Sublattices of the lattice of local clones

TL;DR

This paper explores the structure of the lattice of local clones over an infinite set, showing it contains complex sublattices and has a rich embedding structure.

Contribution

It demonstrates that the lattice includes all algebraic lattices with countably many compact elements as sublattices, and reveals a broader class of embeddable lattices.

Findings

Contains all algebraic lattices with countably many compact elements as sublattices

The class of embeddable lattices into the local clone lattice is strictly larger than the algebraic lattices

Provides insights into the complexity and richness of the lattice of local clones

Abstract

We investigate the complexity of the lattice of local clones over a countably infinite base set. In particular, we prove that this lattice contains all algebraic lattices with at most countably many compact elements as complete sublattices, but that the class of lattices embeddable into the local clone lattice is strictly larger than that.