Topological Structure of Fuzzy PU-new Ideal in PU-algebra

Muhammad Shafiq; Naveed Sheikh; Dawood Khan

doi:doi:10.11648/j.ijssam.20210604.13

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Topological Structure of Fuzzy PU-new Ideal in PU-algebra

Muhammad Shafiq, Naveed Sheikh, Dawood Khan

Published in International Journal of Systems Science and Applied Mathematics (Volume 6, Issue 4)

Received: 6 November 2021 Accepted: 17 December 2021 Published: 31 December 2021

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Abstract

In this manuscript, we consider the fuzzification of the notion PU-new ideal of PU-algebra and define the fuzzy topological terms such that fuzzy topology, τ-open fuzzy set, fuzzy neighborhood, fuzzy interior, sequence of fuzzy sets, fuzzy neighborhood system, fuzzy continuity of a function with respect to PU-new ideal of PU-algebra. We explore the new theorems and related properties of above mention notions with respect to PU-new ideal on PU-algebras. Such that for (Ⱬ, τ) to be a TSFP on Ⱬ and the set Ḟ is a fuzzy in Ⱬ and N_Ḟ be a fuzzy neighborhood system of Ḟ then the finite intersection of elements of ‘N_Ḟ’ is also an element of ‘N_Ḟ’ also any fuzzy set of Ⱬ which contains an element of “N_Ḟ” is also an element of “N_Ḟ”. Furthermore we prove the conditions with respect to fuzzy neighborhood, convergence of a sequence of fuzzy sets, fuzzy interior set of a fuzzy set under which a fuzzy set Ḟ is τ-open. We show that how the function Ψ from (Ⱬ₁,τ) to (Ⱬ₂,ω) is fuzzy continuous. We prove that if Ψ is a fuzzy continuous function then for every fuzzy set Ḟ in Ⱬ₁, inverse of each neighborhood of Ψ(Ḟ) is a neighborhood of a fuzzy set Ḟ.

Published in	International Journal of Systems Science and Applied Mathematics (Volume 6, Issue 4)
DOI	10.11648/j.ijssam.20210604.13
Page(s)	125-130
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2021. Published by Science Publishing Group

Keywords

PU-algebra, Fuzzy PU-new Ideal, Fuzzy Topology, Topology of Fuzzy PU-new Ideal, Fuzzy Continuity

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Muhammad Shafiq, Naveed Sheikh, Dawood Khan. (2021). Topological Structure of Fuzzy PU-new Ideal in PU-algebra. International Journal of Systems Science and Applied Mathematics, 6(4), 125-130. https://doi.org/10.11648/j.ijssam.20210604.13

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ACS Style

Muhammad Shafiq; Naveed Sheikh; Dawood Khan. Topological Structure of Fuzzy PU-new Ideal in PU-algebra. Int. J. Syst. Sci. Appl. Math. 2021, 6(4), 125-130. doi: 10.11648/j.ijssam.20210604.13

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AMA Style

Muhammad Shafiq, Naveed Sheikh, Dawood Khan. Topological Structure of Fuzzy PU-new Ideal in PU-algebra. Int J Syst Sci Appl Math. 2021;6(4):125-130. doi: 10.11648/j.ijssam.20210604.13

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@article{10.11648/j.ijssam.20210604.13,
  author = {Muhammad Shafiq and Naveed Sheikh and Dawood Khan},
  title = {Topological Structure of Fuzzy PU-new Ideal in PU-algebra},
  journal = {International Journal of Systems Science and Applied Mathematics},
  volume = {6},
  number = {4},
  pages = {125-130},
  doi = {10.11648/j.ijssam.20210604.13},
  url = {https://doi.org/10.11648/j.ijssam.20210604.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijssam.20210604.13},
  abstract = {In this manuscript, we consider the fuzzification of the notion PU-new ideal of PU-algebra and define the fuzzy topological terms such that fuzzy topology, τ-open fuzzy set, fuzzy neighborhood, fuzzy interior, sequence of fuzzy sets, fuzzy neighborhood system, fuzzy continuity of a function with respect to PU-new ideal of PU-algebra. We explore the new theorems and related properties of above mention notions with respect to PU-new ideal on PU-algebras. Such that for (Ⱬ, τ) to be a TSFP on Ⱬ and the set Ḟ is a fuzzy in Ⱬ and NḞ be a fuzzy neighborhood system of Ḟ then the finite intersection of elements of ‘NḞ’ is also an element of ‘NḞ’ also any fuzzy set of Ⱬ which contains an element of “NḞ” is also an element of “NḞ”. Furthermore we prove the conditions with respect to fuzzy neighborhood, convergence of a sequence of fuzzy sets, fuzzy interior set of a fuzzy set under which a fuzzy set Ḟ is τ-open. We show that how the function Ψ from (Ⱬ1,τ) to (Ⱬ2,ω) is fuzzy continuous. We prove that if Ψ is a fuzzy continuous function then for every fuzzy set Ḟ in Ⱬ1, inverse of each neighborhood of Ψ(Ḟ) is a neighborhood of a fuzzy set Ḟ.},
 year = {2021}
}

