I have a 3 answers but nobody return me a mathematical structure/category name when I try to classify "the set of integers with respect to subtraction is not a group"
1) Subtraction of integers (and subtraction in general) is just addition with an inverse.
2) In almost every group the operation $gh^{-1}$ is not a group operation since it is not associative. The exception are groups where $h^{-1} = h$. So there is no needed extra structure; the inverse works just fine as it is
3rd answer is more complicated
3) “A group where I’m telling you the group’s subtraction operation rather than the group’s addition operation”.
That is, among many other ways to axiomatize it, one way to define a group is as any two abstract operations $+$ and $−$ on an inhabited set such that the following equations hold in general:
$a+(b+c)=(a+b)+c$
$a+(b−c)=(a+b)−c$
$a−(b+c)=(a−c)−b$ [Note the order reversal here!]
$a−(b−c)=(a+c)−b$ [And again here!]
$(a+b)−b=a$
$(a−b)+b=a$
$(a−a)+b=b$
It turns out that if two groups have the same $+$, they automatically agree on everything else as well. And people often like to refer to groups just by referring to their + operator; so much so that people will call this “the group operation” or such things.
But it also turns out that if two groups have the same $−$, they automatically agree on everything else as well. The − formalized above is an operation just as well available in any group. And so you could just as well refer to groups by referring to their $−$ operator. This is a different way of specifying a group than by specifying its $+$ operator, but it works just as well.
$+$ and $−$ don’t have the same properties as each other, but they’re both equally a part of any group, and they both equally determine everything there is to say about a group.
So the integers with respect to subtraction can be thought of as an instance of this $−$ structure. Which amounts to the same thing as a group, but is described with respect to a different operation. If you like, call it a “soup”, and note that every group comes with a particular corresponding soup and vice versa.
Incidentally, I’ve described everything here using the language of + and −, but a very strong convention is to only use these names when dealing with groups which are commutative (that is, where a+b=b+a), and to otherwise speak in the language of multiplication and division instead. Still, to avoid confusion in this context, it felt best to simply use additive language.
I suspect that the question being asked here is if there exists a theory about sets with one binary operation that is closed and has inverses; if that is the case then the answer is yes, and they are called quasigroups.