Finitely Related Clones: Action Algebras and Applications to Free Algebras

Abstract

For finite-domain constraint satisfaction problems (CSPs), the polymorphism clone provides an algebraic framework for studying the associated constraint language. A central question in clone theory is whether a clone is finitely related, that is, whether it is determined by the operations preserving a finite set of relations. Here, we study this question through the action algebra of a clone on its own n-ary part. We prove that a clone 𝒞 on a finite domain C is finitely related if and only if its action algebra 𝒞̃ = 𝒞 curvearrowright 𝒞⁽ⁿ⁾ is finitely related, provided that n ≥ |C|. We also apply natural clone isomorphisms to obtain a finite-relatedness criterion for free algebras: a finite algebra is finitely related if and only if its free algebra of sufficiently large finite rank is finitely related.

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Cite

BibTeX RIS

BibTeX

@misc{tiwari2026finitely,
  title = {{Finitely Related Clones: Action Algebras and Applications to Free Algebras}},
  author = {Vishwesh Tiwari},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.17762},
  eprint = {2608.17762},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.17762},
}

RIS

TY  - UNPB
TI  - Finitely Related Clones: Action Algebras and Applications to Free Algebras
AU  - Vishwesh Tiwari
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2608.17762
UR  - https://arxiv.org/abs/2608.17762
AB  - For finite-domain constraint satisfaction problems (CSPs), the polymorphism clone provides an algebraic framework for studying the associated constraint language. A central question in clone theory is whether a clone is finitely related, that is, whether it is determined by the operations preserving a finite set of relations. Here, we study this question through the action algebra of a clone on its own n-ary part. We prove that a clone 𝒞 on a finite domain C is finitely related if and only if its action algebra 𝒞̃ = 𝒞 curvearrowright 𝒞^(n) is finitely related, provided that n ≥ |C|. We also apply natural clone isomorphisms to obtain a finite-relatedness criterion for free algebras: a finite algebra is finitely related if and only if its free algebra of sufficiently large finite rank is finitely related.
ER  -