Rees factor semigroup


In mathematics, in semigroup theory, a Rees factor semigroup, 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 of a semigroup is called an ideal of
if both and are subsets of
. Let be an ideal of a semigroup
. The relation in defined by
is an equivalence relation in. The equivalence classes under are the singleton sets with not
in and the set. Since is an ideal of, the relation is a congruence on. The quotient semigroup is, by definition, the Rees factor semigroup of modulo
. For notational convenience the semigroup is also denoted as. The Rees factor
semigroup has underlying set, where is a new element and the product is defined by
The congruence on as defined above is called the Rees congruence on modulo.

Example

Consider the semigroup S = with the binary operation defined by the following Cayley table:

·abcde
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 = which is a subset of S. Since
the set I is an ideal of S. The Rees factor semigroup of S modulo I is the set S/I = with the binary operation defined by the following Cayley table:

·bceI
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.
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.