An (α,β)-Hesitant Fuzzy Set Approach to Ideal Theory in Semigroups

Pairote Yiarayong

doi:10.18778/0138-0680.2022.13

Authors

Pairote Yiarayong Pibulsongkram Rajabhat University, Department of Mathematics, Faculty of Science and Technology, Phitsanulok 65000, Thailand

DOI:

https://doi.org/10.18778/0138-0680.2022.13

Keywords:

^{α}

-hesitant (

_{α}

-hesitant) fuzzy set,

(α, β)

-hesitant fuzzy semigroup,

(α, β)

-hesitant fuzzy ideal,

(α, β)

-hesitant fuzzy semiprime set, regular semigroup

Abstract

The aim of this manuscript is to introduce the $(α, β)$ -hesitant fuzzy set and apply it to semigroups. In this paper, as a generalization of the concept of hesitant fuzzy sets to semigroup theory, the concept of $(α, β)$ -hesitant fuzzy subsemigroups of semigroups is introduced, and related properties are discussed. Furthermore, we define and study $(α, β)$ -hesitant fuzzy ideals on semigroups. In particular, we investigate the structure of $(α, β)$ -hesitant fuzzy ideal generated by a hesitant fuzzy ideal in a semigroup. In addition, we also introduce the concepts of $(α, β)$ -hesitant fuzzy semiprime sets of semigroups, and characterize regular semigroups in terms of $(α, β)$ -hesitant fuzzy left ideals and $(α, β)$ -hesitant fuzzy right ideals. Finally, several characterizations of regular and intra-regular semigroups by the properties of $(α, β)$ -hesitant ideals are given.

References

M. Y. Abbasi, A. F. Talee, S. A. Khan, K. Hila, A hesitant fuzzy set approach to ideal theory in Γ-semigroups, Advances in Fuzzy Systems, vol. 2018 (2018), p. 5738024, DOI: https://doi.org/10.1155/2018/5738024
Google Scholar DOI: https://doi.org/10.1155/2018/5738024

M. Y. Abbasi, A. F. Talee, X. Xie, S. A. Khan, Hesitant fuzzy ideal extension in po-semigroups, Journal of Applied and Engineering Mathematics, vol. 8(2) (2018), pp. 509–521, URL: http://jaem.isikun.edu.tr/web/images/articles/vol.8.no.2/16.pdf
Google Scholar

K. Arulmozhi, V. Chinnadurai, A. Swaminathan, Interval valued bipolar fuzzy ideals in ordered Γ-semigroups, Journal of International Mathematical Virtual Institute, vol. 9(2) (2019), pp. 1–17, URL: http://www.imvibl.org/journal/9_19/journal_imvi_89_2019_1_17.pdf
Google Scholar

M. Cheong, K. Hur, Interval-valued fuzzy ideals and bi-ideals of a semigroup, International Journal of Fuzzy Logic and Intelligent Systems, vol. 11(4) (2019), pp. 259–266, DOI: https://doi.org/10.5391/IJFIS.2011.11.4.259.
Google Scholar DOI: https://doi.org/10.5391/IJFIS.2011.11.4.259

M. Gulistan, M. Shahzad, F. Ahmad, M. Azam, S. Nawaz, Hesitant fuzzy Abel-Grassmann’s groupoids, Science International (Lahore), vol. 28(1) (2016), pp. 19–25, URL: http://www.sci-int.com/pdf/636911097513073130.pdf
Google Scholar

Y. B. Jun, S. S. Ahn, G. Muhiuddin, Hesitant fuzzy soft subalgebras and ideals in BCK/BCI-algebras, The Scientific World Journal, vol. 2014(Article ID 763929) (2014), pp. 1–8, DOI: https://doi.org/10.1155/2014/763929
Google Scholar DOI: https://doi.org/10.1155/2014/763929

Y. B. Jun, K. J. Lee, S. Z. Song, Hesitant fuzzy bi-ideals in semigroups, Communications of the Korean Mathematical Society, vol. 30(3) (2015), pp. 143–154, DOI: https://doi.org/10.4134/CKMS.2015.30.3.143
Google Scholar DOI: https://doi.org/10.4134/CKMS.2015.30.3.143

