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We investigate structures that can be represented by omega-automata, so called omega-automatic structures, and prove that relations defined over such structures in first-order logic expanded by the first-order quantifiers 'there exist at most _0 many', 'there exist finitely many' and 'there exist k modulo m many' are omega-regular. The proof identifies certain algebraic properties of omega-semigroups. As a consequence an omega-regular equivalence relation of countable index has an omega-regulararXiv:0802.2866v1 fatcat:e6u444zcmzhepp56valkfilvqm