  We consider two-variable first-order logic FO2 over infinite words.
Restricting the number of nested negations defines an infinite hierarchy; its
levels are often called the half-levels of the FO2 quantifier alternation
hierarchy. For every level of this hierarchy, we give an effective
characterization. For the lower levels, this characterization is a combination
of an algebraic and a topological property. For the higher levels, algebraic
properties turn out to be sufficient. Within two-variable first-order logic,
each algebraic property is a single ordered identity of omega-terms. The
topological properties are the same as for the lower half-levels of the
quantifier alternation hierarchy without the two-variable restriction (i.e.,
the Cantor topology and the alphabetic topology).
  Our result generalizes the corresponding result for finite words. The proof
uses novel techniques and is based on a refinement of Mal'cev products for
ordered monoids.
