Translations of semi-groups

Transformations of semi-groups that satisfy special conditions: a right translation of a semi-group $S$ is a transformation $\rho$ such that $( xy) \rho = x ( y \rho )$ for any $x, y \in S$; a left translation is defined similarly. For convenience, left translations are often written as left operators. Thus, a left translation of $S$ is a transformation $\lambda$ such that $\lambda ( xy) = ( \lambda x) y$ for any $x, y \in S$. The successive application of two left translations (see Transformation semi-group) is written from right to left. The product of two left (respectively, right) translations of a semi-group is itself a left (respectively, right) translation, so that the set $\Lambda ( S)$( respectively, $\textrm{ P } ( S)$) of all left (respectively, right) translations of $S$ is a sub-semi-group of the symmetric semi-group ${\mathcal T} _ {S}$. For any $a \in S$ the transformation $\lambda _ {a}$( $\rho _ {a}$) defined by $\lambda _ {a} x = ax$( respectively, $x \rho _ {a} = xa$) is the left (respectively, right) translation corresponding to $a$. It is called the inner left (respectively, right) translation. The set $\Lambda _ {0} ( S)$( respectively, $\textrm{ P } _ {0} ( S)$) of all inner left (respectively, right) translations of $S$ is a left ideal in $\Lambda ( S)$( respectively, a right ideal in $\textrm{ P } ( S)$).

A left translation $\lambda$ and a right translation $\rho$ of $S$ are called linked if $x ( \lambda y) = ( x \rho ) y$ for any $x, y \in S$; in this case the pair $( \lambda , \rho )$ is called a bi-translation of $S$. For any $a \in S$, the pair $( \lambda _ {a} , \rho _ {a} )$ is a bi-translation, called the inner bi-translation corresponding to $a$. In semi-groups with a unit, and only in them, every bi-translation is inner. The set $T ( S)$ of all bi-translations of $S$ is a sub-semi-group of the Cartesian product $\Lambda ( S) \times \textrm{ P } ( S)$; it is called the translational hull of $S$. The set $T _ {0} ( S)$ of all inner bi-translations is an ideal in $T ( S)$, called the inner part of $T ( S)$. The mapping $\tau : S \rightarrow T _ {0} ( S)$ defined by $\tau ( a) = ( \lambda _ {a} , \rho _ {a} )$ is a homomorphism of $S$ onto $T _ {0} ( S)$, called the canonical homomorphism. A semi-group $S$ is called weakly reductive if for any $a, b \in S$ the relations $ax = bx$ and $xa = xb$ for all $x \in S$ imply that $a = b$, that is, the canonical homomorphism of $S$ is an isomorphism. If $S$ is weakly reductive, then $T ( S)$ coincides with the idealizer of $T _ {0} ( S)$ in $\Lambda ( S) \times \textrm{ P } ( S)$, that is, with the largest sub-semi-group of $\Lambda ( S) \times \textrm{ P } ( S)$ containing $T _ {0} ( S)$ as an ideal.

Translations of semi-groups, and in particular, translational hulls, play an important role in the study of ideal extensions of semi-groups (cf. Extension of a semi-group). Here the role of the translational hull is to a certain extent similar to that of the holomorph of a group in group theory.

References

[1]	A.H. Clifford, G.B. Preston, "The algebraic theory of semi-groups" , 1 , Amer. Math. Soc. (1967)
[2]	M. Petrich, "Introduction to semigroups" , C.E. Merrill (1973)
[3]	M. Petrich, "The translational hull in semigroups and rings" Semigroup Forum , 1 (1970) pp. 283–360