In
mathematics, in
semigroup theory
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it.
The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy ...
, a Rees factor semigroup (also called Rees quotient semigroup or just Rees factor), named after
David Rees David or Dai Rees may refer to:
Entertainment
* David Rees (author) (1936–1993), British children's author
* Dave Rees (born 1969), American drummer for SNFU and Wheat Chiefs
* David Rees (cartoonist) (born 1972), American cartoonist and televis ...
, is a certain
semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it.
The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ...
constructed using a semigroup and an
ideal of the semigroup.
Let ''S'' be a
semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it.
The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ...
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 David or Dai Rees may refer to:
Entertainment
* David Rees (author) (1936–1993), British children's author
* Dave Rees (born 1969), American drummer for SNFU and Wheat Chiefs
* David Rees (cartoonist) (born 1972), American cartoonist and televis ...
in 1940.
Formal definition
A
subset of a semigroup
is called an ''ideal'' of
if both
and
are subsets of
(where
, and similarly for
). Let
be an ideal of a semigroup
. The relation
in
defined by
: ''x'' ρ ''y'' ⇔ either ''x'' = ''y'' or both ''x'' and ''y'' are in ''I''
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 (here denoted by
) 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:
Let ''I'' = which is a subset of ''S''. Since
:''SI'' = = ⊆ ''I''
:''IS'' = = ⊆ ''I''
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:
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 semigroup
In mathematics, a semigroup is a nonempty set together with an associative binary operation. A special class of semigroups is a class of semigroups satisfying additional properties or conditions. Thus the class of commutative semigroups consists o ...
s, of a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
by a
completely 0-simple semigroup
In mathematics, a semigroup is a nonempty set together with an associative binary operation. A special class of semigroups is a class of semigroups satisfying additional properties or conditions. Thus the class of commutative semigroups consists o ...
, 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.
References
*
{{PlanetMath attribution, id=3517, title=Rees factor
Semigroup theory