Page "Universal algebra" Paragraph 31

It is important to check that this really does capture the definition of a group.

The reason that it might not is that specifying one of these universal groups might give more information than specifying one of the usual kind of group.

After all, nothing in the usual definition said that the identity element e was unique ; if there is another identity element e, then it is ambiguous which one should be the value of the nullary operator e. However, this is not a problem because identity elements can be proved to be always unique.

The same thing is true of inverse elements.

So the universal algebraist's definition of a group really is equivalent to the usual definition.