View source for Translations of semi-groups

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Transformations of semi-groups that satisfy special conditions: a right translation of a [[Semi-group|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|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|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|holomorph of a group]] in group theory.

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