Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Permutation clones that preserve relations

Tim Boykett

TL;DR

The paper investigates permutation clones defined by relational constraints, i.e., relationally defined permutation clones, using Jerábek's weight-co-clone duality to connect component maps with invariant weights. It develops structural results about the lattice of relationally defined permutation clones on finite sets, showing that maximal borrow closed clones are either relationally defined or cancellatively defined, and provides a complete classification for the binary case with exactly 13 such clones. It systematically analyzes how various relational families (unary relations, equivalence relations, affine maps, central relations, and degenerate maps) shape subclones and their closures, connecting to Rosenberg’s maximal clones and the Post lattice in the binary setting. The findings reveal that many infinite clone classes collapse when viewed through the permutation-clone lens and highlight open problems, such as recognizing relations that yield trivial permutation clones. Overall, the work clarifies how relational constraints govern the structure of reversible gate systems and preserves a tight correspondence between relational clone theory and permutation clone dynamics on finite alphabets.

Abstract

Permutation clones generalise permutation groups and clone theory. We investigate permutation clones defined by relations, or equivalently, the automorphism groups of powers of relations. We find many structural results on the lattice of all relationally defined permutation clones on a finite set. We find all relationally defined permutation clones on two element set. We show that all maximal borrow closed permutation clones are either relationally defined or cancellatively defined. Permutation clones generalise clones to permutations of $A^n$. Emil Jeřábek found the dual structure to be weight mappings $A^k\rightarrow M$ to a commutative monoid, generalising relations. We investigate the case when the dual object is precisely a relation, equivalently, that $M={\mathbb B}$, calling these relationally defined permutation clones. We determine the number of relationally defined permutation clones on two elements (13). We note that many infinite classes of clones collapse when looked at as permutation clones.

Permutation clones that preserve relations

TL;DR

The paper investigates permutation clones defined by relational constraints, i.e., relationally defined permutation clones, using Jerábek's weight-co-clone duality to connect component maps with invariant weights. It develops structural results about the lattice of relationally defined permutation clones on finite sets, showing that maximal borrow closed clones are either relationally defined or cancellatively defined, and provides a complete classification for the binary case with exactly 13 such clones. It systematically analyzes how various relational families (unary relations, equivalence relations, affine maps, central relations, and degenerate maps) shape subclones and their closures, connecting to Rosenberg’s maximal clones and the Post lattice in the binary setting. The findings reveal that many infinite clone classes collapse when viewed through the permutation-clone lens and highlight open problems, such as recognizing relations that yield trivial permutation clones. Overall, the work clarifies how relational constraints govern the structure of reversible gate systems and preserves a tight correspondence between relational clone theory and permutation clone dynamics on finite alphabets.

Abstract

Permutation clones generalise permutation groups and clone theory. We investigate permutation clones defined by relations, or equivalently, the automorphism groups of powers of relations. We find many structural results on the lattice of all relationally defined permutation clones on a finite set. We find all relationally defined permutation clones on two element set. We show that all maximal borrow closed permutation clones are either relationally defined or cancellatively defined. Permutation clones generalise clones to permutations of . Emil Jeřábek found the dual structure to be weight mappings to a commutative monoid, generalising relations. We investigate the case when the dual object is precisely a relation, equivalently, that , calling these relationally defined permutation clones. We determine the number of relationally defined permutation clones on two elements (13). We note that many infinite classes of clones collapse when looked at as permutation clones.

Paper Structure

This paper contains 15 sections, 31 theorems, 24 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Some basic properties
  4. Learning from clones
  5. Partial Orders
  6. Unary Relations
  7. Equivalence Relations
  8. Affine Maps
  9. Central Relations
  10. Degenerate or Essentially Unary Maps
  11. $h$-regular // $h$-universal // $h$-generated
  12. Self dual
  13. $\operatorname{Pol}(w_R)$ on a two element set
  14. Conclusions and further work
  15. Acknowledgements

Key Result

lemma 1

Let $C$ be a clone. Then there exists a relational clone ${\cal R}$ such that for all $f \in {\cal O}(A)$, $f \in C$ iff $f \triangleright R$ for all $R \in {\cal R}$.

Figures (1)

  • Figure 1: The inclusion diagram of the clones on two elements that give distinct permutation clones. This is a sublattice of the Post lattice.

Theorems & Definitions (59)

  • definition 1: lau_2006
  • lemma 1
  • theorem 1: Rosenberg
  • lemma 2
  • proof
  • definition 2
  • definition 3
  • definition 4
  • lemma 3
  • proof
  • ...and 49 more