Rees factor semigroup

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. The new semigroup obtained in this way is called the Rees factor semigroup of S modulo I and is denoted by S/I.

The concept of Rees factor semigroup was introduced by David Rees in 1940.

A subset ${\displaystyle I}$ of a semigroup ${\displaystyle S}$ is called an ideal of ${\displaystyle S}$ if both ${\displaystyle SI}$ and ${\displaystyle IS}$ are subsets of ${\displaystyle I}$. Let ${\displaystyle I}$ be an ideal of a semigroup ${\displaystyle S}$. The relation ${\displaystyle \rho }$ in ${\displaystyle S}$ defined by

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