  For some fixed alphabet A, a language L of A* is in the class L(1/2) of the
Straubing-Therien hierarchy if and only if it can be expressed as a finite
union of languages A*aA*bA*...A*cA*, where a,b,...,c are letters. The class
L(1) is defined as the boolean closure of L(1/2). It is known that the classes
L(1/2) and L(1) are decidable. We give a membership criterion for the single
classes of the boolean hierarchy over L(1/2). From this criterion we can
conclude that this boolean hierarchy is proper and that its classes are
decidable.In finite model theory the latter implies the decidability of the
classes of the boolean hierarchy over the class Sigma(1) of the FO(<)-logic.
Moreover we prove a ``forbidden-pattern'' characterization of L(1) of the type:
L is in L(1) if and only if a certain pattern does not appear in the transition
graph of a deterministic finite automaton accepting L. We discuss complexity
theoretical consequences of our results.
