More sublattices of the lattice of local clones

TL;DR

This paper explores the structure of the lattice of local clones over an infinite set, revealing its rich sublattice complexity and embedding properties, including the presence of complex algebraic and non-algebraic lattices.

Contribution

It demonstrates that the local clone lattice contains all algebraic lattices with countably many compact elements and also embeds more complex lattices like M_{2^ω}.

Findings

Contains all algebraic lattices with countably many compact elements.

Embeds complex lattices such as M_{2^ω}.

Shows the lattice's rich and diverse sublattice structure.

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: For example, the lattice $M_{2^\omega}$ is a sublattice of the local clone lattice.