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Let $X$ be an infinite set. The relative rank of a subset $T_{X}$ over a subset $S$ is either uncountable or at most $2$.

I don't understand how to prove this Corollary in case that $T_{X} \setminus S$ is uncountable.

see also(corollary 4.6): http://www3.kfupm.edu.sa/aisys/MATH_ONLY/TechReports_DATA/357.pdf

Mr.Lilly
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    I think you should define the terms in your question properly. What is the relative rank, for example? – James Mitchell Sep 24 '14 at 21:15
  • @nameless: I noticed you haven't accepted answers to any of your questions yet. It's nice that you thank the people who answer your questions, but also to help with the way this site is supposed to work, please accept answers by clicking on the 'tick' next to them. – Tara B Sep 25 '14 at 14:25
  • @TaraB Thanks you for your suggestion. :) – Mr.Lilly Sep 25 '14 at 17:54

1 Answers1

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The result you state follows from the fact that every countable subset of $T_X$ is contained in a 2-generated subsemigroup. See this question for more details:

Evans' theorem of embeddings into 2-generator semigroup.

As for if $T_X\setminus S$ is uncountable, then $\operatorname{rank}(T_X:S)$ is uncountable, this is just plain wrong: $T_X\setminus S_X$ is uncountable but $\operatorname{rank}(T_X:S_X)=2$ where $S_X$ is the symmetric group.

The correct argument goes like this: if there is a countable set that together with $S$ generates $T_X$, then there is a set with at most two elements since every countable set is contained in 2-generated subsemigroup. Hence the relative rank is at most 2 in this case. If there is no countable set that together with $S$ generates $T_X$, then the relative rank is uncountable.

Community
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James Mitchell
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  • You mean that proof of Corollary 4.6 in my link is not correct? By the way, I should to use Evan's theorem to prove this corollary? – Mr.Lilly Sep 24 '14 at 21:31
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    Yes the proof of Corollary 4.6 in the reference you give is wrong. I gave the correct argument in my answer. – James Mitchell Sep 24 '14 at 21:33