Lattices embeddable in three-generated lattices

TL;DR

This paper proves that any finite lattice can be embedded into a three-generated finite lattice and extends this to algebraic lattices with accessible cardinality, strengthening classical results.

Contribution

It introduces new embedding results for finite and algebraic lattices into three-generated lattices, enhancing classical theorems with finiteness, algebraicity, and completeness.

Findings

Finite lattices embed into three-generated finite lattices

Algebraic lattices with accessible cardinality embed into three-generated algebraic lattices

Strengthens classical lattice embedding results with additional properties

Abstract

We prove that every finite lattice L can be embedded in a three-generated finite lattice K. We also prove that every algebraic lattice with accessible cardinality is a complete sublattice of an appropriate algebraic lattice K such that K is completely generated by three elements. Note that ZFC has a model in which all cardinal numbers are accessible. Our results strengthen P. Crawley and R. A. Dean's 1959 results by adding finiteness, algebraicity, and completeness.