Radical in a class of semi-groups
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
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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
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where
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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
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is a radical.
References
[1] | A.H. Clifford, G.B. Preston, "The algebraic theory of semi-groups" , 2 , Amer. Math. Soc. (1967) |
[2] | L.A. Skornyakov, "Radicals of ![]() |
[3] | A.H. Clifford, "Radicals in semigroups" Semigroup Forum , 1 : 2 (1970) pp. 103–127 |
[4] | E.N. Roiz, B.M. Schein, "Radicals of semigroups" Semigroup Forum , 16 : 3 (1978) pp. 299–344 |
Radical in a class of semi-groups. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Radical_in_a_class_of_semi-groups&oldid=48413