Synchronized Shuffle and Regular Languages

New representation results for three families of regular languages are stated, using a special kind of shuffle operation, namely the synchronized shuffle. First, it is proved that the family of regular star languages is the smallest family containing the language (a + bc)* and closed under synchronized shuffle and length preserving morphism. The second representation result states that the family of e-free regular languages is the smallest family containing the language (a + bc)*d and closed under synchronized shuffle, union and length preserving morphism. At last, it is proved that Reg is the smallest family containing the two languages (a+ bb)* and a+(ab)+, closed under synchronized shuffle, union and length preserving morphism.