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TY  - JOUR
T1  - Topological Structure of Fuzzy PU-new Ideal in PU-algebra
AU  - Muhammad Shafiq
AU  - Naveed Sheikh
AU  - Dawood Khan
Y1  - 2021/12/31
PY  - 2021
N1  - https://doi.org/10.11648/j.ijssam.20210604.13
DO  - 10.11648/j.ijssam.20210604.13
T2  - International Journal of Systems Science and Applied Mathematics
JF  - International Journal of Systems Science and Applied Mathematics
JO  - International Journal of Systems Science and Applied Mathematics
SP  - 125
EP  - 130
PB  - Science Publishing Group
SN  - 2575-5803
UR  - https://doi.org/10.11648/j.ijssam.20210604.13
AB  - In this manuscript, we consider the fuzzification of the notion PU-new ideal of PU-algebra and define the fuzzy topological terms such that fuzzy topology, τ-open fuzzy set, fuzzy neighborhood, fuzzy interior, sequence of fuzzy sets, fuzzy neighborhood system, fuzzy continuity of a function with respect to PU-new ideal of PU-algebra. We explore the new theorems and related properties of above mention notions with respect to PU-new ideal on PU-algebras. Such that for (Ⱬ, τ) to be a TSFP on Ⱬ and the set Ḟ is a fuzzy in Ⱬ and NḞ be a fuzzy neighborhood system of Ḟ then the finite intersection of elements of ‘NḞ’ is also an element of ‘NḞ’ also any fuzzy set of Ⱬ which contains an element of “NḞ” is also an element of “NḞ”. Furthermore we prove the conditions with respect to fuzzy neighborhood, convergence of a sequence of fuzzy sets, fuzzy interior set of a fuzzy set under which a fuzzy set Ḟ is τ-open. We show that how the function Ψ from (Ⱬ1,τ) to (Ⱬ2,ω) is fuzzy continuous. We prove that if Ψ is a fuzzy continuous function then for every fuzzy set Ḟ in Ⱬ1, inverse of each neighborhood of Ψ(Ḟ) is a neighborhood of a fuzzy set Ḟ.
VL  - 6
IS  - 4
ER  -

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Author Information

Muhammad Shafiq

Mathematics Department, Faculty of Mathematics Department, University of Balochistan, Quetta, Pakistan
Naveed Sheikh

Mathematics Department, Faculty of Mathematics Department, University of Balochistan, Quetta, Pakistan
Dawood Khan

Mathematics Department, Faculty of Mathematics Department, University of Balochistan, Quetta, Pakistan

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APA Style

Muhammad Shafiq, Naveed Sheikh, Dawood Khan. (2021). Topological Structure of Fuzzy PU-new Ideal in PU-algebra. International Journal of Systems Science and Applied Mathematics, 6(4), 125-130. https://doi.org/10.11648/j.ijssam.20210604.13

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ACS Style

Muhammad Shafiq; Naveed Sheikh; Dawood Khan. Topological Structure of Fuzzy PU-new Ideal in PU-algebra. Int. J. Syst. Sci. Appl. Math. 2021, 6(4), 125-130. doi: 10.11648/j.ijssam.20210604.13

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AMA Style

Muhammad Shafiq, Naveed Sheikh, Dawood Khan. Topological Structure of Fuzzy PU-new Ideal in PU-algebra. Int J Syst Sci Appl Math. 2021;6(4):125-130. doi: 10.11648/j.ijssam.20210604.13

