The homomorphism lattice induced by a finite algebra

TL;DR

This paper investigates which finite lattices can be represented as homomorphism lattices induced by finite algebras, showing that all finite distributive lattices can be realized through quasi-primal algebras.

Contribution

It proves that every finite distributive lattice is realizable as a homomorphism lattice from some quasi-primal algebra, expanding understanding of the structure of such lattices.

Findings

All finite distributive lattices are homomorphism lattices of quasi-primal algebras.

Representation of some classes of lattices as homomorphism lattices, including partition and subspace lattices.

Lattices of the form L⊕1, with L an interval in a subgroup lattice of a finite group, are also realizable.

Abstract

Each finite algebra $\mathbf A$ induces a lattice~$\mathbf L_{\mathbf A}$ via the quasi-order~$\to$ on the finite members of the variety generated by~$\mathbf A$, where $\mathbf B \to \mathbf C$ if there exists a homomorphism from $\mathbf B$ to~$\mathbf C$. In this paper, we introduce the question: `Which lattices arise as the homomorphism lattice $\mathbf L_{\mathbf A}$ induced by a finite algebra $\mathbf A$?' Our main result is that each finite distributive lattice arises as~$\mathbf L_{\mathbf Q}$, for some quasi-primal algebra~$\mathbf Q$. We also obtain representations of some other classes of lattices as homomorphism lattices, including all finite partition lattices, all finite subspace lattices and all lattices of the form $\mathbf L\oplus \mathbf 1$, where $\mathbf L$ is an interval in the subgroup lattice of a finite group.