# React for the algebra enthusiast – Part 1

When I learned to use the react framework, I always had the feeling that it is written in a very mathy way. Since simple googling did not give me any hints if this was a consideration in the design, I thought it might be worth sharing my thoughts on that. I should mention that I am sure others have made the same observations, but it might help algebraist to understand react faster and mathy computer scientiests to remember some algebra.

## Free monoids

In abstract algebra, a monoid is a set M together with a binary operation “ $\cdot$” satisfying these two laws:

• There is a neutral element “e”, such that: $\forall x \in M: x \cdot e = e \cdot x = e$
• The operation is associative, i.e. $\forall x,y,z \in M: x \cdot (y\cdot z) = (x\cdot y) \cdot z$

Here are some examples:

• Any set with exactly one element together with the unique choice of operation on it.
• The natural numbers $\mathbb{N}=\{0,1,2,\dots \}$ with addition.
• The one-based natural numbers $\mathbb{N}_1=\{1,2,3,\dots\}$ with multiplication.
• The Integers $\mathbb Z$ with addition.
• For any set M, the set of maps from M to M is a monoid with composition of maps.
• For any set A, we can construct the set List(A), consisting of all finite lists of elements of A. List(A) is a monoid with concatenation of lists. We will denote lists like this: $[1,2,3,\dots]$

Monoids of the form List(A) are called free. With “of the form” I mean that the elements of the sets can be renamed so that sets and operations are the same. For example, the monoid $\mathbb{N}$ with addition and List({1}) are of the same form, witnessed by the following renaming scheme: $0 \mapsto []$ $1 \mapsto $ $2 \mapsto [1,1]$ $3 \mapsto [1,1,1]$ $\dots$

— so addition and appending lists are the same operation under this identification.

With the exception of $\mathbb{N}_1$, the integers and the monoid of maps on a set, all of the examples above are free monoids. There is also a nice abstract definition of “free”, but for the purpose at hand to describe a special kind of monoid, it is good enough to say, that a monoid M is free, if there is a set A such that M is of the form List(A).

## Action monoids

A react-app (and by that I really mean a react+redux app) has a set of actions. An action always has a type, which is usally a string and a possibly empty list of arguments.

Let us stick to a simple app for now, where each action just has a type and nothing else. And let us further assume, that actions can appear in arbirtrary sequences. That means any action can be fired in any state. The latter simplification will keep us clear from more advanced algebra for now.

For a react-app, sequences of actions form a free monoid. Let us look at a simple example: Suppose our app is a counter which starts with “0” and has an increment (I) and decrement (D) action. Then the sequences of action can be represented by strings like

ID, IIDID, DDD, IDI, …

which form a free monoid with juxtaposition of strings. I have to admit, so far this is not very helpful for a practitioner – but I am pretty sure the next step has at least some potential to help in a complicated situation:

## Quotients

Quotients of sets by an equivalence relation are a very basic tool of modern math. For a monoid, it is not clear if a quotient of its underlying set will still be a monoid with the “same” operations.

Let us look at an example, where everything goes well. In the example from above, the counter should show the same integer if we decrement and then increment (or the other way around). So we could say that the two action sequences

• ID and
• DI

do really nothing and should be considered equivalent to the empty action sequence. So let’s say that any sequence of actions is equivalent to the same sequence with any occurence of “DI” or “ID” deleted. So for example we get:

IIDIIDD $\sim$ I

With this rule, we can reduce any sequence to an equivalent one that is a sequence of Is, a sequence of Ds or empty. So the quotient monoid can be identified with the integers (in two different ways, but that’s ok) and addition corresponds to juxtaposition of action sequences.

The point of this example and the moral of this post is, that we can take a syntactic description (the monoid of action sequences), which is easy to derive from the source code and look at a quotient of the action monoid by a reasonable relation to arrive at some algebraic structure which has a lot to do with the semantic of the app.

So the question remains, if this works just well for an example or if we have a general recipe.

Here is a problem in the general situation: Let $x,y,z\in M$ be elements of a monoid $M$ with operation “ $\cdot$” and $\sim$ be an equivalence relation such that $x$ is identified with $y$. Then, denoting equivalence classes with $[\_]$ it is not clear if $[x] \cdot [y]$ should be defined to be $[x\cdot z]$ or $[y\cdot z]$.

Fortunately problems like that disappear for free monoids like our action monoid and equivalence relations constructed in a specific way. As you can see on wikipedia, it is always ok to take the equivalence relation generated by the same kind of identifications we made above: Pick some pairs of sequences which are known “to do the same” from a semantic point of view (like “ID” and “DI” did the same as the empty sequence) and declare sequences to be equivalent, if they arise by replacing sequences known to be the same.

So the approach is that general: It works for apps, where actions do not have parameters and can be fired in any order and for equivalence relations generated by defining finitely many action sequences to do the same. The “any order” is a real restriction, but this post also has a “Part 1” in the title…

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