Soft Set Theory Applied to Hoops

Borzooei, R. A.; Babaei, E.; Jun, Y. B.; Kologani, M. Aaly; Takallo, M. Mohseni

doi:10.2478/auom-2020-0004

Abstract

In this paper, we introduced the concept of a soft hoop and we investigated some of their properties. Then, we established different types of intersections and unions of the family of soft hoops. We defined two operations ⊙ and → on the set of all soft hoops and we proved that with these operations, it is a hoop and also is a Heyting algebra. Finally we introduced a congruence relation on the set of all soft hoops and we investigated the quotient of it.

References

[1] U. Acar, F. Koyuncu, B. Tanay, Soft sets and soft rings, Computers and Mathematics with Applications, 59 (2010), 3458–3463.10.1016/j.camwa.2010.03.034
Search in Google Scholar Back to article
[2] P. Aglianó, I. M. A. Ferreirim, F. Montagna, Basic hoops: an algebraic study of continuous t-norm, 87(1) (2007), 73-98.10.1007/s11225-007-9078-1
Search in Google Scholar Back to article
[3] H. Aktas, N. Cagman, Soft sets and soft groups, Information Sciences, 177 (2007), 2726–2735.10.1016/j.ins.2006.12.008
Search in Google Scholar Back to article
[4] S. Z. Alavi, R. A. Borzooei, M. Aaly Kologani, Fuzzy filters in pseudo hoops, Journal of Intelligent and Fuzzy Systems, 32 (2017), 1997–2007.10.3233/JIFS-161586
Search in Google Scholar Back to article
[5] S. Z. Alavi, R. A. Borzooei, M. Aaly Kologani, Filter theory of pseudo hoop algebras, Italian Journal of Pure and Applied Mathematics, 37 (2017), 619–632.
Search in Google Scholar Back to article
[6] V. Boicescu, A. Filipoiu, G, Georgescu, S. Rudeano, Lukasiewics-Moisil algebras, Annals of Discrete Mathematics, 49 (1991), 2–15.
Search in Google Scholar Back to article
[7] R. A. Borzooei, M. Aaly Kologani, Filter theory of hoop-algebras, Journal of Advanced Research in Pure Mathematics, 1 (2014), 1–15.
Search in Google Scholar Back to article
[8] R. A. Borzooei, M. Aaly Kologani, Local and perfect semihoops, Journal of Intelligent and Fuzzy Systems, 29 (2015), 223–234.10.3233/IFS-151589
Search in Google Scholar Back to article
[9] R. A. Borzooei, M. Aaly Kologani, On fuzzy filters of hoop algebras, The Journal of Fuzzy Mathematics, 25(1) (2017), 177–195.
Search in Google Scholar Back to article
[10] R. A. Borzooei, M. Aaly Kologani, O. Zahiri, State of hoops, Mathematica Slovaca, 67(1) (2017), 1–16.10.1515/ms-2016-0242
Search in Google Scholar Back to article
[11] B. Bosbach, Komplementäre halbgruppen. axiomatik und arithmetik, Fundamenta Mathematicae, 64 (1969), 257–287.10.4064/fm-64-3-257-287
Search in Google Scholar Back to article
[12] B. Bosbach, Komplementäre halbgruppen. kongruenzen and quotienten, Fundamenta Mathematicae, 69 (1970), 1–14.10.4064/fm-69-1-1-14
Search in Google Scholar Back to article
[13] J. R. Büchi, T. M. Owens, Complemented monoids and hoops, unpublished manuscript, (1975).
Search in Google Scholar Back to article
[14] F. Feng, Y. B. Jun, X. Z. Zhao, Soft semirings, Computers and Mathematics with Applications, 56 (2008), 2621–2628.10.1016/j.camwa.2008.05.011
Search in Google Scholar Back to article
[15] G. Georgescu, L. Leustean, V. Preoteasa, Pseudo-hoops, Journal of Multiple-Valued Logic and Soft Computing, 11(1-2) (2005), 153–184.
Search in Google Scholar Back to article
[16] A. R. Hadipour, A. Borumand Saeid, Fuzzy soft BF-algebra, Indian Journal of Science and Technology, 6(3) (2013), 4199–4204.10.17485/ijst/2013/v6i3.5
Search in Google Scholar Back to article
[17] P. Hájek, Metamathematics of fuzzy logic, Springer, 4 (1998).10.1007/978-94-011-5300-3
Search in Google Scholar Back to article
[18] Y. B. Jun, Soft BCK/BCI-algebras, Computers and Mathematics with Applications, 56 (2008), 1408–1413.10.1016/j.camwa.2008.02.035
Search in Google Scholar Back to article
[19] Y. B. Jun, K. J. Lee, C. H. Park, Fuzzy soft set theory applied to BCK/BCI-algebras, Computers and Mathematics with Applications, 59 (2010), 3180–3192.10.1016/j.camwa.2010.03.004
Search in Google Scholar Back to article
[20] Y. B. Jun, C. H. Park, Applications of soft sets in ideal theory of BCK/BCI-algebras, Information Sciences, 178 (2008), 2466–2475.
Search in Google Scholar Back to article
[21] D. Molodtsov, Soft set theory - First results, Computers and Mathematics with Applications, 37 (1999), 19–31.10.1016/S0898-1221(99)00056-5
Search in Google Scholar Back to article
[22] D. Pei, D. Miaa, From soft sets to information systems, Proceedings of Granular Computing, 2 (2005), 617–621.
Search in Google Scholar Back to article
[23] A. Namdara, R. A. Borzooei, A. Borumand Saeid, M. Aaly Kologani, Some results in hoop algebras, Journal of Intelligent and Fuzzy Systems, 32 (2017), 1805–1813.10.3233/JIFS-152553
Search in Google Scholar Back to article
[24] S. Yamak, O. Kazanci, B. Davvaz, Soft hyper-structure, Computers and Mathematics with Applications, 62 (2011), 797–803.10.1016/j.camwa.2011.06.009
Search in Google Scholar Back to article
[25] J. Zhan, W. A. Dudek, Soft MTL-Algebras based on fuzzy sets, Scientific Bulletin Series A, 74(2) (2012), 41–56.
Search in Google Scholar Back to article
[26] J. Zhan, Y. B. Jun, Soft-algebras based on fuzzy sets, Computers and Mathematics with Applications, 59 (2010), 2037–2046.10.1016/j.camwa.2009.12.008
Search in Google Scholar Back to article

Soft Set Theory Applied to Hoops

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