Groupe de Schützenberger – Wikipédia

In general algebra, and in particular in theory of half-groups, the Group the protection range is a group associated with a

${displaystyle {mathcal {H}}}$

-Classe, in the sense of the Green relationships of a half-group. Schützenberger groups of two

${displaystyle {mathcal {H}}}$

-Classes of the same

${displaystyle {mathcal {D}}}$

-Classe are isomorphic. If a

${displaystyle {mathcal {H}}}$

-Classe is a group, the Schützenberger group of this

${displaystyle {mathcal {H}}}$

-Classe is isomorphic to this class.

There are in fact two groups of Schützenberger associated with a

${displaystyle {mathcal {H}}}$

-class given; They are anti-isomorphic from each other.

Schützenberger groups were described by Marcel-Paul Schützenberger in 1957 ^{[ first ]} . They were named in the book by Alfred H. Clifford and Gordon Preston ^{[ 2 ] , [ 3 ]} .

Either

${displaystyle S}$

A half-group. We define

${displaystyle S^{1}}$

as being equal to

${displaystyle S}$

and

${displaystyle S}$

is a monoid, if not equal to

${displaystyle Scup {1}}$

, Or

${Displaystyle 1}$

is a neutral element added, therefore verifying

${displaystyle a1=1a=a}$

for everything

${displaystyle a}$

of

${displaystyle S^{1}}$

.

The relation de Green

${displaystyle {mathcal {H}}}$

is defined as follows. Be

${displaystyle a}$

And

${displaystyle b}$

two elements of

${displaystyle S}$

. SO

${displaystyle a{mathcal {H}}b}$

If and only if there is

${displaystyle x,y,z,t}$

In

${displaystyle S^{1}}$

such as

${displaystyle xa=yb}$

And

${DisplayStyle az = bt}$

.

The

${displaystyle {mathcal {H}}}$

-class of an element

${displaystyle a}$

is noted

${displaystyle H(a)}$

. This is the set of elements

${displaystyle b}$

of

${displaystyle S}$

such as

${displaystyle a{mathcal {H}}b}$

.

Either

${displaystyle H}$

a

${displaystyle {mathcal {H}}}$

-class of

${displaystyle S}$

. Either

${displaystyle T(H)}$

all the elements

${displaystyle t}$

of

${displaystyle S^{1}}$

such as

${displaystyle Ht}$

is a subset of

${displaystyle H}$

. Each

${displaystyle t}$

of

${displaystyle T(H)}$

defines a transformation, noted

${displaystyle gamma _{t}:Hto H}$

of

${displaystyle H}$

in himself who sends an element

${displaystyle h}$

on

${displaystyle ht}$

:

${displaystyle gamma _{t}:hmapsto ht}$

.

All

${displaystyle Gamma (H)}$

of these transformations is in fact a group for the composition of the functions, considered to be operating on the right (

${DisplayStyle Gamma _ {ts} = Gamma _ {t} Circ gamma _ {s}}}$

). It’s the Group the protection range associated with

${displaystyle {mathcal {H}}}$

-class

${displaystyle H}$

. The other group of Schützenberger is the group of multiplications on the right

${displaystyle delta _{t}:hmapsto th}$

.

All

${displaystyle {mathcal {H}}}$

-class

${displaystyle H}$

To the same cardinality as his Schützenberger group

${displaystyle Gamma (H)}$

.
And

${displaystyle H}$

is a maximum subgroup of a monoid

${displaystyle M}$

, SO

${displaystyle H}$

is a

${displaystyle {mathcal {H}}}$

-Classe and is canonically isomorphic to his group in Schützenberger.

A number of algebraic properties of monoids are reflected in their group of
Schützenberger. Thus, a monoid who has a finite number of ideals on the left and right is finished, or simply in the end of it and only if all his groups of Schützenberger are.