Algebra universalis

, 79:11 | Cite as

Pseudocomplementation and minimal prime ideals in semirings

  • Peyman Nasehpour


In Section 2 of the present paper, we introduce the concept of pseudocomplementation for semirings and show the semiring version of some known results in lattice theory. We also introduce semirings with pc-functions and prove some interesting results for minimal prime ideals of such semirings. In Section 3, some classical results for minimal prime ideals in ring theory are generalized in the context of semiring theory.


Semiring Bounded distributive lattice Minimal prime ideal Pseudocomplemented elements Stone elements Dense elements 

Mathematics Subject Classification

16Y60 06D15 13A15 


  1. 1.
    Ahsan, J., Mordeson, J.N., Shabir, N.: Fuzzy Semirings with Applications to Automata Theory. Springer, Berlin (2012)CrossRefzbMATHGoogle Scholar
  2. 2.
    Cīrulis, J.: Pseudocomplements in sum-ordered partial semirings. Discuss. Math. Gen. Algebra Appl. 27(2), 169–186 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Düntsch, I., Winter, M.: Weak contact structures. In: MacCaull, W., Winter, M., Düntsch, I. (eds.) Relational Methods in Computer Science. RelMiCS 2005. Lecture Notes in Computer Science, vol. 3929, pp. 73–82. Springer, Berlin (2006)CrossRefGoogle Scholar
  4. 4.
    Golan, J.S.: Semirings and Their Applications. Kluwer Academic Publishers, Dordrecht (1999)CrossRefzbMATHGoogle Scholar
  5. 5.
    Golan, J.S.: Semirings and Affine Equations Over Them: Theory and Applications. Kluwer Academic Publishers, Dordrecht (2003)CrossRefzbMATHGoogle Scholar
  6. 6.
    Gondran, M., Minoux, M.: Graphs, Dioids and Semirings. Springer, New York (2008)zbMATHGoogle Scholar
  7. 7.
    Grätzer, G.: Lattice Theory: Foundation. Birkhäuser, Basel (2011)CrossRefzbMATHGoogle Scholar
  8. 8.
    Guttmann, W.: Relation-algebraic verification of prim’s minimum spanning tree algorithm. In: Sampaio, A., Wang, F. (eds.) Theoretical Aspects of Computing ICTAC 2016. Lecture Notes in Computer Science, vol. 9965, pp. 51–68. Springer, Cham (2016)CrossRefGoogle Scholar
  9. 9.
    Hebisch, U., Weinert, H.J.: Semirings—Algebraic Theory and Applications in Computer Science. World Scientific, Singapore (1998)zbMATHGoogle Scholar
  10. 10.
    Huckaba, J.A.: Commutative Rings with Zero Divisors. Marcel Dekker, New York (1988)zbMATHGoogle Scholar
  11. 11.
    Itenberg, I., Mikhalkin, G., Shustin, E.I.: Tropical Algebraic Geometry. Oberwolfach Seminars, vol. 35. Birkhäuser, Basel (2009)CrossRefzbMATHGoogle Scholar
  12. 12.
    Jackson, M., Stockes, T.: Semilattice pseudo-complements on semigroups. Commun. Algebra 32(8), 2895–2918 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kaplansky, I.: Commutative Rings. Allyn and Bacon, Boston (1970)zbMATHGoogle Scholar
  14. 14.
    Manes, E.G., Arbib, M.A.: Algebraic Approaches to Program Semantics. Springer, Berlin (1986)CrossRefzbMATHGoogle Scholar
  15. 15.
    Matlis, E.: The minimal prime spectrum of a reduced ring. Ill. J. Math. 27(3), 353–391 (1983)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Matsumura, H.: Commutative Ring Theory, Transl. from the Japanese by M. Reid., 2nd edn. In: Cambridge Studies in Advanced Mathematics, vol. 8. Cambridge University Press, Cambridge (1989)Google Scholar
  17. 17.
    Mikhalkin, G., Rau, J.: Tropical Geometry, Book in preparation, (preprint) (2015).
  18. 18.
    Monk, J.D.: Introduction to Set Theory. McGraw-Hill Book Co., New York (1969)zbMATHGoogle Scholar
  19. 19.
    Rasiowa, H., Sikorski, R.: The Mathematics of Metamathematics. Panstwowe Wydawnictwo Naukowe, Warszawa (1963)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Engineering ScienceGolpayegan University of TechnologyGolpayeganIran

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