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

TL;DR

This paper introduces a new approach to describing multi-sorted clones on a two-element set, providing structural insights, simplifying proofs of classical results, and establishing a Galois connection between clones and relations.

Contribution

It offers a novel, elementary framework for multi-sorted clones, simplifies the proof of Post's lattice theorem, and demonstrates that all such clones are finitely generated.

Findings

A new class of canonical relations describes all Boolean multi-sorted clones.

Every multi-sorted clone decomposes into a surjective part and smaller clones.

All multi-sorted clones on a two-element set are finitely generated.

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.