Algebra Seminar talk
2023-10-13
Jonathan Theil
Mal'cev conditions
Abstract:
Ever since the middle of the 20th century, universal algebraists have observed the effects of making assumptions about the existence of terms fulfilling certain identities on the structural properties of algebras. More recently however, these same considerations have come to be essential in solving long standing questions in the field of theoretical computer science. The first part of this paper concerns itself with the former, recounting the theorems of Mal’cev, Pixley, J´onsson and Day. Here we look at the lattices of congruence relations in a given variety and how their properties interact with the identities on the respective clone of term operations. In the second part, we give an introduction to CSPs and investigate their connection to universal algebra through polymorphism clones, before stating the algebraic dichotomy theorem.