The Boolean Hierarchy over Level 1/2 of the Straubing-Therien Hierarchy

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.