I've read and fully understand Cayley's theorem for groups, however when I get to the theorem for semigroups I come to a complete stop.
I've figured that the identity and cancellative properties are important but I'm struggling to link together all the different conditions.
So if I have a semigroup that already contains an identity (a monoid) will the difference between a left and right representation change the type of homomorphism, for instance will a right representation produce a isomorphism or just a monomorphism (I have been able to show the monomorphism for an example of right representation).
Then when it comes to a semigroup that doesn't contain an identity already is the only option to fix an identity to the semigroup or does it still work without one. Again does the choice between left or right representation have any impact on the theorem? I read somewhere that the right translation may not always be one-to-one, why is this so? Is there a simple example?