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.