For a distributive lattice L, consider the following two statements:
(A) Every ideal of L is the kernel of at least two different homomorphisms.
(B) L contains no maximal ideal.
Which one of these statements implies the other?
(Every homomorphism φ of L induces an equivalence relation on L: a∼b if and only if a \varphi\equal{}b \varphi. We do not consider two homomorphisms different if they imply the same equivalence relation.)
J. Varlet, E. Fried abstract algebrasuperior algebrasuperior algebra unsolved