Radical in a class of semi-groups

A function $ \rho $ associating to each semi-group $ S $ a congruence $ \rho ( S) $( cf. Congruence (in algebra)) and having the following properties: 1) if $ S $ is isomorphic to $ T $ and $ \rho ( S) = 0 $( 0 denotes the equality relation), then $ \rho ( T) = 0 $; 2) if $ \theta $ is a congruence on $ S $ and $ \rho ( S / \theta ) = 0 $, then $ \rho ( S) \leq \theta $; and 3) $ \rho ( S / \rho ( S) ) = 0 $. If 1) and 3) are satisfied, then 2) is equivalent to

$$ \sup \{ \rho ( S) , \theta \} / \theta \leq \rho ( S / \theta ) $$

for every congruence $ \theta $ on $ S $. A semi-group $ S $ is called $ \rho $- semi-simple if $ \rho ( S) = 0 $. The class of $ \rho $- 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 $ \rho $- semi-simple semi-groups for some radical $ \rho $. If $ \rho ( S) = S \times S $, then $ S $ is called $ \rho $- 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 $ \mathfrak K $ 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 $ S $ the intersection of all congruences $ \theta $ such that $ S / \theta \in \mathfrak K $ turns out to be a radical, called $ \rho _ {\mathfrak K } $. The class $ \mathfrak K $ coincides with the class of $ \rho _ {\mathfrak K } $- semi-simple semi-groups if and only if it is closed relative to subdirect products. In this case $ S / \rho _ {\mathfrak K } ( S) $ is the largest quotient semi-group of $ S $ that lies in $ \mathfrak K $( see Replica).

Example. Let $ \mathfrak K $ be the class of semi-groups admitting a faithful irreducible representation (cf. Representation of a semi-group). Then

$$ \rho _ {\mathfrak K } ( s) = $$

$$ = \ \{ ( a , b ) : a , b \in S , ( a , b ) \in \mu ( as ) \cap \mu ( b s ) \textrm{ for all } s \in S \cup \emptyset \} , $$

where

$$ \mu ( a) = \{ {( x , y ) } : { x , y \in S , a ^ {m} x = a ^ {n} y \textrm{ for some } \ m , n \geq 0 } \} . $$

Radicals defined on a given class of semi-groups that is closed relative to homomorphic images have been studied.

Related to each radical $ \rho $ is the class of left polygons $ \Sigma ( \rho ) $( cf. Polygon (over a monoid)). Namely, if $ A $ is a left $ S $- polygon, then a congruence $ \theta $ on $ S $ is called $ A $- annihilating if $ ( \lambda , \mu ) \in \theta $ implies $ \lambda a = \mu a $ for all $ a \in A $. The least upper bound of all $ A $- annihilating congruences turns out to be an $ A $- annihilating congruence, and is denoted by $ \mathop{\rm Ann} A $. The class $ \Sigma ( \rho ) $, by definition, consists of all left $ S $- polygons $ A $ such that $ \rho ( S / \mathop{\rm Ann} A ) = 0 $, where $ S $ runs through the class of all semi-groups. If $ \theta $ is a congruence on $ S $, then a left $ ( S / \theta ) $- polygon lies in $ \Sigma ( \rho ) $ if and only if it lies in $ \Sigma ( \rho ) $ when considered as a left $ S $- polygon. Conversely, if one is given a class $ \Sigma $ of left polygons with these properties and if $ \Sigma ( s) $ is the class of all left $ S $- polygons in $ \Sigma $, then the function

$$ \rho ( S) = \ \left \{ \begin{array}{ll} S \times S & \textrm{ if } \Sigma ( S) \textrm{ is empty } , \\ \cap _ {A \in \Sigma ( S) } \mathop{\rm Ann} A & \textrm{ otherwise } , \\ \end{array} \right .$$

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 -rings" , Selected problems in algebra and logic , Novosibirsk (1973) pp. 283–299 (In Russian)
[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