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.