Let A be a finite first-order structure and Phi a set of first-order formulas in its language with certain closure properties. We use a result about clonoids to show that the finitary relations on A definable by formulas in Phi are uniquely determined by the definable subsets of arity A^2. This gives new proofs of some finiteness results in Universal Algebraic Geometry. This talk is based on work of Erhard Aichinger and Bernardo Rossi.
Some Finiteness Results in Universal Algebraic Geometry via Clonoids