I've seen copious usage of prepositions like "on" and "over" in mathematical texts with no concrete description of what they mean. Can someone please precisely define these terms for me, as in, what does it really mean when you say "let x be a y on z?
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My general impression is that a foo on $X$ is some kind of function, loosely speaking, with domain $X$ while a foo over $X$ is some kind of function, loosely speaking, with codomain $X$. But these terms don't really have completely precise meanings; you learn how to use them from seeing how other people use them (the same way you learned how to use most of the words you know).