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In this contribution, we fully characterize one- and two-dimensional positively homogeneous aggregation functions and provide a method of checking non-decreasingness of positively homogeneous functions in higher dimensions. We also examine the interaction of positive homogeneity with super-addivity and sub-additivity of functions and link it to the concavity and convexity, respectively, of their generalized opposite diagonal functions. Some remarks on the computation of a linear super-additive transformation of aggregation functions are added. [ABSTRACT FROM AUTHOR]