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Clonoids over vector spaces

Stefano Fioravanti, Michael Kompatscher, Bernardo Rossi

Abstract

Clonoids are sets of finitary operations between two algebraic structures that are closed under composition with their term operations on both sides. We conjecture that, for finite modules $\mathbf A$ and $\mathbf B$ there are only finitely many clonoids from $\mathbf A$ to $\mathbf B$ if and only if $\mathbf A$, $\mathbf B$ are of coprime order. We confirm this conjecture for a broad class of modules $\mathbf A$. In particular we show that, if $\mathbf A$ is a finite $k$-dimensional vector space, then every clonoid from $\mathbf A$ to a coprime module $\mathbf B$ is generated by its $k$-ary functions (and arity $k-1$ does not suffice). In order to prove this results, we investigate `uniform generation by $(\mathbf A,\mathbf B)$-minors', a general criterion, which we show to apply to several other existing classifications results. Based on our analysis, we further prove that the subpower membership problem of certain 2-nilpotent Mal'cev algebras is solvable in polynomial time.

Clonoids over vector spaces

Abstract

Clonoids are sets of finitary operations between two algebraic structures that are closed under composition with their term operations on both sides. We conjecture that, for finite modules and there are only finitely many clonoids from to if and only if , are of coprime order. We confirm this conjecture for a broad class of modules . In particular we show that, if is a finite -dimensional vector space, then every clonoid from to a coprime module is generated by its -ary functions (and arity does not suffice). In order to prove this results, we investigate `uniform generation by -minors', a general criterion, which we show to apply to several other existing classifications results. Based on our analysis, we further prove that the subpower membership problem of certain 2-nilpotent Mal'cev algebras is solvable in polynomial time.
Paper Structure (17 sections, 39 theorems, 43 equations)

This paper contains 17 sections, 39 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Clones and clonoids
  4. Modules and vector spaces
  5. Uniformly generated functions
  6. Basic properties
  7. Uniform generation for clonoids between affine clones
  8. Examples beyond affine clones
  9. Interpolation by uniformly representable operations
  10. The main result
  11. Reduction to functions of singleton support
  12. Generating characteristic functions
  13. Summary
  14. A description of the lattice of clonoids from ${\mathbf F}^k$ to $\mathbf{B}$
  15. Subpower Membership Problem
  16. ...and 2 more sections

Key Result

Theorem 8

Let $\mathcal{A}, \mathcal{B}$ be clones on two finite sets $A$ and $B$. Then $\mathcal{C} \subseteq \mathcal{O}_{A,B}$ is a clonoid from $\mathcal{A}$ to $\mathcal{B}$ if and only if there are relational structures $\mathbb A$, $\mathbb B$ with $\mathcal{C} = \mathop{\mathrm{Pol}}\nolimits(\mathbb

Theorems & Definitions (92)

  • Conjecture 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5: Clonoid
  • Definition 6
  • Definition 7
  • Theorem 8
  • Definition 9
  • Lemma 10
  • ...and 82 more