The substructure generated by a subset I am given the following definition of a substructure generated by a set: "For any subset A ⊆ X of an n-ary structure (X, µ), the family of subsets X' ⊆ X containing A and closed under µ is nonempty: it contains, for example X itself. By Exercise 38, their intersection is closed under µ. The intersection is the smallest subset of X containing A with this property. It will be denoted $\langle A\rangle$ henceforth. The corresponding substructure of (X, µ) is called the substructure of (X, µ) generated by a subset A ⊆ X." To put it in my own terms, the gist of the definition is that you want the smallest possible subset containing your set A that is also closed under the n-ary operation. I am currently trying to think of some examples where A is not equal to the set X in the substructure, and I was wondering what are some good examples of this?

Consider the semigroup $\Bbb Z^+$ under the binary operation of addition. What is the smallest sub-semigroup containing the set $A = \\{2\\}$?