Characterize the invertible elements in a semigroup

Let S be a semigroup with identity e and let a \in S. Show that a is invertible if and only if Sa = aS = S.


Suppose a is invertible; then there exists an element b such that ab = ba = e. Now let s \in S. We have s = se = sba \in Sa and s = es = abs \in aS. So aS = Sa = S.

Conversely, suppose aS = Sa = S. Now e \in S, so there exist elements b,c \in S such that ab = ca = e. Now b = eb = cab = ce = c, so that in fact a is invertible.

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