# 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

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