It would be improper to list yourself as the publisher of a public domain song (we assume that by "traditional" you're referring to PD music). However, you may be able to acquire copyright protection for your unique arrangements.
You can't list yourself as the publisher. A music publisher is a company that owns songwriting copyrights and collects money from people who sell, perform or modify the songs. Because public domain songs are not protected by copyright (they are free to the public) no one can claim ownership of these compositions. Publishers who claim ownership of PD music and try to enforce their claims are subject to lawsuits. One workaround is to add new lyrics to a public domain melody. Elvis and Ken Darby wrote new lyrics (Love Me Tender) for a civil war song (Aura Lee) and their music publisher can stop anyone from using their derivative version (though they can't stop anyone from using the Aura Lee music and lyrics).
You can claim rights to a public domain arrangement. Although you cannot claim ownership of a public domain composition, you may be able to claim copyright of your arrangement. However, in order to assert affirmative rights (that is, go after infringers), the arrangement must be more than a trivial variation of the original. As one case established, "To extend copyrightability to minuscule variations would simply put a weapon for harassment in the hands of mischievous copiers intent on appropriating and monopolizing public domain work." Perhaps more importantly, royalty collection organizations enable you to get paid for your public domain version. For example, the MLC allows you to register public domain works, as do Songtrust, CDBaby and other rights organizations.