A clonoid based approach to some finiteness results in universal algebraic geometry (Q1987546)

The paper applies the theory of clonoids to the study of relations definable by logical formulas of a prescribed type on a finite set. The key insight is that if a set \(R\) of relations on a set \(A\) is closed under intersections, unions and taking minors, then the set of indicator functions of members of \(R\) is a clonoid from \(A\) to the two element lattice. Since the two element lattice has the majority operation, one can then apply a result by \textit{A. Sparks} [Algebra Univers. 80, No. 4, Paper No. 53, 10 p. (2019; Zbl 1436.08005)] that clonoids from \(A\) to the two element lattice are determined by their \(|A|^2\)-ary part. An algebra is called an equational domain if the union of any two algebraic sets is again algebraic. The authors obtain a new proof of the result by \textit{A. G. Pinus} [Algebra Logic 55, No. 6, 501--506 (2017; Zbl 1378.08001); translation from Algebra Logika 55, No. 6, 760--768 (2016)] that on any finite set there are only finitely many equational domains with pairwise different families of algebraic sets. The same argument also gives a new proof of a different result by \textit{A. G. Pinus} [Sib. Math. J. 58, No. 4, 672--675 (2017; Zbl 1420.08002); translation from Sib. Mat. Zh. 58, No. 4, 864--869 (2017)] that there are only finitely many pairwise \(L_0\)-logically inequivalent (giving different sets of relations definable by quantifier free formulas) algebras on a finite set.