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arXiv · 2608.17762

Finitely Related Clones: Action Algebras and Applications to Free Algebras

Vishwesh Tiwari
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Abstract

Constraint Satisfaction Problems (CSPs) provide a framework for expressing complex algorithmic decision problems. For finite-domain CSPs, the polymorphism clone provides a fundamental algebraic invariant governing the associated constraint language. A fundamental structural question in universal algebra is how to determine whether a given clone is finitely related. In this paper, we prove that a clone $\mathcal{C}$ on a finite domain $C$ is finitely related if and only if its action algebra $\widetilde{\mathcal{C}} = \mathcal{C} \curvearrowright \mathcal{C}^{(n)}$ on its $n$-ary part (for $n \ge |C|$) is finitely related. We then apply natural clone isomorphisms to demonstrate that a finite algebra is finitely related if and only if its free algebra of sufficient rank is finitely related.

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Vishwesh Tiwari. 2026-08-18. Finitely Related Clones: Action Algebras and Applications to Free Algebras. https://arxiv.org/abs/2608.17762

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