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

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Abstract

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 \({\mathbf {T}}^{\mathbf {T}}\).

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Acknowledgements

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.

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Authors and Affiliations

  1. Department of Mathematics, Quaid-i-Azam University, Islamabad, 45320, Pakistan

    Zia Bashir & Faisal Afridi

  2. Universal College of Learning, Palmerston North, 4442, New Zealand

    M. G. Abbas Malik

  3. Department of Mathematics, University of Management and Technology, Lahore, 54770, Pakistan

    Tabasam Rashid

Authors
  1. Zia Bashir
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  2. M. G. Abbas Malik
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  3. Faisal Afridi
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  4. Tabasam Rashid
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Correspondence to Tabasam Rashid.

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Communicated by Marcos Eduardo Valle.

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Bashir, Z., Malik, M.G.A., Afridi, F. et al. The algebraic and lattice structures of type-2 intuitionistic fuzzy sets. Comp. Appl. Math. 39, 26 (2020). https://doi.org/10.1007/s40314-019-1008-0

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  • DOI: https://doi.org/10.1007/s40314-019-1008-0

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