In mathematics, in semigroup theory, a Rees factor semigroup (also called Rees quotient semigroup or just Rees factor), named after David Rees, is a certain semigroup constructed using a semigroup and an ideal of the semigroup.Let S be a semigroup and I be an ideal of S. Using S and I one can construct a new semigroup by collapsing I into a single element while the elements of S outside of I retain their identity.