(Quasi, pseudo)-homogeneity of \(\Theta \)–\(\Xi \) functions

Note that \(\Theta \) $\mathrm{\Theta }$ – \(\Xi \) $\mathrm{\Xi }$ 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 \(\Theta \) $\mathrm{\Theta }$ – \(\Xi \) $\mathrm{\Xi }$ functions, which serve as a unified representation of 0-overlap functions and 1-grouping functions, the homogeneity, quasi-homogeneity, and pseudo-homogeneity of \(\Theta \) $\mathrm{\Theta }$ – \(\Xi \) $\mathrm{\Xi }$ functions are comprehensively characterized. Our findings are further illustrated through examples, providing a clear understanding of these functions and their properties.