Submaximal clones over a three-element set up to minor-equivalence

2023/04/25 by Albert Vucaj, Vucaj, Albert, Dmitriy Zhuk +1 · 1 citation
Computer Science · Mathematics · #Advanced Algebra and Logic #Advanced Topics in Algebra #FOS: Mathematics #Rings and Algebras (math.RA) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2304.12807

openalex publication_date 2023/04/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study clones modulo minor homomorphisms, which are mappings from one clone to another preserving arities of operations and respecting permutation and identification of variables. Minor-equivalent clones satisfy the same sets of identities of the form f(x₁,…,x_n)≈ g(y₁,…,y_m), also known as minor identities, and therefore share many algebraic properties. Moreover, it was proved that the complexity of the CSP of a finite structure \mathbbA only depends on the set of minor identities satisfied by the polymorphism clone of \mathbbA. In this article we consider the poset that arises by considering all clones over a three-element set with the following order: we write C \preceq_m D if there exists a minor homomorphism from C to D. We show that the aforementioned poset has only three submaximal elements.

Cited by

• Mal'cev clones over a three-element set up to minor-equivalence

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