Abstract
This paper introduces the concept of conditional monotonicity and other related relaxed monotonicities within the framework of intervals equipped with admissible orders. It generalizes the work of Sesma-Sara et al., who introduced weak/directional monotonicity on intervals endowed with the Kulisch–Miranker order, and the work of Santiago et al., who introduced the notion of g-weak monotonicity in the fuzzy setting. The paper also explores properties of conditional monotonicities, introduce the notion of ordinal sum for a family of functions and examines the connections between conditional monotonicity, ordinal sums and implications.
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Notes
Note that \(X-X = \varvec{0}\) iff X is a degenerated interval.
If \(x \preceq y\), then we also use \(y \succeq x\) as equivalent notation.
I.e., \(\preceq \) is such that if X and Y are elements of \(\mathbb {I}([0,\!1])\) and \(X \preceq _{KM} Y\), then \(X \preceq Y\) holds.
\(\min : \mathbb {I}([0,\!1])^n \rightarrow \mathbb {I}([0,\!1])\) is defined by \(\min (X_1, \ldots , X_n) = X_i\) if \(X_i \preceq X_j\) for all \(j = 1, \ldots , n\). Dually we define the n-ary IV-function maximum
In Santiago et al. (2022), the authors have not attempted to impose the condition \( G(x,y)\ge y \) for all \(x,y \in [0,\!1]\).
This function is the IV-version of the function presented in Example 3 of Miś (2017).
See the definition of t-conorm in Definition 7.3.
Given a n-ary function \(F: \mathbb {I}^n \rightarrow \mathbb {I}\), its dual is the function \(F^{d}: \mathbb {I}^n \rightarrow \mathbb {I}\) defined as
$$\begin{aligned} F^{d}(X_1, \ldots , X_n) = \varvec{1} - F(\varvec{1} - X_1, \ldots , \varvec{1} - X_n). \end{aligned}$$A version for functions on \([0,\!1]\) appears in Bustince et al. (2015) as Corollary 3.
A related result, considering the Kulisch–Miranker order on \( \mathbb {I} \), can be found in Sesma-Sara et al. (2019).
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Acknowledgements
We thank the support for this work received from: (a) the National Council for Scientific and Technological Development (CNPq-Brazil) through the projects 312899/2021-1 and 403167/2022-1; (b) Ministerio de Ciencia e Innovación through the projects PID2020-119478GB-I00 and PID2022-136627NB-I00, supported by MCIN/AEI/10.13039/501100011033/FEDER,UE; (c) the Slovak Academy of Sciences through the projects APVV-20-0069 and VEGA 1/0036/23.
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Monteiro, A.S., Santiago, R., Papčo, M. et al. On conditional monotonicities of interval-valued functions. Comp. Appl. Math. 43, 200 (2024). https://doi.org/10.1007/s40314-024-02715-5
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DOI: https://doi.org/10.1007/s40314-024-02715-5