  Fragments of first-order logic over words can often be characterized in terms
of finite monoids, and identities of omega-terms are an effective mechanism for
specifying classes of monoids. Huschenbett and the first author have shown how
to use infinite Ehrenfeucht-Fraisse games on linear orders for showing that
some given fragment satisfies an identity of omega-terms (STACS 2014). After
revisiting this result, we show that for two-variable logic one can use simpler
linear orders.
