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Since by hypothesis every operation on A elements c with a range of (0, x> is a polynomial, the and d may be taken into polynomial, where e 0 and is any other constant. e by some Thus when inter- preted in B , any constant would have to be equal to 0 . Now we have the following identities in A : x v x =x; (x&O) v (x6l) v ... v (x&(k-D ) = 1 . ) Therefore, in B , b60 = (b60) v (b60) v ... (b60) • v ... = 1 . Thus, not b B 0 , which is a contradiction to the assumption that bH0 .

We also leave to the reader the task of formulating and proving a criterion that separates those partial orders which give distributive congruence lattices of the algebra in the equational class generated by it from those that don't. There is the even older approach of Birkhoff [443, giving, for example, the only subdirectly irreducible distributive lattice, which one might try to generalize. But here again, as in the attempt to apply Jonsson's theorem, we are stymied by the apparent lack of polynomials that would even come close to mimicking either the join and meet operations of lattice theory or the majority trinomials in Jonsson's criterion.

A. , > € C are equal. 1 Assume if at n-1 m * 1 . An elementary is the h-ary relation on the set h of all such that m-l v y 0 1 . , 0 . . , aj? are equal. In other words, m-fold power of the primary h-adic relation any finite set with at least be a function from A The is the h-ary relation on the set O on Assume A onto h h n £ . elements, and let . , • is known as a strong homomorphic preimage of > G n n . (in the parlance of Rosenberg), or to simplify matters, we will say that e is an h-adic relation.