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

$aleph_0$ 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-regular set of representatives. This implies

Blumensath's conjecture that a countable structure with an

$omega$-automatic presentation can be represented using automata

on finite words. This also complements a very recent result of

Hj"orth, Khoussainov, Montalban and Nies showing that there is an

omega-automatic structure which has no injective presentation.