Examples of division outside of numbers

Question

There are many examples of addition and multiplication outside of numbers. In certain cases addition can be seen as essentially combination or merging. Likewise, multiplication can be seen as cloning or duplication.

For letters for example:

\begin{align} a + b &= ab\\ a^2 &= aa\\ a * 3 &= aaa \end{align}

Well, for $+$ you can combine two non-numbers, not sure about multiplication (non-number $*$ number, don't know if there is a non-number $*$ non-number).

Wondering if there is any parallel for division $\div$:

$$ \texttt{nonnumber} \div \texttt{nonnumber} $$

or even just

$$ \texttt{nonnumber} \div \texttt{number} $$

Answer 1

In your specific example, yes, you can define the operation of concatenation and use the $+$ symbol for it. You have the monoid of strings. A monoid is like a group but does not require inverses. You have an identity element, which is the empty string. It is natural to define multiplication by a natural number as repeated addition. Your second and third lines use the operation in that way but use different notation for it. The second uses a power notation while the third uses $*$. Let us use $*$ for your multiplication by a natural and $\cdot$ for multiplication in the naturals. $*$ is a perfectly well defined operation that concatenates a number of copies of the same string, so you could write $ab * 2=abab$ for example. It plays nicely with multiplication in the naturals in that $(ab * 2) * 3=ab *(2 \cdot 3)$ We often use this for groups where we write $na$ with $n \in \Bbb N$ and $a$ in the group as the sum of $n$ copies of $a$.

It is easy to define division by a natural. Just like in the naturals, there may not be a result. You can just take it to be the inverse of $*$ when it exists, so $aaa \div 3 = a$. We can use $\div$ for your division of strings by naturals and $/$ for division in the naturals. Again division in the naturals plays nice with your $*$ in that $(ab*6) \div 2 = ab *(6/2)$

Answer 2

If you are programming or using categories or universal algebra then in some sense the type of the objects does not matter. In particular there is no really reason to conceptualize a notion of "number" you just describe operations on your objects. For example, in functional programming the concept of a flat-map (or map-reduce) is regarded by some as a multipliction metaphor. That is made precise by extracting the abstraction of a monad functor F with a join $F\circ F\to F$. E.g. a List of Lists of integers is easily "joined" together into a single List of integers. For all mathematical purposes that is in a product, even if you wouldn't regard lists as numbers.

As for division then, if division to you means $a\div b$, i.e. a function def div(a:A,b:A):A then its can be as general as any multiplication metaphor, e.g. operating on sets or lists, or trees, elephants. You might then ask what makes it division verses a multiplication, and for that I would suggest that you have it in the company of some other operation you already called multiplication and which is in a meaningful sense reciprocal to your division. E.g. if you have $a\times b$ and $a\div b$ then $(a\times b)\div b=a$ or some such identity.

But as for caring if $a$ and $b$ are "numbers", there is no need for that on a mathematics level at least. Maybe a philosopher would object.