My research paper was published as a conference document. I signed the copyright. In accordance with it the publisher is a copyright owner. Now, I found that my abstracts is written (literally) on the website of some organization which is unknown for me. The abstract on the publisher's website is open and can be copied without restriction and without payment. So, as I understand the abstract is considered as an open access publication and can be borrowed by anyone and without my or publisher's permission. Right? Or there is some rules which can regulate such issue?
If the publisher now holds the copyright, they would have to provide the permission to reproduce it. The "law [that] protects such right" that you referred to in your comment is copyright law.
The publisher can explicitly allow everybody to replicate the work, e.g. through a Creative Commons license (although this often includes the clause to also reproduce the license information), or by a general waiver on their site. So you'd have to look for something like that at your publisher's website, and otherwise you might want to notify them of the reproduction - assuming you don't agree with it.
(Do consider that more publicity for your work might not be a bad thing.)
There are two distinct questions here:
1) Does the website have permission to publish your abstract? You no longer own the copyright to your work, your publisher does. They undoubtedly have many agreements allowing other businesses to publish your work. This website may or may not have permission, unbeknownst to you. In my field it is not uncommon for online search engines to index the papers available at the other online sites.
2) If the website does not have permission, does their usage of your work fall under any of the exemptions to copyright law? This is called "fair use", and allows others to use copyrighted work without explicitly negotiating a license with the copyright holder. It's a recognition that strictly enforcing copyright in all circumstances would inhibit creative expression. When deciding whether or not a particular instance is "fair use" there are four general principles:
- Is the nature of the work non-profit or educational?
- Does the nature of the work have any bearing on how it is used?
- How much of the work is reproduced in relation to the whole?
- How does the unauthorized reproduction affect the market value of the entire work?
(These are not the only factors that may be considered, but they are by and far the most prevalent.)
In your situation, factors 2, 3, and 4 would strongly suggest that there is an argument to be made for fair use in this case. With respect to point 2, the only part of the work that has been reproduced is the abstract. The whole purpose of the abstract is to summarize and advertise the work for the benefit of others. In considering point 3 the abstract is only a small fraction of the overall work. For point 4 it is unlikely that reproducing just the abstract is going to affect the overall market value of the work, as again the purpose of the abstract is to advertise.
As a final note, you should go read the copyright release you signed with the original publisher. Typically these releases say that the publisher has the sole right to pursue action against a copyright infringer, but that they are not obligated to do so. If this is indeed the case, then the only legal recourse you have is to notify the publisher and hope they do something about it. It would be illegal for you to pursue your own remedy (such as filing a DCMA takedown notice).