Order-polynomial completeness of lattices and related problems
by
Miroslav Haviar
M. Bel University, Banská Bystrica, Slovakia

A lattice L is called order-polynomially complete (OPC, for short) if every finitary function on L preserving the order of L is a polynomial function of L. OPC lattices were studied in the 1970s by D. Schweigert, R. Wille, M. Kindermann and D. Dorninger, later on by many others like H. Kaiser, N. Sauer, I. Rival, N. Zaguia, M. Erné, E.T. Schmidt, M. Goldstern, S. Shelah, M. Ploscica and the speaker. It was shown already in the 1970s that a finite lattice is OPC if and only if it is tolerance-free.

There had been a long-standing conjecture saying that there is no infinite OPC lattice. Kaiser and Sauer (1993) proved that an OPC lattice must be bounded and cannot be countably infinite. Ploscica and the speaker (1998) showed that the cardinality of an infinite OPC lattice (if exists) must be at least the first strong limit cardinal. Their method, which will be presented, was later employed by Goldstern and Shelah who first proved that the existence of an infinite OPC lattice implies the existence of an inaccessible cardinal (which cannot be proven from the usual ZF axioms) and then showed that there is no infinite OPC lattice after all.

Some related problems or questions will also be mentioned. According to speaker's knowledge the problem of whether there is an infinite 1-OPC lattice (meaning that every unary order-preserving function is polynomial) is still open.

Date received: March 5, 2003