There are no infinite order polynomially complete lattices after all

Abstract

A lattice L is called opc if every monotone function f : L^n -> L is induced by a polynomial. We show here: If L is a lattice with the interpolation property whose cardinality is a strong limit cardinal of uncountable cofinality, then some finite power L^n has an antichain of size kappa. Using our previous result (math.LO/9707203 in the xxx archive) that the cardinality of an infinite opc lattice must be inaccessible, we can now conclude that there are no infinite opc lattices. However, the existence of strongly amorphous sets implies (in ZF) the existence of infinite opc lattices.