baer levi semigroup

[yunusnida]

Let X be a countable infinite set and let S be the set of one-one maps $\displaystyle \,\alpha\, :\, X\, \rightarrow\, X\,$ with the property that $\displaystyle \, X\, \backslash \, X\alpha\,$ is infinite.

(a) Show that S is a subgroup of $\displaystyle \, \mathcal{T}_X.$

(b) Show that for each $\displaystyle \, \alpha\,$ in S there is a one-one correspondence between $\displaystyle \, X\, \backslash\, X\alpha\,$ and $\displaystyle \, X\alpha\, \backslash\, X\alpha^2\,$.

(c) Deduce that S has no idempotent elements.

S is called the Baer Levi semigroup.

could you please help me to prove this ?

 

[stapel]

Let X be a countable infinite set and let S be the set of one-one maps $\displaystyle \,\alpha\, :\, X\, \rightarrow\, X\,$ with the property that $\displaystyle \, X\, \backslash \, X\alpha\,$ is infinite.

(a) Show that S is a subgroup of $\displaystyle \, \mathcal{T}_X.$

(b) Show that for each $\displaystyle \, \alpha\,$ in S there is a one-one correspondence between $\displaystyle \, X\, \backslash\, X\alpha\,$ and $\displaystyle \, X\alpha\, \backslash\, X\alpha^2\,$.

(c) Deduce that S has no idempotent elements.

S is called the Baer Levi semigroup.

could you please help me to prove this ?

Please reply showing your thoughts and efforts so far. Thank you!

 

[yunusnida]

ıf ı can find how can ı define the  function "a", ı may go further but ı cant find the type of elements of "a"

 

[pka]

ıf ı can find how can ı define the  function "a", ı may go further but ı cant find the type of elements of "a"

Is this a question from work on topological groups? Remember that many authors in top-groups have unique notations.

EXAMPLE: I have no idea that $\displaystyle \mathcal{T}_X$ is. A guess would be a set of mappings. Do you know?

I might guess that $\displaystyle X_{\alpha}$ might denote the $\displaystyle \alpha-$image of $\displaystyle X$.

What is the group (if it is a group) operation? Again a guess is $\displaystyle \alpha^2$ is function composition.

You should have textbook/lecture notes with all this laid out. Unless you can tell us, I see no way to help you,

 

[yunusnida]

Is this a question from work on topological groups? Remember that many authors in top-groups have unique notations.

EXAMPLE: I have no idea that $\displaystyle \mathcal{T}_X$ is. A guess would be a set of mappings. Do you know?

I might guess that $\displaystyle X_{\alpha}$ might denote the $\displaystyle \alpha-$image of $\displaystyle X$.

What is the group (if it is a group) operation? Again a guess is $\displaystyle \alpha^2$ is function composition.

You should have textbook/lecture notes with all this laid out. Unless you can tell us, I see no way to help you,

Bx is all binary relations on a set X. and
The subbset Px of Bx consisting of all partial maps of X and also Px is a subsemigroup of Bx.
Finally the set of Tx is all maps from X into itself.

These sets are not a group they are semigroup.

this is a questions from a semigroup theory book belongs John M. Howie

 

[yunusnida]

Solition


İ wrote it Türkish. Finally This iş a solition.