A clonoid based approach to some finiteness results in universal algebraic geometry

TL;DR

This paper demonstrates that for finite structures, the definable relations are uniquely determined by relations of a specific arity, leading to new proofs of finiteness results in universal algebraic geometry.

Contribution

It introduces a clonoid-based approach to establish finiteness results and provides new proofs for existing theorems in universal algebraic geometry.

Findings

Definable relations are determined by relations of arity |A|^2.

New proofs for finiteness results in universal algebraic geometry.

Establishes a connection between clonoids and definability in finite structures.

Abstract

We prove that for a finite first order structure $\mathbf{A}$ and a set of first order formulas $\Phi$ in its language with certain closure properties, the finitary relations on $A$ that are definable via formulas in $\Phi$ are uniquely determined by those of arity $|A|^{2}$. This yields new proofs for some finiteness results from universal algebraic geometry.