The repository par-alg-rep contains some notes describing a few results that we worked out in the fall of 2016, while I was a postdoc at University of Hawaii, although the main result—a straight-forward proof of the fact that every finite lattice is the congruence lattice of a finite partial algebra—was discovered during a visit to Chapman University in October 2016.
Bill Lampe pointed out that the main result here has been known for a long time. However, our proof seems new and simpler to us.
The note also describes closure operators and reviews some useful facts about them. Finally, we recall a theorem from Berman’s thesis that relates congruence lattices of partial algebras with those of total algebras.
The main purpose of the note is to describe some tools that we plan to exploit in our quest to represent every finite lattice as the congruence lattice of a finite algebra.