I put a large bounty on this question to get a full detailed answer or to at least to draw some attention and get it up-voted. The bounty ends tomorrow and I got none.
What should I do? I feel maybe the question belongs to MO.
That can certainly be frustrating, and is a risk any time you offer a bounty. I don't know what your background is, but please note that MathOverflow is intended primarily for professional mathematicians and graduate students doing cutting-edge research. A sufficiently interesting question from outside that community will sometimes get a useful answer, but it may also be closed, ignored, dismissed, or answered in a way that the person asking will not understand. Have you looked into the sources people suggested in the comments to your post here? I don't know if they will lead you to an answer, but you should probably have a decent understanding of how they relate before you head over to MO.
Don't give up on getting an answer to a question with a bounty until the bitter end—bounty hunters may focus their attention on short-dated bounties.
In the future, consider starting with a smaller bounty. This will allow you to start additional bounties after the first runs out, keeping your question in the spotlight longer.
If you revise the question as Sergio suggested, you might use the occasion to modify the wording, currently in the fifth paragraph, that seems to refer to $\mathcal T(X)/\sim$ as a lattice. Although $\mathcal T(X)$ is a latice, I don't think $\mathcal T(X)/\sim$ is one.
Also, I don't think this question would last long on MO. I'd vote to close it as off-topic, and there are plenty of people there who are more ferocious than I am about closing questions.