The Homomorphism Lattice Induced by a Finite Algebra

Each finite algebra A induces a lattice L A via the quasi-order → on the finite members of the variety generated by A , where B → C if there exists a homomorphism from B to C . In this paper, we introduce the question: ‘Which lattices arise as the homomorphism lattice L A induced by a finite algebra A ?’ Our main result is that each finite distributive lattice arises as L Q , for some quasi-primal algebra 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 L ⊕ 1 , where L is an interval in the subgroup lattice of a finite group.