A semigroup G is a group if it has a left identity and every element has a left inverse. The purpose of this note is to weaken this condition further in two different ways. A semigroup G with an identity e is a group if every element x in G has an inverse. It is well known that this statement can be weakened. A semigroup G with a left identity e is a group if every element x in G has a left inverse [1, 2]. Can this weakened statement be weakened again? This article introduces an element u and two positive integer m, n to weaken the left identity equality ex=x as u[m]x=x, and the left inverse equality x'x=e as x'x=u[n].