 Then every element of "S" has at least one inverse.Suppose that "a" in "S" has two inverses "b" and "c", i.e.,
Then
So, by commuting the pairs of idempotents "ab" & "ac" and "ba" & "ca", the inverse of "a" is shown to be unique.Conversely, it can be shown that any inverse semigroup is a regular semigroup in which idempotents commute.