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.
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.
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.
What does a nonabelian group sound like?
The GroupSound project is about harmonic analysis on finite groups. Classical dsp filtering algorithms can be implemented as operations involving functions (e.g., audio signals) defined on a finite group. That is, the group serves as the domain, or "index set," of the functions. In this project, we explore the idea of using the finite group as an adjustable parameter of a digital audio filter.
The IEProps GitHub repository contains a draft of the article Interval enforceable properties of finite groups, to appear in Communications in Algebra.