Notes on Representing a Finite Lattice as the Congruence Lattice of a Finite Partial Algebra
The main document in this repository is par-alg-rep.pdf. It contains some notes about a few results---not all new---that we worked out in the fall semester of 2016, while the author was a postdoc at University of Hawaii, although the main result was discovered during a visit to Chapman University in October 2016.
This is joint work with Peter Jipsen and Bill Lampe.
Abstract: We give a straight-forward proof of the fact that every finite lattice is the congruence lattice of a finite partial algebra. Bill Lampe pointed out that this has been known for a long time, though our proof is new and seems simpler to us.
We also describe closure operators and review 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 this 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.
For questions, comments, or suggestions please submit an issue.
Thanks for your interest in this work!