Generating some large filters of quasiorder lattices
In 2017, the author proved that the lattice of quasiorders (AKA preorders) of an $n$-element set has a four-element generating set for every natural number $n>1$ with three exceptions. A bit later, his joint paper with J\'ulia Kulin reduced the exceptions from three to one. Here we prove that some large filters of the lattices of quasiorders of finite sets (and some infinite sets) have small generating sets, too (some of these filters in the paper consist of the extensions of the partial orders of forests with few edges). Therefore, these lattices could be used in the author's 2021 secret key cryptographic authentication protocol. Here we point out that the underlying mathematical problem (solving systems of equations in lattices) of the 2021 protocol is likely to be not in P since the opposite assumption leads to P=NP.