# Solutions of the Distributivity Equation $mathcal{I}(mathcal{T}(x,y),z) = mathcal{S}(mathcal{I}(x,z),mathcal{I}(y,z))$ for Some t-Representable T-Norms and T-Conorms

- DOI
- https://doi.org/10.2991/eusflat.2013.81How to use a DOI?
- Keywords
- triangular norm t-norm t-conorm fuzzy implication interval-valued fuzzy sets distributivity equations functional equations
- Abstract
Recently, in [4], [5], [6], and [8] we have discussed the distributivity equation of implications $mathcal{I}(x,mathcal{T}_1(y,z)) = mathcal{T}_2(mathcal{I}(x,y),mathcal{I}(x,z))$ over t-representable t-norms, generated from (classical) continuous Archimedean mbox{t-norms}, in interval-valued fuzzy sets theory. In [7] we discussed similar methods, but for the following distributivity functional equation $mathcal{I}(x,mathcal{S}_1(y,z)) = mathcal{S}_2(mathcal{I}(x,y),mathcal{I}(x,z))$, when $mathcal{S}_1$, $mathcal{S}_2$ are t-representable t-conorms. In this article we continue investigations presented at previous EUSFLAT-LFA 2011, i.e., we will show all solutions for the following distributivity equation $mathcal{I}(mathcal{T}(x,y),z) = mathcal{S}(mathcal{I}(x,z),mathcal{I}(y,z))$, where $mathcal{I}$ is an unknown function, $mathcal{T}$ is a t-representable t-norm on $mathcal{L}^I$ generated from nilpotent t-norms $T_1$, $T_2$ and $mathcal{S}$ is a t-representable t-conorm on $mathcal{L}^I$ generated from strict t-conorms $S_1$, $S_2$.

- Copyright
- © 2013, the Authors. Published by Atlantis Press.
- Open Access
- This is an open access article distributed under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

### Cite this article

TY - CONF AU - Michal Baczynski PY - 2013/08 DA - 2013/08 TI - Solutions of the Distributivity Equation $mathcal{I}(mathcal{T}(x,y),z) = mathcal{S}(mathcal{I}(x,z),mathcal{I}(y,z))$ for Some t-Representable T-Norms and T-Conorms BT - Proceedings of the 8th conference of the European Society for Fuzzy Logic and Technology (EUSFLAT-13) PB - Atlantis Press SP - 574 EP - 580 SN - 1951-6851 UR - https://doi.org/10.2991/eusflat.2013.81 DO - https://doi.org/10.2991/eusflat.2013.81 ID - Baczynski2013/08 ER -