# Addition and multiplication are one and the same…can this be generalized to any operators????

The addition and multiplication operations are essentially one and the same on intergers, i.e. a*b equals the sum of a "b times" over. Now, given certain criteria on the operators and the sets operators' act on can this be generalized?? Does this mean anything and can someone point me in the right direction to study this further.

Here is a sketch of what I'm thinking of...

I apologize in advance for the notation...

Conditions A:

Let G be a set that is infinite and countable.

We define an operator as (+) which acts on any particular elements in G to produce other elements in G which satisfy the following axioms:

Closure- For all a, b, in G the result of the operation a(+)b is also in G

Associativity- For all a, b, c in G the result a (+) [b (+) c] = [a (+) b] (+) c

Identity element- There exists an element in G such that, for every element b in G, the equation I (+) b =b (+) I = b. Such an element is unique.

Inverse element- For each b in G, there exists an element b-1 in G such that b (+) b-1 = b-1 (+) b = I where I is the identity element.

Now let, (+) have an associated relation M(b) which acts on any particular element in set G in such a way as to produce a finite collection of sets Ri = R1, R2, ...Rn, associated with that particular element. Hence M(b) is the collection of sets Ri associated with element b of G.

Properties of M(b)

1)The elements contained in a particular Ri can be duplicated.

2)The number of elements contained in a particular Ri is finite

3)The ordering of the elements in a particular Ri is irrelevant.

4)For a particular operator (+) the collection M(b) is unique and finite for a particular element b in G.

5)The sets that contain the identity element I are not included in M(b).

Examples:

Note that M(b) = <{x1,x2,…,xn}, {y1,y2,….,yn},….,{z}> x1,x2,y1,y2,z are not unique and are elements that are contained in G

Integer: M(4)

4 : <{2,2}> for standard integer multiplication

4 : <{1,1,1,1}, {1,1,2}, {1,3}, {2,2}> for standard integer addition

Now using everything described above we can now relate operators to each other in the following manner:

Operator Axioms:

1) Given an operator (+) that satisfies A and has an associated collection M(b) and suppose there exists a separate operator (%) satisfying conditions A and has an associated V(b). If M(b) = V(b) for all b in G then we say (+) = (%)

2) Suppose (+) and (%) both satisfy axioms A but that M(b) =/ V(b) then we say that (+) and (%) are co-operators. Note that the operators (+) and (%) are NOT equal.

Now I only list two trivial properties however I think many interesting ones could be defined. For example, take the normal addition and multiplication operators acting on the set of integers...We could say something like multiplilcation is just a higher order operater representation of the addition operator since for any particular integer b, the collections M(b) for addition has more members than the collection V(b) for multiplication.

Since you're asking for directions, I'll try to sum up what you wrote (perhaps more rigourously) and then ask some follow-up questions I'd ask myself when trying to see the overall structure I came up with.
I used mostly your notation.

$(G, \oplus)$ is an infinite countable group and $M$ is a function rather than relation (just for the simplicity, I know these two are the same thing). What $M$ does is it uniquely assigns to each element $b\in G\setminus\{I\}$ a finite set of finite multisets of $G$ in such a way that the identity element of $G$ never shows up.

Now, from your examples it looks like you want to note some special cases of this function $M$. We could say, for example, that $M$ partitions the group $(G, \oplus)$ if:

for each $b\in G\setminus\{I\}$, if $B\in M(b)$, then $\bigoplus B = b$,

where $\bigoplus B$ means "summing" all the elements in $B$. You could also add a condition that demands $M(b)$ to have all such possible multisets $B$, so that we could call it a maximal partition ;).
Note also, that for prime elements of the group we could have $M(b) = \emptyset$ (e.g. $M(5)$ in $(\mathbb{Z}, \cdot)$), but that still fits the definition (the implication is trivially true).

You then consider other possible operators on $G$ and axiomatize the uniqueness of this "partitioning" (if you choose to define it the way I did above, then it's rather a statement to be proved that an axiom). Two distinct operators are called co-operators, when they both partition the group $G$.

Have you tried to produce some trivial cases of such partitions?
Perhaps you could drop the condition of $G$ being infinite and consider some small examples like $\mathbb{Z}$ modulo $5$ or $11$?
In above cases, can you find - using partitions - a third (apart from $+$ and $\cdot$) operator on $\mathbb{Z}_5, \mathbb{Z}_{11}$?
Can you change an operaton of taking power so that it could form a group? How does its partition look like and how does it relate to sum and product? Hint: for $\mathbb{Z}_{11}$ you can use $a\odot b = a^{\log{b}}$, where $log$ is $2$-based.
Then I'd go back for $\mathbb{Z}$ to see if I can produce any other operators. That would require to deduce the partition from the two you already have and by observing its nature in finite cases.
See also how the whole thing works in other finite groups - symmetric group $S_4$, for example. I'd love to see that, if it's not trivial ;).

Good luck!