Clifford's order based on non-commutative operations

Document Type : Research Paper

Author

School of Mathematical Sciences, Sichuan Normal University

Abstract

Based on the classical works of Clifford inducing partial order from semigroups, recently, Gupta and Jayaram explored the order $\sqsubseteq_{F}$ from an associative operation $F$ through \emph{local left identity} (\textbf{LLI}). Inspired by their works, we further present an order $\sqsubseteq^{*}_F$ obtained from non-commutative operation $F$ which has the \emph{local right identity} (\textbf{LRI}) since the non-commutativity of $F$ implies that the local left and right identity may be different for each element, which means that both orders may not coincide in the same domain. Firstly, we determine an equivalent characterization for two orders induced by non-commutative operation $F$. Secondly, we investigate both orders induced by semi-t-operators and deeply study their properties. Finally, we characterize both orders obtained from semi-uninorm (resp. semi-nullnorm) under the condition that semi-uninorm (resp. semi-nullnorm) is locally continuous.

Keywords

Main Subjects

References

[1] E. A¸sıcı, An order induced by nullnorms and its properties, Fuzzy Sets and Systems, 325 (2017), 35-46. https:
//doi.org/10.1016/j.fss.2016.12.004
[2] E. A¸sıcı, F. Kara¸cal, On the T-partial order and properties, Information Sciences, 267 (2014), 323-333. https:
//doi.org/10.1016/j.ins.2014.01.032
[3] A. H. Clifford, Naturally totally ordered commutative semigroups, American Journal of Mathematics, 76 (1954),
631-646. https://doi.org/10.2307/2372706
[4] J. Drewniak, P. Dryga´s, E. Rak, Distributivity between uninorms and nullnorms, Fuzzy Sets and Systems, 159
(2008), 1646-1657. https://doi.org/10.1016/j.fss.2007.09.015
[5] P. Dryga´s, Distributivity between semi-t-operators and semi-nullnorms, Fuzzy Sets and Systems, 264 (2015), 100-
109. https://doi.org/10.1016/j.fss.2014.09.003
[6] U. Ertu˘grul, M. N. Kesicio˘glu, F. Kara¸cal, ¨ Ordering based on uninorms, Information Sciences, 330 (2016), 315-327.
https://doi.org/10.1016/j.ins.2015.10.019
[7] B. W. Fang, B. Q. Hu, Semi-t-operators on bounded lattices, Information Sciences, 490 (2019), 191-209. https:
//doi.org/10.1016/j.ins.2019.03.077
[8] G. Gr¨atzer, Lattice theory: Foundation, Birkh¨auser, 2011.
[9] V. K. Gupta, B. Jayaram, Importation lattices, Fuzzy Sets and Systems, 405 (2021), 1-17. https://doi.org/10.
1016/j.fss.2020.04.003
[10] V. K. Gupta, B. Jayaram, Order based on associative operations, Information Sciences, 566 (2021), 326-346.
https://doi.org/10.1016/j.ins.2021.02.020
[11] V. K. Gupta, B. Jayaram, Clifford’s order obtained from uninorms on bounded lattices, Fuzzy Sets and Systems,
462 (2023), 108384. https://doi.org/10.1016/j.fss.2022.08.016
[12] F. Kara¸cal, M. N. Kesicio˘glu, A T-partial order obtained from t-norms, Kybernetika, 47 (2011), 300-314.
[13] F. Kara¸cal, R. Mesiar, Ordering based on implications, Information Sciences, 276 (2014), 377-386. https://doi.
org/10.1016/j.ins.2013.12.047
[14] M. N. Kesicio˘glu, Some notes on the partial orders induced by a uninorm and a nullnorm in a bounded lattice,
Fuzzy Sets and Systems, 346 (2018), 55-71. https://doi.org/10.1016/j.fss.2017.08.010
[15] M. N. Kesicio˘glu, U. Ertu˘grul, F. Kara¸cal, ¨ An equivalence relation based on the U-partial order, Information
Sciences, 411 (2017), 39-51. https://doi.org/10.1016/j.ins.2017.05.020
[16] M. N. Kesicio˘glu, U. Ertu˘grul, F. Kara¸cal, ¨ Some notes on U-partial order, Kybernetika, 55 (2019), 518-530.
