4 problems

Sectionally complemented Congruence Lattice conjecture. Every distributive semilattice with zero is isomorphic to the congruence semilattice of a sectionally complemented lattice.

Let $D$ be a finite distributive lattice. Condition (a) of the lemma is that there is an abstract illuminated system $(\mathcal S;\leq)$ isomorphic to $(\operatorname{J}(D);\leq)$.…

Let $L$ be a finite lattice with $n$ elements, and write ${\rm Con}(L)$ for its congruence lattice. Let ${\cal L}_k$, ${\cal L}_3\boxplus {\cal L}_5$, and…

The fifth-largest-value conjecture. The set of lattices attaining the fifth-largest congruence count is