A soft relation approach to approximate the spherical fuzzy ideals of semigroups

Rabia Mazhar; Shahida Bashir; Muhammad Shabir; Mohammed Al-Shamiri; Rabia Mazhar; Shahida Bashir; Muhammad Shabir; Mohammed Al-Shamiri

doi:10.3934/math.2025173

AIMS Mathematics

2025, Volume 10, Issue 2: 3734-3758. doi: 10.3934/math.2025173

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Research article

A soft relation approach to approximate the spherical fuzzy ideals of semigroups

1.
Department of Mathematics, University of Gujrat, Gujrat 50700, Pakistan
2.
Department of Mathematics, Quaid-i-Azam University, Islamabad 45320, Pakistan
3.
Department of Mathematics, Faculty of Science and Arts, King Khalid University, Abha, Saudi Arabia

Received: 03 November 2024 Revised: 18 January 2025 Accepted: 06 February 2025 Published: 25 February 2025
MSC : 03E72, 18B40

Our main objective of this work was to study rough approximations of spherical fuzzy ideals by using soft relations that are free from all the complexities that are faced by many scientists. Furthermore, lower and upper approximations of spherical fuzzy subsemigroups, spherical fuzzy left (right) ideals, spherical fuzzy interior ideals, and spherical fuzzy bi-ideals of semigroups were studied using soft relations. Mainly, we proved, for a spherical fuzzy ideal of the universe, that upper and lower approximations are spherical fuzzy soft ideals but the converse may not hold, as shown by examples. Compatible relations and complete relations are needed for upper approximations and lower approximations, respectively. Also, using examples, we showed that the conditions of complete relations were necessary for lower approximations. Last, a comparison study and conclusions of the introduced technique are given, demonstrating how our work is superior and efficient in contrast to other techniques.
- spherical fuzzy set,
- soft set,
- spherical fuzzy subsemigroup,
- spherical fuzzy ideal,
- soft relation
Citation: Rabia Mazhar, Shahida Bashir, Muhammad Shabir, Mohammed Al-Shamiri. A soft relation approach to approximate the spherical fuzzy ideals of semigroups[J]. AIMS Mathematics, 2025, 10(2): 3734-3758. doi: 10.3934/math.2025173

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Abstract

Our main objective of this work was to study rough approximations of spherical fuzzy ideals by using soft relations that are free from all the complexities that are faced by many scientists. Furthermore, lower and upper approximations of spherical fuzzy subsemigroups, spherical fuzzy left (right) ideals, spherical fuzzy interior ideals, and spherical fuzzy bi-ideals of semigroups were studied using soft relations. Mainly, we proved, for a spherical fuzzy ideal of the universe, that upper and lower approximations are spherical fuzzy soft ideals but the converse may not hold, as shown by examples. Compatible relations and complete relations are needed for upper approximations and lower approximations, respectively. Also, using examples, we showed that the conditions of complete relations were necessary for lower approximations. Last, a comparison study and conclusions of the introduced technique are given, demonstrating how our work is superior and efficient in contrast to other techniques.

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