# Article

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Keywords:
bounded lattice; t-norm; t-conorm; ordinal sum
Summary:
In this study, we introduce new methods for constructing t-norms and t-conorms on a bounded lattice $L$ based on a priori given t-norm acting on $[a,1]$ and t-conorm acting on $[0,a]$ for an arbitrary element $a\in L\backslash \{0,1\}$. We provide an illustrative example to show that our construction methods differ from the known approaches and investigate the relationship between them. Furthermore, these methods are generalized by iteration to an ordinal sum construction for t-norms and t-conorms on a bounded lattice.
References:
[1] Aşıcı, E., Karaçal, F.: Incomparability with respect to the triangular order. Kybernetika 52 (2016), 1, 15-27. DOI 10.14736/kyb-2016-1-0015 | MR 3482608
[2] Aşıcı, E.: On the properties of the F-partial order and the equivalence of nullnorms. Fuzzy Sets and Systems 346 (2018), 72-84. DOI 10.1016/j.fss.2017.11.008 | MR 3812758
[3] Aşıcı, E.: An extension of the ordering based on nullnorms. Kybernetika 55 (2019), 2, 217-232. DOI 10.14736/kyb-2019-2-0217
[4] Birkhoff, G.: Lattice Theory. American Mathematical Society Colloquium Publ., Providence 1967. DOI 10.1090/coll/025 | MR 0227053 | Zbl 0537.06001
[5] Butnariu, D., Klement, E. P.: Triangular Norm-Based Measures and Games with Fuzzy Coalitions. Kluwer Academic Publishers, Dordrecht 1993. DOI 10.1007/978-94-017-3602-2 | MR 2867321
[6] Clifford, A.: Naturally totally ordered commutative semigroups. Am. J. Math. 76 (1954), 631-646. DOI 10.2307/2372706 | MR 0062118
[7] Çaylı, G. D., Karaçal, F., Mesiar, R.: On a new class of uninorms on bounded lattices. Inform. Sci. 367-368 (2016), 221-231. DOI 10.1016/j.ins.2016.05.036 | MR 3684677
[8] Çaylı, G. D., Karaçal, F.: Construction of uninorms on bounded lattices. Kybernetika 53 (2017), 3, 394-417. DOI 10.14736/kyb-2017-3-0394 | MR 3684677
[9] Çaylı, G. D.: Characterizing ordinal sum for t-norms and t-conorms on bounded lattices. In: Advances in Fuzzy Logic and Technology 2017. IWIFSGN 2017, EUSFLAT 2017. Advances in Intelligent Systems and Computing (J. Kacprzyk, E. Szmidt, S. Zadrozny, K. Atanassov, M. Krawczak, eds.), vol. 641 Springer, Cham 2018, pp. 443-454. DOI 10.1007/978-3-319-66830-7_40
[10] Çaylı, G. D.: On a new class of t-norms and t-conorms on bounded lattices. Fuzzy Sets and Systems 332 (2018), 129-143. DOI 10.1016/j.fss.2017.07.015 | MR 3732255
[11] Çaylı, G. D.: On the structure of uninorms on bounded lattices. Fuzzy Sets and Systems 357 (2019), 2-26. DOI 10.1016/j.fss.2018.07.012 | MR 3913056
[12] Deschrijver, G., Kerre, E. E.: Uninorms in $L^{\ast }$-fuzzy set theory. Fuzzy Sets and Systems 148 (2004), 243-262. DOI 10.1016/j.fss.2003.12.006 | MR 2100198
[13] Drossos, C. A., Navara, M.: Generalized t-conorms and closure operators. In: EUFIT 96, Aachen 1996.
[14] Drossos, C. A.: Generalized t-norm structures. Fuzzy Sets Systems 104 (1999), 53-59. DOI 10.1016/s0165-0114(98)00258-9 | MR 1685809
[15] Drygaś, P.: On properties of uninorms with underlying t-norm and t-conorm given as ordinal sums. Fuzzy Sets and Systems 161 (2010), 149-157. DOI 10.1016/j.fss.2009.09.017 | MR 2566236 | Zbl 1191.03039
[16] Esteva, F., Godo, L.: Monoidal t-norm based logic: towards a logic for left-continuous t-norms. Fuzzy Sets and Systems 124 (2001), 271-288. DOI 10.1016/s0165-0114(01)00098-7 | MR 1860848
[17] Ertuğrul, Ü., Karaçal, F., Mesiar, R.: Modified ordinal sums of triangular norms and triangular conorms on bounded lattices. Int. J. Intell. Systems 30 (2015), 807-817. DOI 10.1002/int.21713
[18] Goguen, J. A.: L-fuzzy sets. J. Math. Anal. Appl. 18 (1967), 145-174. DOI 10.1016/0022-247x(67)90189-8 | MR 0224391
[19] Goguen, J. A.: The fuzzy Tychonoff theorem. J. Math. Anal. Appl. 43 (1973), 734-742. DOI 10.1016/0022-247x(73)90288-6 | MR 0341365
[20] Grabisch, M., Nguyen, H. T., Walker, E. A.: Fundamentals of Uncertainty Calculi with Applications to Fuzzy Inference. Kluwer Academic Publishers, Dordrecht 1995. MR 1472733
[21] Höhle, U.: Probabilistische Topologien. Manuscr. Math. 26 (1978), 223-245. DOI 10.1007/bf01167724 | MR 0515397
[22] Höhle, U.: Commutative, residuated SOH-monoids, Non-classical logics and their applications to fuzzy subsets. In: A handbook of the mathematical foundations of fuzzy set theory, theory and decision library series B: mathematical and statistical methods (K. Höhle, ed.), vol. 32. The Netherlands Kluwer, Dordrecht 1995. MR 1345641
[23] Klement, E. P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer Acad. Publ., Dordrecht 2000. DOI 10.1007/978-94-015-9540-7 | MR 1790096 | Zbl 1087.20041
[24] Medina, J.: Characterizing when an ordinal sum of t-norms is a t-norm on bounded lattices. Fuzzy Sets and Systems 202 (2012), 75-88. DOI 10.1016/j.fss.2012.03.002 | MR 2934787
[25] Saminger, S.: On ordinal sums of triangular norms on bounded lattices. Fuzzy Sets and Systems 157 (2006), 10, 1403-1416. DOI 10.1016/j.fss.2005.12.021 | MR 2226983 | Zbl 1099.06004
[26] Schweizer, B., Sklar, A.: Probabilistic Metric Spaces. North-Holland, New York 1983. MR 0790314 | Zbl 0546.60010

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