A. Khan, M. Z. A. Y. B. Jun, Characterizations of ordered semigroups in terms of (∈,∈∨q)-fuzzy interior ideals, Neural Computing and Applications, vol. 21 (2012), pp. 433–440, DOI: https://doi.org/10.1007/s00521-010-0463-8
Google Scholar DOI: https://doi.org/10.1007/s00521-010-0463-8

H. U. Khan, N. H. Sarmin, A. Khan, F. M. Khan, Classification of ordered semigroups in terms of generalized interval-valued fuzzy interior ideals, Journal of Intelligent Systems, vol. 25(2) (2016), pp. 297–318, DOI: https://doi.org/10.1515/jisys-2015-0035
Google Scholar DOI: https://doi.org/10.1515/jisys-2015-0035

P. Mosrijai, A. Satirad, A. Iampan, New types of hesitant fuzzy sets on UP-algebras, Mathematica Moravica, vol. 22(2) (2018), pp. 29–39, DOI: https://doi.org/10.5937/MatMor1802029M
Google Scholar DOI: https://doi.org/10.5937/MatMor1802029M

G. Muhiuddin, S. Aldhafeeri, Subalgebras and ideals in BCK/BCI-algebras based on Uni-hesitant fuzzy set theory, European Journal of Pure and Applied Mathematics, vol. 11(2) (2018), pp. 417–430, DOI: https://doi.org/10.29020/nybg.ejpam.v11i2.3246
Google Scholar DOI: https://doi.org/10.29020/nybg.ejpam.v11i2.3246

R. M. Rodriguez, L. Martinez, F. Herrera, Hesitant fuzzy linguistic term sets for decision making, IEEE Transactions on Fuzzy Systems, vol. 20(1) (2012), pp. 109–119, DOI: https://doi.org/10.1109/TFUZZ.2011.2170076
Google Scholar DOI: https://doi.org/10.1109/TFUZZ.2011.2170076

S. Z. Song, H. Bordbar, Y. B. Jun, A new type of hesitant fuzzy subalgebras and ideals in BCK/BCI-algebras, Journal of Intelligent and Fuzzy Systems, vol. 32 (2017), pp. 2009–2016, DOI: https://doi.org/10.3233/JIFS-161601
Google Scholar DOI: https://doi.org/10.3233/JIFS-161601

A. F. Talee, M. Y. Abbasi, S. A. Khan, Hesitant fuzzy ideals in semigroups with two frontiers, Journal of Basic and Applied Engineering Research, vol. 4(6) (2017), pp. 385–388.
Google Scholar

J. Tang, B. Davvaz, X. Xie, A study on (fuzzy) quasi-Γ-hyperideals in ordered Γ-semihypergroups, Journal of Intelligent and Fuzzy Systems, vol. 32 (2017), pp. 3821–3838, DOI: https://doi.org/10.3233/IFS-162117
Google Scholar DOI: https://doi.org/10.3233/IFS-162117

F. Q. Wang, X. Li, X. H. Chen, Hesitant fuzzy soft set and its applications in multicriteria decision making, Journal of Applied Mathematics, vol. 2014(Article ID 643785) (2014), pp. 1–10, DOI: https://doi.org/10.1155/2014/643785
Google Scholar DOI: https://doi.org/10.1155/2014/643785

P. Yairayong, Applications of hesitant fuzzy sets to completely regular semigroups, IAENG International Journal of Computer Science, vol. 47(1) (2020), pp. 107–118, URL: http://www.iaeng.org/IJCS/issues_v47/issue_1/IJCS_47_1_13.pdf
Google Scholar

N. Yaqoob, Interval valued intuitionistic fuzzy ideals of regular LA-semigroups, Thai Journal of Mathematics, vol. 11(3) (2013), pp. 683–695, URL: http://thaijmath.in.cmu.ac.th/index.php/thaijmath/article/view/735/566
Google Scholar

P. Yiarayong, Applications of hesitant fuzzy sets to ternary semigroups, Heliyon, vol. 6(4) (2020), p. e03668, DOI: https://doi.org/10.1016/j.heliyon.2020.e03668
Google Scholar DOI: https://doi.org/10.1016/j.heliyon.2020.e03668

P. Yiarayong, Semigroup characterized by picture fuzzy sets, International Journal of Innovative Computing, Information and Control, vol. 16(6) (2020), pp. 2121–2130, DOI: https://doi.org/10.24507/ijicic.16.06.2121
Google Scholar