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The complexity of homomorphism factorization


We investigate the computational complexity of the problem of deciding if an algebra homomorphism can be factored through an intermediate algebra. Specifically, we fix an algebraic language, \(\mathcal L\), and take as input an algebra homomorphism \(f:X\rightarrow Z\) between two finite \(\mathcal L\)-algebras X and Z, along with an intermediate finite \(\mathcal L\)-algebra Y. The decision problem asks whether there are homomorphisms \(g:X\rightarrow Y\) and \(h:Y\rightarrow Z\) such that \(f=hg\). We show that this problem is NP-complete for most languages.

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Correspondence to Kevin M. Berg.

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This material is based upon work supported by the National Science Foundation Grant no. DMS 1500254.

Communicated by Presented by R. Willard.

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Berg, K.M. The complexity of homomorphism factorization. Algebra Univers. 82, 47 (2021).

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  • Homomorphism factorization
  • NP-complete
  • Computational complexity
  • Finite structures

Mathematics Subject Classification

  • 68Q25
  • 03C05
  • 08A05