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The homomorphism lattice induced by a finite algebra
Each finite algebra induces a lattice~ via
the quasi-order~ on the finite members of the variety generated
by~, where if there exists a homomorphism
from to~. In this paper, we introduce the question:
`Which lattices arise as the homomorphism lattice
induced by a finite algebra ?' Our main result is that each finite
distributive lattice arises as~, for some quasi-primal
algebra~. 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 , where is an interval in the subgroup lattice of a finite group