Let $I$ be an ideal of a semigroup $S$ Define a congruence $\sim$ by $x \sim y$ iff $x = y$ or $x, y \in I$

Then the Rees factor of $S$ by $I$ is the quotient $S/\sim$ As a matter of notation, the congruence $\sim$ is normally suppressed, and the quotient is simply written $S/I$

Note that a Rees factor always has a zero element. Intuitively, the quotient identifies all element in $I$ and the resulting element is a zero element.