Onω-Languages whose syntactic monoid is trivial

For a languageL the syntactic monoid SynL is trivial if and only if indeedL itself is trivial, that isL = Ø orL=X*. As a surprise one realizes that the syntactic monoid SynL of an ω-languageL being trivial by no means implies thatL be trivial. This situation is analyzed in this paper. The results may help clarify the difference between deterministically and nondeterministically finite state acceptable ω-languages.