# The Unapologetic Mathematician

## Structures related to groups

We’ve said a lot about groups, but there are are a number of related structures. I’ve delayed these simpler structures because most of the easiest examples to give are actually groups anyway.

One of the most important is a monoid. A monoid is like a group, but without the requirement that each element have an inverse. Any group is thus a monoid — just forget the fact that there actually are inverses. Like groups, monoids may or may not be commutative. Between monoids we have monoid homomorphisms preserving the composition and identity element.

Many of the constructions for groups work for monoids too. For example, given a set $X$ we can form the free monoid $F(X)$. This is just like the free group, but without adding inverses for the generators. Also similar to the free group, any function from $X$ to the underlying set of a monoid $M$ extends to a unique monoid homomorphism from $F(X)$ to $M$. There are also direct and free products of monoids, which can be defined with the same sorts of universal properties as their analogues for groups.

Another construction that’s sort of interesting is the free group on a monoid. Given a monoid $M$, take another copy of it and call it $M'$. The element of $M'$ corresponding to an element $m$ of $M$ is $m'$. Now make a group by putting $M$ together with $M'$, identifying the identity elements $e$ and $e'$, and making $m'$ the inverse of $m$. This group $F(M)$ has the property that any monoid homomorphism from $M$ to the underlying monoid of a group $G$ extends to a unique group homomorphism from $F(M)$ to $G$.

The other related structure to mention is a semigroup. This doesn’t even have an identity element — just an associative composition. Again, there are semigroup homomorphisms, products and free products of semigroups, and so on. You should be able to give proper definitions of all these by analogy with monoids and groups. I don’t find semigroups to be as interesting as groups, but it’s a nice, concise term to have ready when it does come up.

One more structure that’s often mentioned in this context is a groupoid. I actually don’t want to go into groupoids now because there’s a much more natural way to describe them. To those of you who like them, rest assured I’ll get there eventually.

March 10, 2007 - Posted by | Algebra, Group theory

1. I know this is probably irrational, but I hate the term “monoid”. It bothers because it doesn’t sound at all related to groups, while “semigroup” does. It also breaks the analogy with rings. We don’t need special words for “ring without 1” and “ring with 1”, because the theories aren’t so different that you can’t just specify which you mean. The theories of semigroups and monoids just aren’t that different.

Comment by Walt | March 15, 2007 | Reply

2. Walt, you make a good point. In fact, (switching to high gear) when you consider what the left adjoint to the forgetful functor from monoid objects in a category to semigroup objects is, it’s pretty silly.

However, the word is there and we’re sort of stuck with it. We could no more change from “field” to “body” (like the rest of the world does) than start saying “semigroup with identity”.

On the other hand, outside of rings I’m hard pressed to come up with a natural semigroup that isn’t a monoid already. Monoids do seem to be more prevalent in the wild.

Comment by John Armstrong | March 16, 2007 | Reply

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