Copyright for abstract 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.)