Constructing a Hoop Using Rough Filters

Rajab Ali Borzooei; Elham Babaei

doi:10.18778/0138-0680.2022.10

Authors

Rajab Ali Borzooei Shahid Beheshti University, Department of Mathematics, Faculty of Mathematical Sciences, Tehran 1983963113, Iran Author https://orcid.org/0000-0001-7538-7885
Elham Babaei Shahid Beheshti University, Department of Mathematics, Faculty of Mathematical Sciences, Tehran 1983963113, Iran Author

DOI:

https://doi.org/10.18778/0138-0680.2022.10

Keywords:

hoop, rough set, rough approximations (lower and upper), rough filter

Abstract

When it comes to making decisions in vague problems, rough is one of the best tools to help analyzers. So based on rough and hoop concepts, two kinds of approximations (Lower and Upper) for filters in hoops are defined, and then some properties of them are investigated by us. We prove that these approximations- lower and upper- are interior and closure operators, respectively. Also after defining a hyper operation in hoops, we show that by using this hyper operation, set of all rough filters is monoid. For more study, we define the implicative operation on the set of all rough filters and prove that this set with implication and intersection is made a hoop.

References

M. Aaly Kologani, R. A. Borzooei, On ideal theory of hoops, Mathematica Bohemica, vol. 145(2) (2020), pp. 141–162, DOI: https://doi.org/10.21136/MB.2019.0140-17 DOI: https://doi.org/10.21136/MB.2019.0140-17

M. Aaly Kologani, S. Z. Song, R. A. Borzooei, Y. B. Jun, Constructing some logical algebras with hoops, Mathematics, vol. 7 (2019), p. 1243, DOI: https://doi.org/10.3390/math7121243 DOI: https://doi.org/10.3390/math7121243

P. Aglianò, I. M. A. Ferreirim, F. Montagna, Basic hoops: An algebraic study of continuous t-norms, Studia Logica, vol. 87 (2007), pp. 73–98, DOI: https://doi.org/10.1007/s11225-007-9078-1 DOI: https://doi.org/10.1007/s11225-007-9078-1

R. Biswas, S. Nanda, Rough groups and rough subgroups, Bulletin of the Polish Academy of Sciences. Mathematics, vol. 42(3) (1994), pp. 251–254.

R. A. Borzooei, M. Aaly Kologani, Results on hoops, Journal of Algebraic Hyperstructures and Logical Algebras, vol. 1(1) (2020), pp. 61–77, DOI: https://doi.org/10.29252/HATEF.JAHLA.1.1.5 DOI: https://doi.org/10.29252/hatef.jahla.1.1.5

R. A. Borzooei, E. Babaei, Y. B. J. nad M. Aaly Kologani, M. Mohseni Takallo, Soft set theory applied to hoops, Analele Universitatii Ovidius Constanta-Seria Matematica, vol. 28(1) (2020), pp. 61–79, DOI: https://doi.org/10.2478/auom-2020-0004 DOI: https://doi.org/10.2478/auom-2020-0004

R. A. Borzooei, M. Sabetkish, E. H. Roh, M. Aaly Kologani, Int-soft filters in hoops, International Journal of Fuzzy Logic and Intelligent Systems, vol. 19(3) (2019), pp. 213–222, DOI: https://doi.org/10.5391/IJFIS.2019.19.3.213 DOI: https://doi.org/10.5391/IJFIS.2019.19.3.213

B. Bosbach, Komplementäre Halbgruppen. Kongruenzen and Quotienten, Fundamenta Mathematicae, vol. 69(1) (1970), pp. 1–14, URL: http://matwbn.icm.edu.pl/ksiazki/fm/fm69/fm6911.pdf DOI: https://doi.org/10.4064/fm-69-1-1-14

G. Georgescu, L. Leustean, V. Preoteasa, Pseudo-hoops, Journal of Multiple-Valued Logic and Soft Computing, vol. 11(1–2) (2005), pp. 153–184, URL: http://www.oldcitypublishing.com/journals/mvlsc-home/mvlsc-issue-contents/mvlsc-volume-11-number-1-2-2005/mvlsc-11-1-2-p-153-184/

P. Hájek, Metamathematics of Fuzzy Logic, Springer, vol. 4 (1998), DOI: https://doi.org/10.1007/978-94-011-5300-3 DOI: https://doi.org/10.1007/978-94-011-5300-3

T. B. Iwiński, Algebraic approach to rough sets, Bulletin of the Polish Academy of Sciences, vol. 42(3) (1994), pp. 251–254.

I. M. James, Introduction to Uniform Spaces, London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge (2013), DOI: https://doi.org/10.1017/CBO9780511721519 DOI: https://doi.org/10.1017/CBO9780511721519

Y. B. Jun, Roughness of ideals in BCK-algebras, Scientiae Mathematicae Japonicae Online, vol. 7 (2002), pp. 115–119, URL: https://www.jams.jp/scm/contents/Vol-7-2/7-13.pdf

Y. B. Jun, K. H. Kim, Rough set theory applied to BCC-algebras, International Mathematical Forum, vol. 2(41–44) (2007), pp. 2023–2029. DOI: https://doi.org/10.12988/imf.2007.07182

N. Kuroki, Rough ideals in semigroups, Information Sciences, vol. 100(1–4) (1997), pp. 139–163, DOI: https://doi.org/10.1016/S0020-0255(96)00274-5 DOI: https://doi.org/10.1016/S0020-0255(96)00274-5

N. Kuroki, J. Mordeson, Structure of rough sets and rough groups, Journal of Fuzzy Mathematics, vol. 5 (1997), pp. 183–191.

R. Rasoul, B. Davvaz, Roughness in MV-Algebra, Information Siences, vol. 180(5) (2010), pp. 737–747, DOI: https://doi.org/10.1016/j.ins.2009.11.008 DOI: https://doi.org/10.1016/j.ins.2009.11.008