The homomorphism lattice induced by a finite algebra

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.