Rees factor

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.