A constructive approach to the finite congruence lattice representation problem (Q1866816)

A finite lattice is representable if it is isomorphic to the congruence lattice of a finite algebra. The author develops methods by which one can construct new representable lattices from known ones. The main results: Theorem 4.2. Every finite lattice which contains no 3-element antichains is representable. Theorem 5.1. Let \(A\) be a finite set. There is a set \(P\) of polynomials of \(\langle \text{Eq}(A); \vee ,\wedge \rangle \) so that any \(0\)-\(1\) lattice \({\mathcal L}\) of equivalence relations on \(A\) is the congruence lattice of an algebra on \(A\) if and only if \({\mathcal L}\) is closed under the operations in \(P\). Theorem 5.2. Suppose \(L\) is a finite representable lattice. If \(L\) is order polynomially complete then every diagonal subdirect power of \(L\) is also representable.