Tamás Waldhauser (University of Szeged (Hungary), Solution sets and polymorphism-homogeneity

Tue, 1 Dec 2020, 1 pm MDT

Let A be a finite algebra and let C be the clone of term operations of A. A set of tuples is called and algebraic set, if it is the set of all solutions of a system of equations over A. These algebraic sets are the main objects of study in the so-called universal algebraic geometry initiated by Boris Plotkin. It is straightforward to verify that algebraic sets are always closed under C*, the centralizer of the clone C. We are interested in algebras for which the converse is also true, i.e., where solution sets of systems of equations can be characterized as C*-closed sets of tuples. We prove that this holds if and only if the algebra is polymorphism-homogeneous, and we determine such algebras in several classes of algebras, namely

  • two-element algebras,
  • finite lattices,
  • finite semilattices,
  • finite abelian groups,
  • finite monounary algebras.
This is joint work with Endre Tóth (University of Szeged).

[slides] [video]