| Title:
|
On a functional equation connected to the distributivity of fuzzy implications over triangular norms and conorms (English) |
| Author:
|
Baczyński, Michał |
| Author:
|
Szostok, Tomasz |
| Author:
|
Niemyska, Wanda |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 (print) |
| ISSN:
|
1805-949X (online) |
| Volume:
|
50 |
| Issue:
|
5 |
| Year:
|
2014 |
| Pages:
|
679-695 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Distributivity of fuzzy implications over different fuzzy logic connectives have a very important role to play in efficient inferencing in approximate reasoning, especially in fuzzy control systems (see [9, 15] and [4]). Recently in some considerations connected with these distributivity laws, the following functional equation appeared (see [5]) $$ f(\min(x+y,a))=\min(f(x)+f(y),b), $$ where $a,b>0$ and $f\colon[0,a]\to[0,b]$ is an unknown function. In this paper we consider in detail a generalized version of this equation, namely the equation $$ f(m_1(x+y))=m_2(f(x)+f(y)), $$ where $m_1,m_2$ are functions defined on some intervals of ${\mathbb R}$ satisfying additional assumptions. We analyze the cases when $m_2$ is injective and when $m_2$ is not injective. (English) |
| Keyword:
|
fuzzy connectives |
| Keyword:
|
fuzzy implication |
| Keyword:
|
distributivity |
| Keyword:
|
functional equations |
| MSC:
|
03B52 |
| MSC:
|
03E72 |
| MSC:
|
39B05 |
| MSC:
|
39B22 |
| MSC:
|
39B99 |
| idZBL:
|
Zbl 06410697 |
| idMR:
|
MR3301854 |
| DOI:
|
10.14736/kyb-2014-5-0679 |
| . |
| Date available:
|
2015-01-13T09:22:48Z |
| Last updated:
|
2016-01-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144100 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
|
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| Reference:
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| Reference:
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| Reference:
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