I think the following analogies are too interesting to be ignored:
Union = Least Common Multiple
If $G_1,...,G_n$ denote a number of sets of points (either linear or in any number of dimensions), the set which contains every point that belongs to one or more of the given sets is called their greatest common measure, and is denoted by $M(G_1,...,G_n)$. In case no two of the given sets have a point in common, the common measure of the sets may be denoted by $G_1 + G_2 + ...$ and it may be spoken of as their sum. By some writers the term "sum" is employed for the greatest common measure.
Intersection = Greatest Common Divisor
That set which contains all those points which belong to every one of the given sets is called their greatest common divisor, and it may be denoted by $D(G_1,...,G_n)$.
Symmetric Difference
A common operation in sets is called "symmetric difference". It is used in a special type of set called a boolean algebra. The basic operation on two sets is to subtract the intersection from the union. To do this in analogy to the integers is to divide the lcm by the gcd.
Prime numbers = singleton sets
Composite Number = Set with subsets (Divisor?)
If all the points of a set H are points of a set G, H is said to be contained in G, or to be a part, or component, of G.
Remainder = Complement?
The set $G$ is said to contain $H$. Those points of $G$ that do not belong to $H$ form a set which may be denoted by $G-H$. The set $G-H$ is said to be the complement of $H$ with respect to $G$, and is sometimes denoted by $C_G(H)$.
Derivative of a Set
Returning to the case of a set $G$ in a finite interval or cell, we observe that the limiting points of $G$ form a set of points which may be finite or infinite; this set is called the derived set, or first derivative of $G$, and may be denoted by $G'$. In case the set $G$ contains an infinite number of points, it possesses itself a derivative set $G''$, which is called the second derivative of $G$. If we proceed in this manner, we may obtain a series $G',G'',...$ of derivatives of $G$. If the n'th derivative $G^n$ contains a finite number only of points, then these have no limiting point, and we may say that $G^{n+1}=0$
$e^x$ = Perfect Set
A set $G$ which is both closed and dense in itself is said to be perfect*. Thus a perfect set $G$ is identical with its derivative.
Does anybody have anymore of these? A reference perhaps? Comments? Can this be made more structured?
I don't really understand the denseness argument, but it sounds very similar to gauge invariance / or the additive constant of integration or something:
If, from any set $G$, we remove those points which also belong to its derivative, the remainder is an isolated set; thus $G - D(G,G')$ forms an isolated set. Any set $G$ may be regarded as the sum of an isolated set and of a component of the derivative $G$ . If a component $H$ of the set $G$ is such that every point of $G$ is a limiting point of $H$, the set $H$ is said to be dense in $G$.
A few people have mentioned, if you study abstract algebra, then you know that integers form a ring under addition and multiplication. If you go on further to study some category theory then you will see that it is not surprising because, to put it vaguely, you can try to find bijections between the sets of different algebraic structures. Category theory formalises these patterns.
And yes, you can take other algebraic structures as well. Here is a link to the first paper written on this, if you're interested in reading it. I don't know what your background is but it might be helpful to know something about more than one algebraic structure (vector spaces, groups, rings etc.) so you have pet examples to examine this concept in more detail.
What you start with is something called a category. A category takes some notions of structure we see and formalises them. For example a group, or natural numbers, or a set. But then this structure itself can be seen as an object to be put in a category called a functor. So the set of sets, or the set of groups with a certain property. You can then take something called a natural transformation. As wikipedia says: "Abstracting yet again, some diagrammatic and/or sequential constructions are often "naturally related" – a vague notion, at first sight. This leads to the clarifying concept of natural transformation, a way to "map" one functor to another. Many important constructions in mathematics can be studied in this context. "Naturality" is a principle, like general covariance in physics, that cuts deeper than is initially apparent. An arrow between two functors is a natural transformation when it is subject to certain naturality or commutativity conditions."