There are no infinite order polynomially complete lattices, after all (Q5950779)

A lattice \(L\) is order polynomially complete (OPC) if for every integer \(n\geq 1\) every monotone function \(L^n\rightarrow L\) is induced by a lattice polynomial. Although finite OPC lattices were described by D. Schweigert, the question of infinite OPC lattices remained open. In a previous paper by the same authors [Algebra Univers. 39, No. 3-4, 197-209 (1998; Zbl 0935.06007)], it was shown that if an infinite OPC lattice \(L\) exists, then the cardinality of \(L\) must be a strongly inaccessible cardinal. Now the result is completed by showing (in ZFC) that this cardinality cannot be a strongly inaccessible cardinal, i.e. no infinite OPC lattice exists.