(∈, ∈ Vq)-Fuzzy Ideals in Semirings

  • Javed Ahsan
  • John N. Mordeson
  • Mohammad Shabir
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 278)


Following Bhakat and Das [21], Dudek et. al. [47], Ma and Zhan [106] defined (∈, ∈ Vq)-fuzzy ideals in semirings. In this chapter we have studied properties of these fuzzy ideals. In Section 1, we look at (α,β)-fuzzy ideals, and show that a fuzzy left (right) ideal of a semiring is an (∈, ∈)-fuzzy left (right) ideal. Section 2 provides a characterization of (∈, ∈ Vq)-fuzzy left (right) ideals, quasi and bi-ideals of semirings. Section 3 presents characterizations of regular semirings involving these ideals and Section 4 contains characterizations of regular and intra regular semirings by these ideals. Section 5 presents a study of (∈, ∈ Vq)-fuzzy k-ideals, k-quasi-ideals and k-bi-ideals of semirings. A study of k-regular and k-intra regular semirings in this context is separately made in Sections 6 and 7.

Throughout this chapter R is a semiring with zero element.


Computational Intelligence Positive Real Number Nonempty Subset Left Ideal Fuzzy Subset 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag GmbH Berlin Heidelberg 2012

Authors and Affiliations

  • Javed Ahsan
    • 1
  • John N. Mordeson
    • 2
  • Mohammad Shabir
    • 3
  1. 1.Department of Mathematics Wah Engineering CollegeUniversity of WahWah CanttPakistan
  2. 2.Department of MathematicsCreighton UniversityOmahaUSA
  3. 3.Department of MathematicsQuad-i-Azam UniversityIslamabadPakistan

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