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Translations of semi-groups

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Transformations of semi-groups that satisfy special conditions: a right translation of a semi-group is a transformation such that for any ; a left translation is defined similarly. For convenience, left translations are often written as left operators. Thus, a left translation of is a transformation such that for any . 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 (respectively, ) of all left (respectively, right) translations of is a sub-semi-group of the symmetric semi-group . For any the transformation () defined by (respectively, ) is the left (respectively, right) translation corresponding to . It is called the inner left (respectively, right) translation. The set (respectively, ) of all inner left (respectively, right) translations of is a left ideal in (respectively, a right ideal in ).

A left translation and a right translation of are called linked if for any ; in this case the pair is called a bi-translation of . For any , the pair is a bi-translation, called the inner bi-translation corresponding to . In semi-groups with a unit, and only in them, every bi-translation is inner. The set of all bi-translations of is a sub-semi-group of the Cartesian product ; it is called the translational hull of . The set of all inner bi-translations is an ideal in , called the inner part of . The mapping defined by is a homomorphism of onto , called the canonical homomorphism. A semi-group is called weakly reductive if for any the relations and for all imply that , that is, the canonical homomorphism of is an isomorphism. If is weakly reductive, then coincides with the idealizer of in , that is, with the largest sub-semi-group of containing 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
How to Cite This Entry:
Translations of semi-groups. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Translations_of_semi-groups&oldid=49018
This article was adapted from an original article by L.N. Shevrin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article