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

Albert Vucaj, Dmitriy Zhuk

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 $f(x_1,\dots,x_n)\approx g(y_1,\dots,y_m)$, also known as minor identities, and therefore share many algebraic properties. Moreover, it was proved that the complexity of the $\operatorname{CSP}$ of a finite structure $\mathbb{A}$ only depends on the set of minor identities satisfied by the polymorphism clone of $\mathbb{A}$. In this article we consider the poset that arises by considering all clones over a three-element set with the following order: we write $\mathcal{C} \preceq_{\mathrm{m}} \mathcal{D}$ if there exists a minor homomorphism from $\mathcal{C}$ to $\mathcal{D}$. We show that the aforementioned poset has only three submaximal elements.

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

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 , also known as minor identities, and therefore share many algebraic properties. Moreover, it was proved that the complexity of the of a finite structure only depends on the set of minor identities satisfied by the polymorphism clone of . In this article we consider the poset that arises by considering all clones over a three-element set with the following order: we write if there exists a minor homomorphism from to . We show that the aforementioned poset has only three submaximal elements.
Paper Structure (11 sections, 19 theorems, 56 equations, 1 figure)

This paper contains 11 sections, 19 theorems, 56 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A Galois connection for clones
  4. The pp-constructability poset
  5. The unique coatom
  6. Auxiliary theorems
  7. Submaximal elements of $\mathfrak P_3$
  8. Generalized minority operations
  9. Totally symmetric operations of every arity
  10. The main result
  11. Conclusion

Key Result

Theorem 2.1

Let $F$ be a set of operations over a finite set. The following equality holds: $\mathop{\mathrm{Pol}}\nolimits(\mathop{\mathrm{Inv}}\nolimits(F)) = \langle F \rangle$.

Figures (1)

  • Figure 1: The structure $\mathbb{K}\coloneqq (K; {\color{blue}R}, {\color{red}S})$.

Theorems & Definitions (40)

  • Theorem 2.1: BoKaKoRoGeiger
  • Theorem 2.2: BoKaKoRoGeiger
  • Theorem 2.3: BoKaKoRoGeiger
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Remark 2.8
  • Theorem 2.9: wonderland
  • Corollary 2.10: wonderland
  • ...and 30 more