## java on linux

This post details the steps I use to update or install the latest version of the Java Development Kit (JDK) on my Linux systems.

## The Palfy-Saxl Problem

Suppose the lattice shown below is a congruence lattice of an algebra.

Conjecture: If the three $\alpha_i$'s pairwise permute, then all pairs in the lattice permute.

## 3-SAT and partition lattices

It is not hard to see that 3-SAT reduces to the problem of deciding whether all coatoms in a certain partition lattice are contained in the union of a collection of certain principal filters. Therefore, the latter problem, which we will call the covered coatoms problem (CCP), is NP-complete. In this post we simply define CCP. Later we check that 3-SAT reduces to CCP, and then develop some ideas about constructing a feasible algorithm to solve CCP.