Soju Filters in Hoop Algebras

Rajab Ali Borzooei; Gholam Reza Rezaei; Mona Aaly Kologhani; Young Bae  Jun

doi:10.18778/0138-0680.2020.28

Authors

Rajab Ali Borzooei Shahid Beheshti University, Department of Mathematics, Tehran, Iran http://orcid.org/0000-0001-7538-7885
Gholam Reza Rezaei University of Sistan and Baluchestan, Department of Mathematics, Zahedan, Iran http://orcid.org/0000-0001-5693-402X
Mona Aaly Kologhani Hatef Higher Education Institute Iran, Zahedan, Iran http://orcid.org/0000-0002-5234-2876
Young Bae Jun Shahid Beheshti University, Department of Mathematics, Tehran, Iran; Gyeongsang National University, Department of Mathematics Education, Jinju 52828, Korea http://orcid.org/0000-0002-0181-8969

DOI:

https://doi.org/10.18778/0138-0680.2020.28

Keywords:

Soju sub-hoop, soju filter, implicative soju filter

Abstract

The notions of (implicative) soju filters in a hoop algebra are introduced, and related properties are investigated. Relations between a soju sub-hoop, a soju filter and an implicative soju filter are discussed. Conditions for a soju filter to be implicative are displayed, and characterizations of an implicative soju filters are considered. The extension property of an implicative soju filter is established.

References

[1] K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, vol. 20(1) (1986), pp. 87–96, DOI: https://doi.org/10.1016/S0165-0114(86)80034-3
Google Scholar DOI: https://doi.org/10.1016/S0165-0114(86)80034-3

[2] K. T. Atanassov, New operations defined over the intuitionistic fuzzy sets, Fuzzy Sets and Systems, vol. 61(2) (1994), pp. 137–142, DOI: https://doi.org/10.1016/0165-0114(94)90229-1
Google Scholar DOI: https://doi.org/10.1016/0165-0114(94)90229-1

[3] W. J. Blok, I. M. A. Ferreirim, Hoops and their implicational reducts, Logical Computer Sciences, vol. 28 (1993), pp. 219–230.
Google Scholar DOI: https://doi.org/10.4064/-28-1-219-230

[4] W. J. Blok, I. M. A. Ferreirim, On the structure of hoops, Algebra Universalis, vol. 43 (2000), pp. 233–257, DOI: https://doi.org/10.1007/s000120050156
Google Scholar DOI: https://doi.org/10.1007/s000120050156

[5] R. A. Borzooei, M. A. Kologani, Local and perfect semihoops, Journal of Intelligent and Fuzzy Systems, vol. 29(1) (2015), pp. 223–234, DOI: https://doi.org/10.3233/IFS-151589
Google Scholar DOI: https://doi.org/10.3233/IFS-151589

[6] R. A. Borzooei, M. A. Kologani, M. S. Kish, Y. B. Jun, Fuzzy positive implicative filters of hoops based on fuzzy points, Mathematics, vol. 7 (2019), p. 566, DOI: https://doi.org/10.3390/math7060566
Google Scholar DOI: https://doi.org/10.3390/math7060566

[7] R. A. Borzooei, H. R. Varasteh, K. Borna, Fundamental hoop-algebras, Ratio Mathematica, vol. 29 (2015), pp. 25–40.
Google Scholar

[8] B. Bosbach, Komplementäre Halbgruppen. Axiomatik und Arithmetik, Fundamenta Mathematicae, vol. 64(3) (1969), pp. 257–287, DOI: https://doi.org/10.4064/FM-64-3-257-287
Google Scholar DOI: https://doi.org/10.4064/fm-64-3-257-287

[9] B. Bosbach, Komplementäre Halbgruppen. Kongruenzen und Quotienten, Fundamenta Mathematicae, vol. 69(1) (1970), pp. 1–14.
Google Scholar DOI: https://doi.org/10.4064/fm-69-1-1-14

[10] G. Georgescu, L. Leustean, V. Preoteasa, Pseudo-hoops, Journal of Multiple-Valued Logic and Soft Computing, vol. 11(1–2) (2005), pp. 153–184.
Google Scholar

[11] Y. B. Jun, S. Z. Song, E. H. Roh, Soju structures with applications in BCK=BCI-algebras, Afrika Matematika, (submitted).
Google Scholar

[12] M. A. Kologani, R. A. Borzooei, On ideal theory of hoops, Mathematica Bohemica, vol. 145 (2019), pp. 1–22, DOI: https://doi.org/10.21136/MB.2019.0140-17
Google Scholar DOI: https://doi.org/10.21136/MB.2019.0140-17

[13] M. Kondo, Some types of filters in hoops, Multiple-Valued Logic, (IS-MVL), (2011), pp. 50–53, DOI: https://doi.org/10.1109/ISMVL.2011.9
Google Scholar DOI: https://doi.org/10.1109/ISMVL.2011.9

[14] P. K. Maji, R. Biswas, A. R. Roy, Soft set theory, Computers and Mathematics with Applications, vol. 45(4–5) (2003), pp. 555–562, DOI: https://doi.org/10.1016/S0898-1221(03)00016-6
Google Scholar DOI: https://doi.org/10.1016/S0898-1221(03)00016-6

[15] D. Molodtsov, Soft set theory – First results, Computers and Mathematics with Applications, vol. 37(4–5) (1999), pp. 19–31, DOI: https://doi.org/10.1016/S0898-1221(99)00056-5
Google Scholar DOI: https://doi.org/10.1016/S0898-1221(99)00056-5

[16] X. L. Xin, R. A. Borzooei, Y. B. Jun, Positive implicative soju ideals in BCK-algebras , Bulletin of the Section of Logic, vol. 48(1) (2019), pp. 1–18, DOI: https://doi.org/10.18778/0138-0680.48.1.01
Google Scholar DOI: https://doi.org/10.18778/0138-0680.48.1.01