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The Homomorphism Lattice Induced by a Finite Algebra

  • Brian A. Davey
  • Charles T. Gray
  • Jane G. Pitkethly

DOI: 10.1007/s11083-017-9426-3

Cite this article as:
Davey, B.A., Gray, C.T. & Pitkethly, J.G. Order (2017). doi:10.1007/s11083-017-9426-3


Each finite algebra A induces a lattice LA via the quasi-order → on the finite members of the variety generated by A, where BC if there exists a homomorphism from B to C. In this paper, we introduce the question: ‘Which lattices arise as the homomorphism lattice LA induced by a finite algebra A?’ Our main result is that each finite distributive lattice arises as LQ, 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 L1, where L is an interval in the subgroup lattice of a finite group.


Homomorphism order Finitely generated variety Quasi-primal algebra Distributive lattice Covering forest 

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsLa Trobe UniversityVictoriaAustralia

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