i \leq n

) )

Problem Statement