Cardinality and counting quantifiers on omega-automatic structures

Abstract

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.