Strong Folner condition(SFC) implies the existence of a left Følner sequence.
I got stuck with this problem while reading Density in Arbitrary Semigroups by Hindman and Strauss. It says:
Problem: If $S$ is a countable semigroup. Then SFC on $S$ implies the existence of a left Følner sequence.
Definition of SFC: $forall; Hin mathscr{P}_f(S),; forall; epsilon>0,; exists; Kin mathscr{P}_f(S)$ such that $forall ; sin H, ; |KDelta sK|<epsilon|K|$.
Definition of left Folner sequence: Let $S$ be a semigroup. A left Følner sequence in $mathscr{P}_f(S)$ is a sequence ${F_n}_{ninmathbb{N}}$ in $mathscr{P}_f(S)$ such that for each $sin S, lim_{nrightarrowinfty}frac{|sF_nDelta F_n|}{|F_n|}=0$
($mathscr{P}_f(S)$ stands for the set of all finite subsets of $S$).
Thanks for any help.
ergodic-theory semigroups ramsey-theory amenability
asked Nov 29 at 15:19
Surajit
5689
add a comment |
I got stuck with this problem while reading Density in Arbitrary Semigroups by Hindman and Strauss. It says:
Problem: If $S$ is a countable semigroup. Then SFC on $S$ implies the existence of a left Følner sequence.
Definition of SFC: $forall; Hin mathscr{P}_f(S),; forall; epsilon>0,; exists; Kin mathscr{P}_f(S)$ such that $forall ; sin H, ; |KDelta sK|<epsilon|K|$.
Definition of left Folner sequence: Let $S$ be a semigroup. A left Følner sequence in $mathscr{P}_f(S)$ is a sequence ${F_n}_{ninmathbb{N}}$ in $mathscr{P}_f(S)$ such that for each $sin S, lim_{nrightarrowinfty}frac{|sF_nDelta F_n|}{|F_n|}=0$
($mathscr{P}_f(S)$ stands for the set of all finite subsets of $S$).
Thanks for any help.
ergodic-theory semigroups ramsey-theory amenability
asked Nov 29 at 15:19
Surajit
5689
add a comment |
I got stuck with this problem while reading Density in Arbitrary Semigroups by Hindman and Strauss. It says:
Problem: If $S$ is a countable semigroup. Then SFC on $S$ implies the existence of a left Følner sequence.
Definition of SFC: $forall; Hin mathscr{P}_f(S),; forall; epsilon>0,; exists; Kin mathscr{P}_f(S)$ such that $forall ; sin H, ; |KDelta sK|<epsilon|K|$.
Definition of left Folner sequence: Let $S$ be a semigroup. A left Følner sequence in $mathscr{P}_f(S)$ is a sequence ${F_n}_{ninmathbb{N}}$ in $mathscr{P}_f(S)$ such that for each $sin S, lim_{nrightarrowinfty}frac{|sF_nDelta F_n|}{|F_n|}=0$
($mathscr{P}_f(S)$ stands for the set of all finite subsets of $S$).
Thanks for any help.
ergodic-theory semigroups ramsey-theory amenability
asked Nov 29 at 15:19
Surajit
5689
I got stuck with this problem while reading Density in Arbitrary Semigroups by Hindman and Strauss. It says:
Problem: If $S$ is a countable semigroup. Then SFC on $S$ implies the existence of a left Følner sequence.
Definition of SFC: $forall; Hin mathscr{P}_f(S),; forall; epsilon>0,; exists; Kin mathscr{P}_f(S)$ such that $forall ; sin H, ; |KDelta sK|<epsilon|K|$.
Definition of left Folner sequence: Let $S$ be a semigroup. A left Følner sequence in $mathscr{P}_f(S)$ is a sequence ${F_n}_{ninmathbb{N}}$ in $mathscr{P}_f(S)$ such that for each $sin S, lim_{nrightarrowinfty}frac{|sF_nDelta F_n|}{|F_n|}=0$
($mathscr{P}_f(S)$ stands for the set of all finite subsets of $S$).
