The class of Unambiguous Star-Free Regular Languages (UL) was defined by Schutzenberger as the class of languages defined by Unambiguous Polynomials. UL has been variously characterized (over finite words) by logics such as TL[Xa,Ya], UITL, TL[F,P], FO2[<], the variety DA of monoids, as well as partially-ordered two-way DFA (po2dfa). We revisit this language class with emphasis on notion of una...