Skip to main content

Ideals Associated to Poset Homomorphisms: A Survey

  • Chapter
  • First Online:
Homological and Computational Methods in Commutative Algebra

Part of the book series: Springer INdAM Series ((SINDAMS,volume 20))

Abstract

In this survey, we give an overview to the various known algebraic properties and invariants of ideals of poset homomorphisms. A particular attention lies on classical related notions that occur as special cases.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. A. Aramova, J. Herzog, T. Hibi, Finite lattices and lexicographic Gröbner bases. Eur. J. Combin. 21(4), 431–439 (2000)

    Article  MATH  Google Scholar 

  2. M. Bigdeli, J. Herzog, T. Hibi, A.A. Qureshi, A. Shikama, Isotonian algebras (2015). arXiv preprint arXiv:1512.01973

    Google Scholar 

  3. G. Birkhoff, Lattice Theory. American Mathematical Society Colloquium Publications, Vol. 25, 3rd edn. (American Mathematical Society, Providence, 1979), vi+418 pp

    Google Scholar 

  4. A. D’Alì, G. Fløystad, A. Nematbakhsh, Resolutions of co-letterplace ideals and generalizations of bier spheres (2016). arXiv preprint arXiv:1601.02793

    Google Scholar 

  5. A. D’Alì, G. Fløystad, A. Nematbakhsh, Resolutions of letterplace ideals of posets (2016). arXiv preprint arXiv:1601.02792

    Google Scholar 

  6. V. Ene, T. Hibi, The join-meet ideal of a finite lattice. J. Commut. Algebra 5(2), 209–230 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. V. Ene, J. Herzog, F. Mohammadi, Monomial ideals and toric rings of Hibi type arising from a finite poset. Eur. J. Comb. 32(3), 404–421 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. V. Ene, J. Herzog, T. Hibi, S. Saeedi Madani, Pseudo-gorenstein and level hibi rings. J. Algebra 431, 138–161 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. V. Ene, J. Herzog, S.S. Madani, A note on the regularity of hibi rings. Manuscripta Mathematica 148(3), 501–506 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. G. Fløystad, B.M. Greve, J. Herzog, Letterplace and co-letterplace ideals of posets (2015). arXiv preprint arXiv:1501.04523

    Google Scholar 

  11. J. Herzog, T. Hibi, Distributive lattices, bipartite graphs and alexander duality. J. Algebraic Combin. 22(3), 289–302 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. J. Herzog, T. Hibi, Monomial Ideals (Springer, Berlin, 2010)

    MATH  Google Scholar 

  13. J. Herzog, M. Steurich, Golodideale der Gestalt \(\mathfrak{a} \cap \mathfrak{b}\). J. Algebra 58(1), 31–36 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  14. J. Herzog, V. Reiner, V. Welker, Componentwise linear ideals and Golod rings. Mich. Math. J. 46(2), 211–223 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  15. J. Herzog, A.A. Qureshi, A. Shikama, Alexander duality for monomial ideals associated with isotone maps between posets. J. Algebra Appl. (2015, to appear). arXiv preprint arXiv:1504.01520

    Google Scholar 

  16. J. Herzog, A.A. Qureshi, A. Shikama, On the relations of isotonian algebras. Proc. Amer. Math. Soc. (2017). doi:https://doi.org/10.1090/proc/13502

  17. T. Hibi, Level rings and algebras with straightening laws. J. Algebra 117(2), 343–362 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  18. M. Jöllenbeck, On the multigraded Hilbert and Poincaré–Betti series and the Golod property of monomial rings. J. Pure Appl. Algebra 207(2), 261–298 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. M. Juhnke-Kubitzke, L. Katthän, S. Saeedi Madani, Algebraic properties of ideals of poset homomorphisms. J. Algebraic Combin. 44, 1–28 (2016)

    Article  MATH  Google Scholar 

  20. M. Miyazaki, A sufficient condition for a hibi ring to be level and levelness of schubert cycles. Commun. Algebra 35(9), 2894–2900 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. I. Peeva, Graded Syzygies (Springer, Berlin, 2010)

    MATH  Google Scholar 

  22. R.P. Stanley, Hilbert functions of graded algebras. Adv. Math. 28(1), 57–83 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  23. R.P. Stanley, Combinatorics and Commutative Algebra, 2nd edn. (Birkhäuser, Boston, 1996)

    MATH  Google Scholar 

Download references

Acknowledgements

Both authors were supported by the German Research Council DFG-GRK 1916.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Martina Juhnke-Kubitzke .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Juhnke-Kubitzke, M., Madani, S.S. (2017). Ideals Associated to Poset Homomorphisms: A Survey. In: Conca, A., Gubeladze, J., Römer, T. (eds) Homological and Computational Methods in Commutative Algebra. Springer INdAM Series, vol 20. Springer, Cham. https://doi.org/10.1007/978-3-319-61943-9_8

Download citation

Publish with us

Policies and ethics