Commutative and additive monoids

Question

I am unfamiliar with higher levels of topology and number theory but find myself working on a project in which I need an understanding of some of those topics. Is anyone able to provide information on the difference between a commutative and additive monoid? Or are they the same type of monoid and additive is just a property?

Answer 1

A monoid is a set $M$ together with a binary operation $\times\colon M\times M\to M$ (usually denoted using infix notation), which is:

1. Assocative: for all $x,y,z\in M$, $(x\times y)\times z = x\times(y\times z)$.
2. Has an identity: there exists $e\in M$ such that for all $x\in M$, $e\times x = x\times e = x$.

The monoid is “commutative” if for all $x,y\in M$, we have $x\times y = y\times x$.

An example of a monoid is to take a set $X$ and let $M$ be the set of all functions $X\to X$; and letting the operation be composition of functions. The identity element is the identity function. This is generally not a commutative monoid.

An example of a commutative monoid is the positive integers under multiplication. Another example is the nonnegative integers under addition.

Often, instead of $\times$ we simply use juxtaposition, so that instead of writing “$a\times b$”, we write $ab$ or $a\cdot b$. We also use “$1$” to denote the identity.

However, it is also common that when the operation is commutative, we use “additive notation”, and use $+$ for the operation symbol and $0$ for the name of the identity.

It sometimes happens that additive notation is used even if the monoid is not commutative, but that is rare and the authors will tell you they are doing so explicitly and prominently.