Thanks for any help.
ergodic-theory semigroups ramsey-theory amenability
ergodic-theory semigroups ramsey-theory amenability
asked Nov 29 at 15:19
Surajit
5689
asked Nov 29 at 15:19
Surajit
5689
asked Nov 29 at 15:19
Surajit
5689
asked Nov 29 at 15:19
Surajit
5689
asked Nov 29 at 15:19
Surajit
5689
5689
add a comment |
add a comment |
1 Answer
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Hint Since $S$ is countable, you can construct an increasing sequence $H_1 subset H_2 subset ... subset H_n subset ...$ of finite subsets such that
$$S= bigcup_n H_n$$
For each $H_n$ you can find some $F_n$ such that
$$
frac{|sF_nDelta F_n|}{|F_n|} < frac{1}{n} qquad forall s in H_n
$$
Show that $F_n$ is a left Følner sequence.
answered Nov 29 at 15:27
N. S.
101k5109204
I almost tried in this way. But for each $H_m$, I was considering a Folner sequence $F_n^{(m)}$, and I thought, may be $F_n^{(n)}$ would work. Anyway, thank you very much for your help. Upvoted and accepted.
– Surajit
Nov 29 at 15:37
add a comment |
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1 Answer
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1 Answer
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active
oldest
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Hint Since $S$ is countable, you can construct an increasing sequence $H_1 subset H_2 subset ... subset H_n subset ...$ of finite subsets such that
$$S= bigcup_n H_n$$
For each $H_n$ you can find some $F_n$ such that
$$
frac{|sF_nDelta F_n|}{|F_n|} < frac{1}{n} qquad forall s in H_n
$$
Show that $F_n$ is a left Følner sequence.
answered Nov 29 at 15:27
N. S.
101k5109204
I almost tried in this way. But for each $H_m$, I was considering a Folner sequence $F_n^{(m)}$, and I thought, may be $F_n^{(n)}$ would work. Anyway, thank you very much for your help. Upvoted and accepted.
– Surajit
Nov 29 at 15:37
add a comment |
Hint Since $S$ is countable, you can construct an increasing sequence $H_1 subset H_2 subset ... subset H_n subset ...$ of finite subsets such that
$$S= bigcup_n H_n$$
For each $H_n$ you can find some $F_n$ such that
$$
frac{|sF_nDelta F_n|}{|F_n|} < frac{1}{n} qquad forall s in H_n
$$
Show that $F_n$ is a left Følner sequence.
answered Nov 29 at 15:27
N. S.
101k5109204
I almost tried in this way. But for each $H_m$, I was considering a Folner sequence $F_n^{(m)}$, and I thought, may be $F_n^{(n)}$ would work. Anyway, thank you very much for your help. Upvoted and accepted.
– Surajit
Nov 29 at 15:37
add a comment |
Hint Since $S$ is countable, you can construct an increasing sequence $H_1 subset H_2 subset ... subset H_n subset ...$ of finite subsets such that
$$S= bigcup_n H_n$$
For each $H_n$ you can find some $F_n$ such that
$$
frac{|sF_nDelta F_n|}{|F_n|} < frac{1}{n} qquad forall s in H_n
$$
Show that $F_n$ is a left Følner sequence.
answered Nov 29 at 15:27
N. S.
101k5109204
Hint Since $S$ is countable, you can construct an increasing sequence $H_1 subset H_2 subset ... subset H_n subset ...$ of finite subsets such that
$$S= bigcup_n H_n$$
For each $H_n$ you can find some $F_n$ such that
$$
frac{|sF_nDelta F_n|}{|F_n|} < frac{1}{n} qquad forall s in H_n
$$
Show that $F_n$ is a left Følner sequence.
answered Nov 29 at 15:27
N. S.
101k5109204
answered Nov 29 at 15:27
N. S.
101k5109204
answered Nov 29 at 15:27
N. S.
101k5109204
answered Nov 29 at 15:27
N. S.
101k5109204
101k5109204
I almost tried in this way. But for each $H_m$, I was considering a Folner sequence $F_n^{(m)}$, and I thought, may be $F_n^{(n)}$ would work. Anyway, thank you very much for your help. Upvoted and accepted.
– Surajit
Nov 29 at 15:37
add a comment |
I almost tried in this way. But for each $H_m$, I was considering a Folner sequence $F_n^{(m)}$, and I thought, may be $F_n^{(n)}$ would work. Anyway, thank you very much for your help. Upvoted and accepted.
– Surajit
Nov 29 at 15:37
I almost tried in this way. But for each $H_m$, I was considering a Folner sequence $F_n^{(m)}$, and I thought, may be $F_n^{(n)}$ would work. Anyway, thank you very much for your help. Upvoted and accepted.
– Surajit
Nov 29 at 15:37
I almost tried in this way. But for each $H_m$, I was considering a Folner sequence $F_n^{(m)}$, and I thought, may be $F_n^{(n)}$ would work. Anyway, thank you very much for your help. Upvoted and accepted.
– Surajit
Nov 29 at 15:37
add a comment |
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