I'm curious as to whether a certain mathematical structure has a name.
Let's say I have two sets, S and D, and I define a "difference" operation on elements in S which results in an element in D, and an addition operation taking an element of S and D to S, as well as an addition operation on D:
- : S x S -> D
+1 : S x D -> S
+2 : D x D -> D
This pattern fits for several real-world examples:
* The difference between two dates is a time interval. You can add a date and an interval to get a date. You can add two intervals to get an interval.
* The difference between two absolute file paths is a relative path. You can add an absolute path and a relative path to get an absolute path.
* In the C programming language, you can subtract two pointers and get a value of "pointer difference" type. The same concept exists for array indices and an array size type.
This concept seems similar to a metric space, but yet different in that D is not the set of reals, and that elements of D can be somehow "negative".
So my question is: Is there a name for this mathematical concept. Do S and D and the defined operations satisfy the rules for some sort of "space" that has a well-defined name in the mathematical community?
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