How to understand the convention on describing the "position" of mathematical objects When studying algebra, I find that it seems like a convention that we use positional words like on, under, over etc. to describe relations between things. For example:

$V$ is a vector space over $F$.


$G$ is a group under multiplication.


the operation on $G$ is defined by the usual addition.

Positional words are also used in other situations, for example we say:

Analysis on complex manifolds.

My observation is that such convention prefer to put structures/objects/operations on top of the (in some sense "more basic") things what define them.
To me, they are somewhat natural to say. But as I'm not a naive speaker, they are still pretty vague, for example, why we choose to say vector space over field and operation on set but not the other way around.
So, may I ask for the precise reasons why we "visualize" them as such?
 A: Although each preposition has a core meaning or set of (generally closely related) core meanings, prepositional usage is highly idiomatic, and not just in English: this is true in general. Once you get away from those core meanings, choice of preposition is largely a matter of idiom.
For instance, there is no outstandingly obvious choice of preposition to express the relationship between a group and its operation; the use of under, as in a group under multiplication, is simply idiomatic. It makes some intuitive sense if we think of the operation as something imposed on the underlying set, but one could make cases for other ways of thinking about the relationship.
Some of these idioms are already present in non-mathematical language. For instance, it is entirely idiomatic to speak of operating on or performing operations on things in general; to speak of addition, say, as an operation on the real numbers is just an instance of this existing idiom. In other cases a mathematical prepositional usage may not have clear non-mathematical parallels, and there may be no way to tell how it got started; presumably someone started using it, others picked it up, and it eventually became a standard idiom.
A: As in many languages, prepositions in English are a complete mess for non-native speakers. Any given preposition can have many different meanings (and the groupings of these meanings varies from language to language).
So my best answer to your question is that, while "over", "under", and "on" can all refer to positional relationships, they don't necessarily refer to positional relationships. Here are a few examples of many different usages of "over", "under", and "on". To me, the following example sentences from those links come closer to how these prepositions are used in mathematics than interpreting them as positional:

*

*"There was a lot of discussion over who should get the job." (connected with)

*"Children will educate themselves under [the] right conditions." (an aspect of)

*"We lived on an estate." (in a setting)