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@article{10.11648/j.ijssam.20210604.13,
  author = {Muhammad Shafiq and Naveed Sheikh and Dawood Khan},
  title = {Topological Structure of Fuzzy PU-new Ideal in PU-algebra},
  journal = {International Journal of Systems Science and Applied Mathematics},
  volume = {6},
  number = {4},
  pages = {125-130},
  doi = {10.11648/j.ijssam.20210604.13},
  url = {https://doi.org/10.11648/j.ijssam.20210604.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijssam.20210604.13},
  abstract = {In this manuscript, we consider the fuzzification of the notion PU-new ideal of PU-algebra and define the fuzzy topological terms such that fuzzy topology, τ-open fuzzy set, fuzzy neighborhood, fuzzy interior, sequence of fuzzy sets, fuzzy neighborhood system, fuzzy continuity of a function with respect to PU-new ideal of PU-algebra. We explore the new theorems and related properties of above mention notions with respect to PU-new ideal on PU-algebras. Such that for (Ⱬ, τ) to be a TSFP on Ⱬ and the set Ḟ is a fuzzy in Ⱬ and NḞ be a fuzzy neighborhood system of Ḟ then the finite intersection of elements of ‘NḞ’ is also an element of ‘NḞ’ also any fuzzy set of Ⱬ which contains an element of “NḞ” is also an element of “NḞ”. Furthermore we prove the conditions with respect to fuzzy neighborhood, convergence of a sequence of fuzzy sets, fuzzy interior set of a fuzzy set under which a fuzzy set Ḟ is τ-open. We show that how the function Ψ from (Ⱬ1,τ) to (Ⱬ2,ω) is fuzzy continuous. We prove that if Ψ is a fuzzy continuous function then for every fuzzy set Ḟ in Ⱬ1, inverse of each neighborhood of Ψ(Ḟ) is a neighborhood of a fuzzy set Ḟ.},
 year = {2021}
}

Copy | Download

TY  - JOUR
T1  - Topological Structure of Fuzzy PU-new Ideal in PU-algebra
AU  - Muhammad Shafiq
AU  - Naveed Sheikh
AU  - Dawood Khan
Y1  - 2021/12/31
PY  - 2021
N1  - https://doi.org/10.11648/j.ijssam.20210604.13
DO  - 10.11648/j.ijssam.20210604.13
T2  - International Journal of Systems Science and Applied Mathematics
JF  - International Journal of Systems Science and Applied Mathematics
JO  - International Journal of Systems Science and Applied Mathematics
SP  - 125
EP  - 130
PB  - Science Publishing Group
SN  - 2575-5803
UR  - https://doi.org/10.11648/j.ijssam.20210604.13
AB  - In this manuscript, we consider the fuzzification of the notion PU-new ideal of PU-algebra and define the fuzzy topological terms such that fuzzy topology, τ-open fuzzy set, fuzzy neighborhood, fuzzy interior, sequence of fuzzy sets, fuzzy neighborhood system, fuzzy continuity of a function with respect to PU-new ideal of PU-algebra. We explore the new theorems and related properties of above mention notions with respect to PU-new ideal on PU-algebras. Such that for (Ⱬ, τ) to be a TSFP on Ⱬ and the set Ḟ is a fuzzy in Ⱬ and NḞ be a fuzzy neighborhood system of Ḟ then the finite intersection of elements of ‘NḞ’ is also an element of ‘NḞ’ also any fuzzy set of Ⱬ which contains an element of “NḞ” is also an element of “NḞ”. Furthermore we prove the conditions with respect to fuzzy neighborhood, convergence of a sequence of fuzzy sets, fuzzy interior set of a fuzzy set under which a fuzzy set Ḟ is τ-open. We show that how the function Ψ from (Ⱬ1,τ) to (Ⱬ2,ω) is fuzzy continuous. We prove that if Ψ is a fuzzy continuous function then for every fuzzy set Ḟ in Ⱬ1, inverse of each neighborhood of Ψ(Ḟ) is a neighborhood of a fuzzy set Ḟ.
VL  - 6
IS  - 4
ER  -

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