On some equation related to the distributivity laws of fuzzy implications. Jensen equation extended to the infinity. (English) Zbl 1423.39033
Summary: Recently, in several articles related to the distributivity laws of fuzzy implications over triangular norms and conorms, the following functional equation appeared \(f(\min(x + y, a)) = \min(f(x) + f(y), b)\), where \(a, b\) are finite or infinite nonnegative constants. In our earlier papers, we have considered a generalized version of this equation, i.e., the equation \(f(m_1(x + y)) = m_2(f(x) + f(y))\). Firstly, we analyzed the situation when both functions \(m_1\), \(m_2\) are defined on some finite intervals of \(\mathbb{R}\). We also investigated the situation when both functions \(m_1\), \(m_2\) have finite or infinite domains and codomains, but they satisfy several additional assumptions. In this article, we consider the above equation when \(m_1\), \(m_2\) are defined on some finite or infinite intervals and satisfy only one additional assumption: \(m_2\) is injective. Our proofs are based on the Jensen equation, therefore, we also present the detailed analysis of this functional equation when the domain or codomain are extended to the infinity and/or are bounded.
MSC:
| 39B52 | Functional equations for functions with more general domains and/or ranges |
| 03B52 | Fuzzy logic; logic of vagueness |
References:
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