# Article

Full entry | PDF   (0.5 MB)
Keywords:
extensions; algebraic completely distributive lattices; fuzzy connectives
Summary:
The main goal of this paper is to construct fuzzy connectives on algebraic completely distributive lattice(ACDL) by means of extending fuzzy connectives on the set of completely join-prime elements or on the set of completely meet-prime elements, and discuss some properties of the new fuzzy connectives. Firstly, we present the methods to construct t-norms, t-conorms, fuzzy negations valued on ACDL and discuss whether De Morgan triple will be kept. Then we put forward two ways to extend fuzzy implications and also make a study on the behaviors of \$R\$-implication and reciprocal implication. Finally, we construct two classes of infinitely \$\bigvee\$-distributive uninorms and infinitely \$\bigwedge\$-distributive uninorms.
References:
[1] Baczyński, M., Jayaram, B.: Fuzzy Implications. Studies in Fuzziness and Soft Computing, Springer-Verlag, Berlin Heidelberg 2008. MR 2428086 | Zbl 1293.03012
[2] Çaylı, G. D.: On a new class of t-norms and t-conorms on bounded lattices. Fuzzy Sets Syst. 332 (2018), 129-143. DOI 10.1016/j.fss.2017.07.015 | MR 3732255
[3] Çaylı, G. D., Drygaś, P.: Some properties of idempotent uninorms on a special class of bounded lattices. Inform. Sci. 422 (2018), 352-363. DOI 10.1016/j.ins.2017.09.018 | MR 3709474
[4] Çaylı, G. D., Karaçal, F.: Construction of uninorms on bounded lattices. Kybernetika 53 (2017), 394-417. DOI 10.14736/kyb-2017-3-0394 | MR 3684677
[5] 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
[6] Davey, B. A., Priestley, H. A.: Introduction to lattices and Order. Cambridge University Press, Cambridge 1990. MR 1058437
[7] Baets, B. De, Mesiar, R.: Triangular norms on product lattices. Fuzzy Sets Syst. 104 (1999), 61-75. DOI 10.1016/s0165-0114(98)00259-0 | MR 1685810 | Zbl 0935.03060
[8] Deschrijver, G.: Uninorms which are neither conjunctive nor disjunctive in interval-valued fuzzy set theory. Inform. Sci. 244 (2013), 48-59. DOI 10.1016/j.ins.2013.04.033 | MR 3068360
[9] Jenei, S., Baets, B. De: On the direct decomposability of t-norms on product lattices. Fuzzy Sets Syst. 139 (2003), 699-707. DOI 10.1016/s0165-0114(03)00125-8 | MR 2015162
[10] Karaçal, F., Ertuğrul, Ü., Mesiar, R.: Characterization of uninorms on bounded lattices. Fuzzy Sets Syst. 308 (2017), 54-71. DOI 10.1016/j.fss.2016.05.014 | MR 3579154
[11] Karaçal, F., Mesiar, R.: Uninorms on bounded lattices. Fuzzy Sets Syst. 261 (2015), 33-43. DOI 10.1016/j.fss.2014.05.001 | MR 3291484
[12] Karaçal, F., Sağiroğlu, Y.: Infinitely \$\bigvee\$-distributive t-norms on complete lattices and pseudo-complements. Fuzzy Sets Syst. 160 (2009), 32-43. DOI 10.1016/j.fss.2008.03.022 | MR 2469428
[13] Klement, E. P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer Academic Publishers, Dordrecht 2000. MR 1790096 | Zbl 1087.20041
[14] Palmeira, E. S., Bedregal, B. C.: Extensions of fuzzy logic operators defined on bounded lattices via retraction. Comput. Math. Appl. 63 (2012), 1026-1038. DOI 10.1016/j.camwa.2011.12.007 | MR 2892746
[15] Palmeira, E. S., Bedregal, B. C.: On the extension of lattice-valued implications via retractions. Fuzzy Sets Syst. 240 (2014), 66-85. DOI 10.1016/j.fss.2013.07.023 | MR 3167513
[16] Palmeira, E. S., Bedregal, B. C., Mesiar, R., Fernandez, J.: A new way to extend t-norms, t-conorms and negations. Fuzzy Sets Syst. 240 (2014), 1-21. DOI 10.1016/j.fss.2013.05.008 | MR 3167509
[17] Saminger-Platz, S.: On ordinal sums of triangular norms on bounded lattices. Fuzzy Sets Syst. 157 (2006), 1403-1416. DOI 10.1016/j.fss.2005.12.021 | MR 2226983
[18] Saminger-Platz, S., Klement, E. P., Mesiar, R.: On extensions of triangular norms on bounded lattices. Indag. Math. 19 (2008), 1, 135-150. DOI 10.1016/s0019-3577(08)80019-5 | MR 2466398
[19] Wang, Z. D., Fang, J. X: On the direct decomposability of pseudo-t-norms, t-norms and implication operators on product lattices. Fuzzy Sets Syst. 158 (2007), 2494-2503. DOI 10.1016/j.fss.2007.06.011 | MR 2361663
[20] Wang, Z. D., Yu, Y. D: Pseudo-t-norms and implication operators on a complete Brouwerian lattice. Fuzzy Sets Syst. 132 (2002), 113-124. DOI 10.1016/s0165-0114(01)00210-x | MR 1936220
[21] Yılmaz, Ş., Kazancı, O.: Constructions of triangular norms on lattices by means of irreducible elements. Inform. Sci. 397-398 (2017), 110-117. DOI 10.1016/j.ins.2017.02.041

Partner of