Abstract
In this paper, we focus on combining the theories of interval-valued fuzzy soft sets over ternary semigroups, and establishing a new framework for interval-valued fuzzy soft ternary semigroups. The aim of this manuscript is to apply interval-valued fuzzy soft set for dealing with several kinds of theories in ternary semigroups. First, we present the concepts of interval-valued fuzzy soft sets, interval-valued fuzzy soft ternary semigroups and interval-valued fuzzy soft ideals. Meanwhile, some illustrative examples are given to show the rationality of the definitions introduced in this paper. Second, several new kinds of generalized fuzzy soft sets over ternary semigroups are proposed, and related properties and mutual relationships are also investigated. Finally, our goal is to establish a relation between different types of ternary subsemigroups (left ideal, right ideal, lateral ideal, two-sided ideal, ideal) of a ternary semigroup and interval-valued fuzzy soft subsemigroups (interval-valued fuzzy soft left ideal, interval-valued fuzzy soft right ideal, interval-valued fuzzy soft lateral ideal, interval-valued fuzzy soft two-sided ideal, interval-valued fuzzy soft two-sided ideal) over a ternary semigroup and also characterize these notions in terms of soft sets in ternary semigroups.
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REFERENCES
M. Y. Abbasi and S. A. Khan, ‘‘On some generalized ideals in ternary semigroups,’’ Quasigroups Relat. Syst. 25, 181–188 (2017).
M. Y. Abbasi, S. A. Khan, and A. Basar, ‘‘On generalised quasi-ideals in ordered ternary semigroups,’’ KYUNGPOOK Math. J. 57, 545–558 (2017).
M. Y. Abbasi, K. P. Shum, S. A. Khan, and A. F. Talee, ‘‘Generalised bi-ideals in ordered ternary semigroups,’’ Euras. Bull. Math. 1 (1), 1–23 (2018).
A. Chronowski, ‘‘Congruences on ternary semigroups,’’ Ukr. Math. J. 56, 662–681 (2004).
B. Davvaz, ‘‘Fuzzy hyperideals in ternary semihyperrings,’’ Iran. J. Fuzzy Syst. 6 (4), 21–36 (2009).
B. Davvaz, ‘‘On the lower level sets and anti fuzzy ternary polygroups,’’ Bull. Georgian Natl. Acad. Sci. 8 (2), 19–28 (2014).
B. Davvaz, F. Dehghan, and M. Farshi, ‘‘Ternary hypergroups of type \(U\) on the right,’’ Kragujevac J. Math. 42, 177–191 (2018).
M. Dehghanian and M. S. Modarres, ‘‘Ternary \(\gamma\)-homomorphisms and ternary \(\gamma\)-derivations on ternary semigroups,’’ J. Inequal. Appl. 2012 (34) (2012).
T. K. Dutta, S. Kar, and K. Das, ‘‘Power ternary semirings,’’ Afr. Mat. 26, 1483–1494 (2015).
M. Farooq, A. Khan, M. Izhar, and B. Davvaz, ‘‘On soft ordered ternary semihypergroups,’’ Int. J. Algebra Stat. 7, 27–39 (2018).
S. Kar and B. K. Maity, ‘‘Some ideals of ternary semigroups,’’ An. Sti. Univ. Al I Cuza din Iasi—Mat. 57, 247–258 (2011).
A. Khan and M. Sarwar, ‘‘Uni-soft ideals of ternary semigroups,’’ Int. J. Math. Sci. Comput. 5, 60–65 (2014).
K. Hila, B. Davvaz, and K. Naka, ‘‘On hyperideal structure of ternary semihypergroups,’’ Iran. J. Math. Sci. Inform. 9, 81–98 (2015).
K. Hila, K. Naka, V. L. Fotea, and S. Sadiku, ‘‘Algebraic hyperstructures of soft sets associated with ternary semihypergroups,’’ Ital. J. Pure Appl. Math. 30, 349–372 (2013).
A. Iampan, ‘‘On ordered ideal extensions of ordered ternary semigroups,’’ Lobachevskii J. Math. 31 (1), 13–17 (2010).
K. Naka and K. Hila, ‘‘On the structure of quasi-hyperideals and bi-hyperideals in ternary semihypergroups,’’ Afr. Mat. 26, 1573–1591 (2015).
K. Naka and K. Hila, ‘‘Regularity of ternary semihypergroups,’’ Quasigroups Relat. Syst. 25, 291–306 (2017).
K. Naka and K. Hila, ‘‘Some properties of hyperideals in ternary semihypergroups,’’ Math. Slov. 63, 449–468 (2013).
T. S. Rao and B. S. Kumar, ‘‘On regular, ideals in partially ordered soft ternary semi groups,’’ Int. J. Civil Eng. Technol. 9, 830–837 (2018).
N. Rehman, J. G. Kang, and Y. B. Jun, ‘‘A generalization of \((\in,\in\vee q)\)-fuzzy ternary subsemigroups,’’ Afr. Mat. 29, 887–898 (2018).
M. L. Santiago and S. S. Bala, ‘‘Ternary semigroups,’’ Semigroup Forum 81, 380–388 (2010).
M. K. Sen and A. K. Bhuniya, ‘‘On semirings whose additive reduct is a semilattice,’’ Semigroup Forum 82, 131–140 (2011).
M. Shabir and A. Ahmad, ‘‘On soft ternary semigroups,’’ Ann. Fuzzy Math. Inform. 3, 39–59 (2012).
A. F. Talee, M. Y. Abbasi, and S. A. Khan, ‘‘An investigatiozn on prime and semiprime fuzzy hyperideals in po-ternary semihypergroups,’’ Appl. Appl. Math. 13, 1184–1199 (2018).
N. Yaqoob, M. Aslam, and K. Hila, ‘‘Rough fuzzy hyperideals in ternary semihypergroups,’’ Adv. Fuzzy Syst. 2012, 595687 (2012).
N. Yaqoob, M. Khan, M. Akram, and A. Khan, ‘‘Interval valued intuitionistic \((\bar{S},\bar{T})\)-fuzzy ideals of ternary semigroups,’’ Indian J. Sci. Technol. 6 (911), 5418–5428 (2013).
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Yiarayong, P. On Interval-Valued Fuzzy Soft Set Theory Applied To Ternary Semigroups. Lobachevskii J Math 42, 222–230 (2021). https://doi.org/10.1134/S1995080221010273
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DOI: https://doi.org/10.1134/S1995080221010273