  Fragments of first-order logic over words can often be characterized in terms
of finite monoids or finite semigroups. Usually these algebraic descriptions
yield decidability of the question whether a given regular language is
definable in a particular fragment. An effective algebraic characterization can
be obtained from identities of so-called omega-terms. In order to show that a
given fragment satisfies some identity of omega-terms, one can use
Ehrenfeucht-Fraisse games on word instances of the omega-terms. The resulting
proofs often require a significant amount of book-keeping with respect to the
constants involved. In this paper we introduce Ehrenfeucht-Fraisse games on
omega-terms. To this end we assign a labeled linear order to every omega-term.
Our main theorem shows that a given fragment satisfies some identity of
omega-terms if and only if Duplicator has a winning strategy for the game on
the resulting linear orders. This allows to avoid the book-keeping. As an
application of our main result, we show that one can decide in exponential time
whether all aperiodic monoids satisfy some given identity of omega-terms,
thereby improving a result of McCammond (Int. J. Algebra Comput., 2001).
