Order polynomially complete lattices must be LARGE

Details Order polynomially complete lattices must be LARGE Order polynomially complete lattices must be LARGE

1997-07-16

arxiv.org/abs/math/9707203v1

Mathematics

Logic

This is paper number GoSh:633 in Shelah's list. The paper is to appear in Algebra Universalis.

Scientific paper

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.

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