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On the lattice of multi-sorted relational clones on a two-element set

Vojtěch David, Dmitriy Zhuk

TL;DR

The paper extends clone theory to multi-sorted relational clones on a Boolean domain by developing canonical relations and elementary operations that yield tractable, pp/qpp-based descriptions. It provides a short, elementary proof of Post's lattice theorem, proves that every multi-sorted clone decomposes into a surjective part defined by canonical relations and 2k small (k−1)-clones, and embeds the clone lattice into a simple poset, leading to a finite generation result for all k-clones. A central contribution is the Galois connection between surjective multi-sorted clones and quantified relational clones, extended to the multi-sorted setting with rigorous closure characterizations. Together, these results give structural insight, enabling inductive analyses of the lattice and enabling applications to concrete problems in the study of Boolean multi-sorted relational clones.

Abstract

We introduce a new approach to the description of multi-sorted clones (sets of $k$-tuples of operations of the same arity, closed under coordinatewise composition and containing all projection tuples) on a two-element domain. Leveraging the well-known Galois connection between operations and relations, we define a small class of canonical relations sufficient to describe all Boolean multi-sorted clones up to non-surjective operations. Furthermore, we introduce elementary operations on relations, which are less cumbersome than general formulas and have many useful properties. Using these tools, we provide a new and elementary proof of the famous Post's lattice theorem. We also show that every multi-sorted clone of $k$-tuples of operations decomposes into a surjective part described by canonical relations and $2k$ clones of $(k-1)$-tuples of operations. This structural understanding allows us to describe an embedding of the lattice of multi-sorted clones into a well-understood poset. In particular, we rederive - by a simpler method - a result of V. Taimanov originally from 1983, showing that every multi-sorted clone on a two-element domain is finitely generated. Finally, we also give a concise proof of the Galois connection between (surjective) multi-sorted clones and the corresponding closed sets of relations.

On the lattice of multi-sorted relational clones on a two-element set

TL;DR

The paper extends clone theory to multi-sorted relational clones on a Boolean domain by developing canonical relations and elementary operations that yield tractable, pp/qpp-based descriptions. It provides a short, elementary proof of Post's lattice theorem, proves that every multi-sorted clone decomposes into a surjective part defined by canonical relations and 2k small (k−1)-clones, and embeds the clone lattice into a simple poset, leading to a finite generation result for all k-clones. A central contribution is the Galois connection between surjective multi-sorted clones and quantified relational clones, extended to the multi-sorted setting with rigorous closure characterizations. Together, these results give structural insight, enabling inductive analyses of the lattice and enabling applications to concrete problems in the study of Boolean multi-sorted relational clones.

Abstract

We introduce a new approach to the description of multi-sorted clones (sets of -tuples of operations of the same arity, closed under coordinatewise composition and containing all projection tuples) on a two-element domain. Leveraging the well-known Galois connection between operations and relations, we define a small class of canonical relations sufficient to describe all Boolean multi-sorted clones up to non-surjective operations. Furthermore, we introduce elementary operations on relations, which are less cumbersome than general formulas and have many useful properties. Using these tools, we provide a new and elementary proof of the famous Post's lattice theorem. We also show that every multi-sorted clone of -tuples of operations decomposes into a surjective part described by canonical relations and clones of -tuples of operations. This structural understanding allows us to describe an embedding of the lattice of multi-sorted clones into a well-understood poset. In particular, we rederive - by a simpler method - a result of V. Taimanov originally from 1983, showing that every multi-sorted clone on a two-element domain is finitely generated. Finally, we also give a concise proof of the Galois connection between (surjective) multi-sorted clones and the corresponding closed sets of relations.
Paper Structure (19 sections, 31 theorems, 132 equations, 1 figure, 1 table)

This paper contains 19 sections, 31 theorems, 132 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction and main results
  2. Introduction
  3. Main results
  4. Applications
  5. Structure of the paper
  6. Main definitions and preliminaries
  7. Canonical relations
  8. Closedness criteria and Post's lattice theorem
  9. Elementary operations
  10. 1-sorted case and Post's lattice theorem
  11. Lattice of multi-sorted clones
  12. Embedding of the lattice
  13. Inductive structure of the lattice
  14. Galois connection
  15. Closed sets of operations
  16. ...and 4 more sections

Key Result

Theorem 2.8

Let $k\in\mathbb{N}$ and let $A$ be a nonempty finite set. For all $\mathcal{F}'\subseteq\mathcal{F}\subseteq \mathcal{O}_A^k$ and all $S'\subseteq S \subseteq R_A^k$, we have and

Figures (1)

  • Figure 1: Hasse diagram of all 1-sorted quantified relational clones on $\{0,\,1\}$. Each vertex denotes a quantified relational clone generated by all the relations below it.

Theorems & Definitions (85)

  • Definition 2.2: Operations, clones
  • Definition 2.3: Relations, relational clones
  • Remark 2.5
  • Definition 2.6: Polymorphisms, invariant relations
  • Theorem 2.8
  • Definition 2.9: Linear equations
  • Theorem 2.10
  • Definition 2.11
  • Definition 3.1
  • Theorem 3.2
  • ...and 75 more