The range of possible meanings can be so broad, that sometimes we could even say that the preposition doesn't have any meaning.
little semantic motivation of the preposition on
The apple hung on the tree.
The painting hung on the wall.
The book is on the table.
He did it on his behalf.
On the other side it just say nothing, this land was made for you and me.
This author would say that basic meanig of on is upon. One object is positioned horizontally upon another object. However apples in autumn are actually hanging ~in the trees. It is hard to see upon what they are hanging. A sentence like "Ripe apples are hanging on a branch" is still more strange, because this is not compatible with the gravity force. It is to suppose that the apples are under the branch and not upon the branch. However the position of a painting that hangs on the wall is completely different. It is positioned vertically on the wall. In "He did it on his behalf" on doesn't have any meaning. The only reason to use on in this context is the fact that there is no preposition better suited. ~with his behalf, ~ for his behalf, ~by his behalf wouldn't be better. The same thing is true for on the other side.
Many Portuguese nouns connect other nouns with the preposition a. There is no need to try to find a logic in that. There is none.
Arte é uma alternativa a um mundo repleto de violência.
Art is an alternative to a world full of violence.
A reação aos medicamentos é uma das principais causas de visitas ao consultório do alergista.
The reaction to drugs is one of the main reasons to visit an allergist.
O cheiro a mofo é extremamente desagradável.
The mouldy odour is very unpleasant.
O medo aos exames, como a maior parte das fobias, funcionam contra a lógica.
The fear of exams lacks, as most of phobia, any logic.