Rees factor semigroup

In

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.^[1][2]

Formal definition

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 ${\displaystyle I}$ (where ${\displaystyle SI=\{sx\mid s\in S{\text{ and }}x\in I\}}$, and similarly for ${\displaystyle IS}$). Let ${\displaystyle I}$ be an ideal of a semigroup ${\displaystyle S}$. The relation ${\displaystyle \rho }$ in ${\displaystyle S}$ defined by

x ρ y  ⇔  either x = y or both x and y are in I

is an equivalence relation in ${\displaystyle S}$. The equivalence classes under ${\displaystyle \rho }$ are the singleton sets ${\displaystyle \{x\}}$ with ${\displaystyle x}$ not in ${\displaystyle I}$ and the set ${\displaystyle I}$. Since ${\displaystyle I}$ is an ideal of ${\displaystyle S}$, the relation ${\displaystyle \rho }$ is a

${\displaystyle S}$

quotient semigroup

${\displaystyle S/{\rho }}$ is, by definition, the Rees factor semigroup of ${\displaystyle S}$ modulo ${\displaystyle I}$. For notational convenience the semigroup ${\displaystyle S/\rho }$ is also denoted as ${\displaystyle S/I}$. The Rees factor semigroup^[4] has underlying set ${\displaystyle (S\setminus I)\cup \{0\}}$, where ${\displaystyle 0}$ is a new element and the product (here denoted by ${\displaystyle *}$) is defined by

${\displaystyle s*t={\begin{cases}st&{\text{if }}s,t,st\in S\setminus I\\0&{\text{otherwise}}.\end{cases}}}$

The congruence ${\displaystyle \rho }$ on ${\displaystyle S}$ as defined above is called the Rees congruence on ${\displaystyle S}$ modulo ${\displaystyle I}$.

Example

Consider the semigroup S = { a, b, c, d, e } with the binary operation defined by the following Cayley table:

·	a	b	c	d	e
a	a	a	a	d	d
b	a	b	c	d	d
c	a	c	b	d	d
d	d	d	d	a	a
e	d	e	e	a	a

Let I = { a, d } which is a subset of S. Since

SI = { aa, ba, ca, da, ea, ad, bd, cd, dd, ed } = { a, d } ⊆ I

IS = { aa, da, ab, db, ac, dc, ad, dd, ae, de } = { a, d } ⊆ I

the set I is an ideal of S. The Rees factor semigroup of S modulo I is the set S/I = { b, c, e, I } with the binary operation defined by the following Cayley table:

·	b	c	e	I
b	b	c	I	I
c	c	b	I	I
e	e	e	I	I
I	I	I	I	I

Ideal extension

A semigroup S is called an ideal extension of a semigroup A by a semigroup B if A is an ideal of S and the Rees factor semigroup S/A is isomorphic to B. ^[5]

Some of the cases that have been studied extensively include: ideal extensions of

cancellation by a group with added zero. In general, the problem of describing all ideal extensions of a semigroup is still open.^[6]