The present thesis consists of two parts.
Based on syntactic closure axioms of formula sets, the first part gives a formal definition of logic fragments. It also shows the versatileness of this notion of logic fragments, inter alia, giving, C-variety descriptions of logic fragments and abstractly investigating the influence of certain predicates on the expressiveness of logic fragments.
The second part considers two-variable first-order logic FO2. A combinatorial description in terms of so-called rankers is given for all full levels as well as all half levels of the quantifier alternation hierarchy of FO2 over the order predicate, both with and without the successor predicate.
Also in the second part, effective algebraic criteria describing all full levels as well as all half levels of the quantifier alternation hierarchy of FO2 over several signatures are given, yielding in particular decidability of the definability problem.
