Translations of semi-groups
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 |
Translations of semi-groups. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Translations_of_semi-groups&oldid=49018