Order polynomially complete lattices must be LARGE

If L is an order polynomially complete lattice, (that is: every monotone function from L^n to L is induced by a lattice-theoretic polynomial) then the cardinality of L is a strongly inaccessible cardinal. In particular, the existence of such lattices is not provable in ZFC, nor from ZFC+GCH. Although the problem originates in algebra, the proof is purely set-theoretical. The main tools are partition and canonisation theorems. It is still open if the existence of infinite o.p.c. lattices can be refuted in ZFC.