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I am currently tinkering with an idea and have exhausted the possible exploration on the matter by me. I would like to present what I am doing and where I am at and hopefully ask people to show me other avenues to pursue or if they can find something interesting in it worth exploring. Maybe, they can think of how to usefully proceed in my exploration, better than I can with my "Mathematician's block".

What is the appropriate way of going about this on Math.SE since I do not actually have a question to ask with a concrete answer?

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In short, such open-ended "questions" are not a very good fit here, or anywhere on the StackExchange network.

If you can narrow down your concerns to particular questions with concrete answers, then those questions might be on topic here, or perhaps on MathOverflow (which is aimed at math researchers).

For general guidance or suggestions, I recommend talking to someone in real life. If you are at a school and know math faculty, then I'm sure you can get/be directed to someone who will entertain questions.

Since I know nothing of what level or topic you are asking about, I can't say much more. The chatroom here (or at MathOverflow) may be able to help. Other notable online communities include those mentioned in Useful mathematical fora and, heavily depending on your subject matter, reddit's /r/math and /r/mathematics.

At the moment, there is no cannonical place for people to say "Here's how far I got into this or that problem. What are some avenues to explore?" Well, some people can use blogs. But this requires that there be a blog that you can write to and which people read.

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  • $\begingroup$ I am playing around with the idea of $f(x,y) f(y,z) = f(x,z)$. It's mostly playing around with abstract algebra. I am about to hit the wall on what is possible with this idea. I just want to know what other interesting things I can do with this. It's not an idea interesting enough to bother the math faculty with. It's cutting edge research or anything. $\endgroup$ – XYZT Dec 5 '14 at 7:08
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    $\begingroup$ @NikhilMahajan You might want to look up "rectangular bands". If $S$ is a set, then a rectangular band is a semigroup $T:=S\times S$ with multiplication given by $(a, b)(c, d)=(a, d)$. $\endgroup$ – user1729 Dec 5 '14 at 9:33

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