Skip to main content
SpringerLink
Log in

The complexity of homomorphism factorization

  • Published:
Algebra universalis Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. Bergman, C., Slutzki, G.: Complexity of some problems concerning varieties and quasi-varieties of algebras. SIAM J. Comput. 30(2), 359–382 (2000)

    Article  MathSciNet  Google Scholar 

  2. Booth, K.S.: Isomorphism testing for graphs, semigroups, and finite automata are polynomially equivalent problems. SIAM J. Comput. 7(3), 273–279 (1978)

    Article  MathSciNet  Google Scholar 

  3. Distler, A., Mitchell, J.D.: The number of nilpotent semigroups of degree 3. Electron. J. Comb. 19(2), P51 (2012)

  4. Flum, J., Grohe, M.: Parameterized Complexity Theory. Texts in Theoretical Computer Science. Springer, Berlin (2006)

    MATH  Google Scholar 

  5. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. A Series of Books in the Mathematical Sciences. W. H. Freeman and Co., San Francisco (1979)

    MATH  Google Scholar 

  6. Hedrlín, Z., Pultr, A.: On full embeddings of categories of algebras. Ill. J. Math. 10(3), 392–406 (1966)

    MathSciNet  MATH  Google Scholar 

  7. Hell, P., Nešetřil, J.: Graphs and Homomorphisms. Oxford Lecture Series in Mathematics and Its Applications. Oxford University Press, Oxford (2004)

    Book  Google Scholar 

  8. Hoffmann, C.M.: Group-Theoretic Algorithms and Graph Isomorphism. Lecture Notes in Computer Science, vol. 136. Springer, Berlin (1982)

    Book  Google Scholar 

  9. Jackson, M., McKenzie, R.: Interpreting graph colorability in finite semigroups. Int. J. Algebra Comput. 16(01), 119–140 (2006)

    Article  MathSciNet  Google Scholar 

  10. Torán, J.: Reductions to graph isomorphism. Theory Comput. Syst. 47(1), 288–299 (2010)

    Article  MathSciNet  Google Scholar 

  11. Van Name, J.: Is there a good computer program for searching for endomorphisms between finite algebras which make diagrams commute? Is this problem NP–complete? https://mathoverflow.net/questions/261966/. Accessed 14 Feb 2018

Download references

Author information

Author notes

  1. Kevin M. Berg

    Present address: Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic

Authors and Affiliations

  1. Department of Mathematics, University of Colorado, Boulder, CO, USA

    Kevin M. Berg

Authors
  1. Kevin M. Berg
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kevin M. Berg.

Additional information

Communicated by Presented by R. Willard.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Berg, K.M. The complexity of homomorphism factorization. Algebra Univers. 82, 47 (2021). https://doi.org/10.1007/s00012-021-00742-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00012-021-00742-5

Keywords

Mathematics Subject Classification

Search

Navigation