I have learned (in naive set theory) that intersection, union, difference, cartesian product, etc. are defined as operations on sets. But now when I'm studying abstract algebra, I came to know that an algebraic structure is actually, sets with operations defined on it. But then I thought that any arbitrary set has the operations like the ones mentioned above, defined on them. So is it possible that there exists an algebraic structure where the underlying set is the set of all sets while the operations defined on this structure are intersection, union, difference, cartesian product, etc? If this is true can you suggest me any source from where I can find further details about this concept?
Also when we define the operation (also functions and relations) we use the concept of the cartesian product, so this raises a question in my mind that how can we define operation using an operation itself? This process appears a bit cyclic to me.