im reading a book and it says:
An operation O is right invertible or left invertible in the set K if for any two elements x and y of the set K there always exists an element z of K such that x=yOz or x=zOy.
An operation O which is both right and left invertible is simply invertible in the class K.
K is a group with respect to O if this K is closed under O and O is associative and invertible in K.
"To qualify as a group the set and operation, (G, •), must satisfy four requirements:
Closure: For all a, b in G, the result of the operation, a•b, is also in G.
Associativity: For all a, b and c in G, (a•b)•c = a•(b • c).
Identity element: There exists an element e in G such that, for every element a in G, the equation e•a=a•e=a holds.
Inverse element: For each a in G, there exists an element b in G, such that a•b=b•a=e, where e is the identity element."
I don't really get it, the closure property and the associativity property are the same in both definitions of group, but how right invertible and left invertible are the same of "identity element" or "inverse element"?