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 we can form the free monoid . This is just like the free group, but without adding inverses for the generators. Also similar to the free group, any function from to the underlying set of a monoid extends to a unique monoid homomorphism from to . 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 , take another copy of it and call it . The element of corresponding to an element of is . Now make a group by putting together with , identifying the identity elements and , and making the inverse of . This group has the property that any monoid homomorphism from to the underlying monoid of a group extends to a unique group homomorphism from to .
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.