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