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.