The algebraic and lattice structures of type-2 intuitionistic fuzzy sets

  • Zia Bashir
  • M. G. Abbas Malik
  • Faisal Afridi
  • Tabasam RashidEmail author


Type-2 intuitionistic fuzzy sets are proposed as functions from non empty set U to \({\mathbf {T}}^{\mathbf {T}}\) where \({\mathbf {T}}=\{(\mu ,\nu ):\mu +\nu \le 1,\mu \ge 0,\nu \ge 0\}\) and \({\mathbf {T}}^{\mathbf {T}}\) is the set of all mappings from \({\mathbf {T}}\) to \({\mathbf {T}}\). The members of \({\mathbf {T}}^{\mathbf {T}}\) are called intuitionistic fuzzy values (IFV). In this paper, we develop a mathematical framework for IFVs by defining a set of generalized operations on \({\mathbf {T}}^{\mathbf {T}}\) and proved it to be an algebra. The other important properties like convexity, normality of IFVs and many important subalgebras are also explored and studied. Furthermore, two partial orders based on generalized operations are defined, which enable us to study the lattices in \({\mathbf {T}}^{\mathbf {T}}\).


Type-2 fuzzy sets Type-2 intuitionistic fuzzy sets Intuitionistic fuzzy values Algebra 

Mathematics Subject Classification

03B52 47S40 46S40 



The authors would like to thank the editors and the anonymous reviewers, whose insightful comments and constructive suggestions helped us to significantly improve the quality of this paper.


This study is not funded.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


It is submitted with the consent of all the authors.


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Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019

Authors and Affiliations

  • Zia Bashir
    • 1
  • M. G. Abbas Malik
    • 2
  • Faisal Afridi
    • 1
  • Tabasam Rashid
    • 3
    Email author
  1. 1.Department of MathematicsQuaid-i-Azam UniversityIslamabadPakistan
  2. 2.Universal College of LearningPalmerston NorthNew Zealand
  3. 3.Department of MathematicsUniversity of Management and TechnologyLahorePakistan

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