On two functional equations connected with distributivity of fuzzy implications

Roman Ger, Marcin Emil Kuczma, and Wanda Niemyska. "On two functional equations connected with distributivity of fuzzy implications." Commentationes Mathematicae 55.2 (2015): null. <http://eudml.org/doc/292387>.

@article{RomanGer2015,
abstract = {The distributivity law for a fuzzy implication $I\colon [0,1]^2 \rightarrow [0,1]$ with respect to a fuzzy disjunction $S\colon [0,1]^2 \rightarrow [0,1]$ states that the functional equation $ I(x,S(y,z))=S(I(x,y),I(x,z)) $ is satisfied for all pairs $(x,y)$ from the unit square. To compare some results obtained while solving this equation in various classes of fuzzy implications, Wanda Niemyska has reduced the problem to the study of the following two functional equations: $ h(\min (xg(y),1)) = \min (h(x)+ h(xy),1)$, $x \in (0,1)$, $y \in (0,1]$, and $ h(xg(y)) = h(x)+ h(xy)$, $x,y \in (0, \infty )$, in the class of increasing bijections $h\colon [0,1] \rightarrow [0,1]$ with an increasing function $g\colon (0,1] \rightarrow [1, \infty )$ and in the class of monotonic bijections $h\colon (0, \infty ) \rightarrow (0, \infty )$ with a function $g\colon (0, \infty ) \rightarrow (0, \infty )$, respectively. A description of solutions in more general classes of functions (including nonmeasurable ones) is presented.},
author = {Roman Ger, Marcin Emil Kuczma, Wanda Niemyska},
journal = {Commentationes Mathematicae},
keywords = {fuzzy implication; distributivity; functional equation; t-conorm},
language = {eng},
number = {2},
pages = {null},
title = {On two functional equations connected with distributivity of fuzzy implications},
url = {http://eudml.org/doc/292387},
volume = {55},
year = {2015},
}

TY - JOUR
AU - Roman Ger
AU - Marcin Emil Kuczma
AU - Wanda Niemyska
TI - On two functional equations connected with distributivity of fuzzy implications
JO - Commentationes Mathematicae
PY - 2015
VL - 55
IS - 2
SP - null
AB - The distributivity law for a fuzzy implication $I\colon [0,1]^2 \rightarrow [0,1]$ with respect to a fuzzy disjunction $S\colon [0,1]^2 \rightarrow [0,1]$ states that the functional equation $ I(x,S(y,z))=S(I(x,y),I(x,z)) $ is satisfied for all pairs $(x,y)$ from the unit square. To compare some results obtained while solving this equation in various classes of fuzzy implications, Wanda Niemyska has reduced the problem to the study of the following two functional equations: $ h(\min (xg(y),1)) = \min (h(x)+ h(xy),1)$, $x \in (0,1)$, $y \in (0,1]$, and $ h(xg(y)) = h(x)+ h(xy)$, $x,y \in (0, \infty )$, in the class of increasing bijections $h\colon [0,1] \rightarrow [0,1]$ with an increasing function $g\colon (0,1] \rightarrow [1, \infty )$ and in the class of monotonic bijections $h\colon (0, \infty ) \rightarrow (0, \infty )$ with a function $g\colon (0, \infty ) \rightarrow (0, \infty )$, respectively. A description of solutions in more general classes of functions (including nonmeasurable ones) is presented.
LA - eng
KW - fuzzy implication; distributivity; functional equation; t-conorm
UR - http://eudml.org/doc/292387
ER -