Universality of the lattice of transformation monoids

TL;DR

This paper proves that the lattice of all transformation monoids on an infinite set is universal for closed sublattices, meaning it can represent all such lattices with a bounded number of compact elements.

Contribution

It establishes the universality of the lattice of transformation monoids for closed sublattices, characterizing its structure in terms of algebraic lattices.

Findings

The lattice Mon(λ) is a complete algebraic lattice with 2^λ compact elements.

Closed sublattices of Mon(λ) correspond to all complete algebraic lattices with ≤ 2^λ compact elements.

Mon(λ) is universal for these sublattices, up to isomorphism.

Abstract

The set of all transformation monoids on a fixed set of infinite cardinality \lambda, equipped with the order of inclusion, forms a complete algebraic lattice Mon(\lambda) with 2^{\lambda} compact elements. We show that this lattice is universal with respect to closed sublattices, i.e., the closed sublattices of Mon(\lambda) are, up to isomorphism, precisely the complete algebraic lattices with at most 2^{\lambda} compact elements.