Geometric and conditional geometric equivalences of algebras

The basis for classifying universal algebras is conventionally formed by one or other type of equivalence relations between the algebras, such as isomorphism, elementary equivalence, equivalence of algebras in other logical languages, geometric equivalence, and so on. Of crucial importance in this event are results that reduce one of such equivalences to another. The most important example (enjoying multiple applications) is the theorem stating that every two algebras are elementarily equivalent iff their ultrapowers with respect to some ultrafilters are isomorphic. Similar results are established for different equivalences of algebras relevant to algebraic geometry of universal algebras.