[17] M. N. Kesicio˘glu, F. Kara¸cal, R. Mesiar, Order-equivalent triangular norms, Fuzzy Sets and Systems, 268 (2015),
59-71. https://doi.org/10.1016/j.fss.2014.10.006
[18] E. P. Klement, R. Mesiar, E. Pap, Triangular norms, Springer Netherlands, 2000. https://doi.org/10.1007/
978-94-015-9540-7
[19] M. Komorn´ıkov´a, R. Mesiar, Aggregation functions on bounded partially ordered sets and their classification, Fuzzy
Sets and Systems, 175 (2011), 48-56. https://doi.org/10.1016/j.fss.2011.01.015
[20] Z. Li, Y. Su, On linearly ordered index sets for ordinal sums in the sense of A. H. Clifford yielding uninorms,
Iranian Journal of Fuzzy Systems, 20(2) (2023), 161-166. https://doi.org/10.22111/ijfs.2023.7563
[21] H. W. Liu, Semi-uninorms and implications on a complete lattice, Fuzzy Sets and Systems, 191 (2012), 72-82.
https://doi.org/10.1016/j.fss.2011.08.010
[22] Z. Q. Liu, An order induced by extended t-norms on convex normal functions, Fuzzy Sets and Systems, 465 (2023),
108530. https://doi.org/10.1016/j.fss.2023.108530
[23] Z. Q. Liu, X. P. Wang, Partial orders induced by the smallest and greatest nullnorms on bounded lattices, Soft
Computing, 26 (2022), 8245-8252. https://doi.org/10.1007/s00500-022-07256-9
[24] J. Lu, K. Wang, B. Zhao, Equivalence relations induced by the U-partial order, Fuzzy Sets and Systems, 334
(2018), 73-82. https://doi.org/10.1016/j.fss.2017.07.013
[25] M. Mas, G. Mayor, J. Torrens, t-operators, International Journal of Uncertainty, Fuzziness and Knowledge-Based
Systems, 7 (1999), 31-50. https://doi.org/10.1142/S0218488599000039
[26] A. Mesiarov´a-Zem´ankov´a, Natural partial order induced by a commutative, associative and idempotent function,
Information Sciences, 545 (2021), 499-512. https://doi.org/10.1016/j.ins.2020.09.028
[27] A. Mesiarov´a-Zem´ankov´a, Representation of non-commutative, idempotent, associative functions by pair-orders,
Fuzzy Sets and Systems, 475 (2023), 108759. https://doi.org/10.1016/j.fss.2023.108759
[28] H. Mitsch, A natural partial order for semigroups, Proceedings of the American Mathematical Society, 97 (1986),
384-388. https://doi.org/10.2307/2046222
[29] K. S. Nambooripad, The natural partial order on a regular semigroup, Proceedings of the Edinburgh Mathematical
Society, 23 (1980), 249-260. https://doi.org/10.1017/S0013091500003801
[30] K. Nanavati, B. Jayaram, Order from non-associative operations, Fuzzy Sets and Systems, 467 (2023), 108484.
https://doi.org/10.1016/j.fss.2023.02.005
[31] J. S. Qiao, On binary relations induced from overlap and grouping functions, International Journal of Approximate
Reasoning, 106 (2019), 155-171. https://doi.org/10.1016/j.ijar.2019.01.006
[32] F. Qin, Distributivity between semi-uninorms and semi-t-operators, Fuzzy Sets and Systems, 299 (2016), 66-88.
https://doi.org/10.1016/j.fss.2015.10.012
[33] Y. Su, W. W. Zong, H. W. Liu, P. J. Xue, Migrativity property for uninorms and semi t-operators, Information
Sciences, 325 (2015), 455-465. https://doi.org/10.1016/j.ins.2015.07.030
[34] Z. D. Wang, J. X. Fang, Residual operations of left and right uninorms on a complete lattice, Fuzzy Sets and
Systems, 160 (2009), 22-31. https://doi.org/10.1016/j.fss.2008.03.001
[35] R. R. Yager, A. Rybalov, Uninorm aggregation operators, Fuzzy Sets and Systems, 80 (1996), 111-120. https:
//doi.org/10.1016/0165-0114(95)00133-6
Iranian Journal of Fuzzy Systems
Volume 21, Issue 3
May and June 2024
Pages 77-90

Files

Share

How to cite

Statistics