The homomorphism lattice induced by a finite algebra

2016/03/01 by Brian A. Davey, Davey, Brian A., Charles T. Gray +3
Mathematics · #06B15 #08A40 #08B25 #FOS: Mathematics #Rings and Algebras (math.RA) #math.RA #msc:06B15 #msc:08A40 #msc:08B25

paper · pdf · doi:10.48550/arxiv.1603.00129

arxiv created 2016/12/19 · arxiv updated 2016/12/20

Abstract

Each finite algebra \mathbf A induces a lattice~\mathbf L_{\mathbf A} via the quasi-order~→ on the finite members of the variety generated by~\mathbf A, where \mathbf B → \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⊕ \mathbf 1, where \mathbf L is an interval in the subgroup lattice of a finite group.

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