Kenneth Rosen

Question

Let (A,*) be a semigroup. Furthermore let there be an element a in A such that for every x in A there exists u and v in A satisfying the relation-

a*u=v*a=x.

Prove that there is an identity element in A

Answer 1

Let x=a.

Thus, a*u=v*a=a.

Thus, u and v are identity elements of any element 'a' that belongs to A.