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.

Formal definition
A subset $$I$$ of a semigroup $$S$$ is called an ideal of $$S$$ if both $$SI$$ and $$IS$$ are subsets of $$I$$ (where $$SI = \{sx \mid s \in S \text{ and } x \in I\}$$, and similarly for $$IS$$). Let $$I$$ be an ideal of a semigroup $$S$$. The relation $$\rho$$ in $$S$$ defined by


 * x &rho; y ⇔  either x = y or both x and y are in I

is an equivalence relation in $$S$$. The equivalence classes under $$\rho$$ are the singleton sets $$\{x\}$$ with $$x$$ not in $$I$$ and the set $$I$$. Since $$I$$ is an ideal of $$S$$, the relation $$\rho$$ is a congruence on $$S$$. The quotient semigroup $$S/{\rho}$$ is, by definition, the Rees factor semigroup of $$S$$ modulo $$I$$. For notational convenience the semigroup $$S/\rho$$ is also denoted as $$S/I$$. The Rees factor semigroup has underlying set $$(S \setminus I) \cup \{0\}$$, where $$0$$ is a new element and the product (here denoted by $$*$$) is defined by

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

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

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

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


 * SI = { aa, ba, ca, da, ea, ad, bd, cd, dd, ed } = { a, d } &sube; I
 * IS = { aa, da, ab, db, ac, dc, ad, dd, ae, de } = { a, d } &sube; 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:

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.

Some of the cases that have been studied extensively include: ideal extensions of completely simple semigroups, of a group by a completely 0-simple semigroup, of a commutative semigroup with cancellation by a group with added zero. In general, the problem of describing all ideal extensions of a semigroup is still open.