On Freese's technique

TL;DR

This paper investigates Freese's technique for embedding specific lattices into the congruence lattices of algebras, providing new conditions for certain lattice families to be admissible as sublattices within algebraic varieties.

Contribution

It extends previous unpublished results by establishing sufficient conditions for rods and snakes lattices to be sublattices of varieties generated by a given algebra.

Findings

Sufficient conditions for rods and snakes lattices to be sublattices.

Extension of Freese and Lipparini's unpublished results.

Application of Freese's technique to identify lattice embeddings.

Abstract

In this paper we explore some applications of a certain technique (that we call the Freese's technique), which is a tool for identifying certain lattices as sublattices of the congruence lattice of a given algebra. In particular we will give sufficient conditions for two family of lattices (called the rods and the snakes) to be admissible as sublattices of a variety generated by a given algebra, extending an unpublished result of R. Freese and P. Lipparini.