 The difficulty was to find enough structure in congruence lattices of non-congruence-permutable lattices.We shall denote by ε the `parity function' on the natural numbers, that is, ε("n")="n" mod 2, for any natural number "n".We let "L" be an algebra possessing a structure of semilattice ("L",∨) such that every congruence of "L" is also a congruence for the operation ∨ .