On the problem of dimension versus size for lattices (Q1916062)

The author reduces the intriguing conjecture \(\dim (L) = o(|L|)\) for lattices to an extremal set-theoretical conjecture: \(\lim_{d \to \infty} d(\lambda(d))^{-1} = 0\), where \(d\) is the order dimension of a lattice \(L\) and \(\lambda(d)\) is the minimum size of a lattice of dimension \(d\). The paper shows a possible way of solving this conjecture by reducing it to an extremal set-theoretical one. Reduction Theorem. Let \(L\) be a lattice. Then there exists a lattice \(H\) with the following three properties: (i) \(\dim(L) \leq \dim(H) \leq \dim(L) + 1\); (ii) The set of all (meet or join) irreducible elements of \(H\) is of height 1; (iii) \(|L|\leq |H|\leq 2 \cdot |L|+ 1\). This theorem can be used to obtain lower bounds for the sizes of special lattices.