A function
associating to each
semi-group
a congruence
(cf.
Congruence (in algebra))
and having the following properties: 1) if
is isomorphic to
and
(0 denotes the equality relation), then
;
2) if
is a congruence on
and
,
then
;
and 3)
.
If 1) and 3) are satisfied, then 2) is equivalent to
for every congruence
on
.
A semi-group
is called
-semi-simple
if
.
The class of
-semi-simple
semi-groups contains the one-element semi-group and is closed
relative to isomorphism and subdirect products. Conversely, each class
of semi-groups having these properties is the class of
-semi-simple
semi-groups for some radical
.
If
,
then
is called
-radical.
In contrast to rings, in semi-groups the radical is not determined by
the corresponding radical class. If in the definition of a radical
the discussion is limited to congruences defined by ideals, then
another concept of a radical arises, where the corresponding function chooses an
ideal
in each semi-group.
If
is a class of semi-groups that is closed relative to
isomorphisms and that contains the one-element semi-group, then
the function that associates to each semi-group
the intersection of all congruences
such that
turns out to be a radical, called
.
The class
coincides with the class of
-semi-simple
semi-groups if and only if it is closed relative to subdirect products. In this case
is the largest quotient semi-group of
that lies in
(see
Replica).
Example.
Let
be the class of semi-groups admitting a faithful irreducible representation (cf.
Representation of a semi-group).
Then
where
Radicals defined on a given class of semi-groups that
is closed relative to homomorphic images have been studied.
Related to each radical
is the class of left polygons
(cf.
Polygon (over a monoid)).
Namely, if
is a left
-polygon,
then a congruence
on
is called
-annihilating
if
implies
for all
.
The least upper bound of all
-annihilating
congruences turns out to be an
-annihilating
congruence, and is denoted by
.
The class
,
by definition, consists of all left
-polygons
such that
,
where
runs through the class of all semi-groups. If
is a congruence on
,
then a left
-polygon
lies in
if and only if it lies in
when considered as a left
-polygon.
Conversely, if one is given a class
of left polygons with these properties and if
is the class of all left
-polygons
in
,
then the function
is a radical.