Abstract
Note that \(\Theta \)–\(\Xi \) functions and aggregation functions are two interrelated concepts rather than mutually inclusive ones. On the other hand, 0-overlap functions and 1-grouping functions are common subsets of these two broader categories. Although the homogeneity and quasi-homogeneity properties of 0-overlap functions and 1-grouping functions have been thoroughly described through existing research on aggregation functions, it is noteworthy that our approach offers a direct proof of their homogeneity, rather than treating them as a mere special case of quasi-homogeneity. Then the pseudo-homogeneity of these functions is extended to aggregation functions with continuous diagonals, and the necessary and sufficient conditions for such aggregation functions are obtained by utilizing the pseudo-inverses of unary functions. Finally, by exploring the structures of the \(\Theta \)–\(\Xi \) functions, which serve as a unified representation of 0-overlap functions and 1-grouping functions, the homogeneity, quasi-homogeneity, and pseudo-homogeneity of \(\Theta \)–\(\Xi \) functions are comprehensively characterized. Our findings are further illustrated through examples, providing a clear understanding of these functions and their properties.
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Acknowledgements
This work was supported by the National Natural Science Foundation of China (Nos. 12371459, 12331016), the Jiangxi Provincial Natural Science Foundation (Nos. 20232ACB201002, 20224BCD41001), and the Guangdong Philosophy and Social Science Planning Project (No. GD23XSH03).
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Liu, C., Qin, F., Qiao, J. et al. (Quasi, pseudo)-homogeneity of \(\Theta \)–\(\Xi \) functions. Comp. Appl. Math. 44, 11 (2025). https://doi.org/10.1007/s40314-024-02971-5
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DOI: https://doi.org/10.1007/s40314-024-02971-5