On nonpermutational transformation semigroups with an application to   syntactic complexity
  We give an upper bound of $n((n-1)!-(n-3)!)$ for the possible largest size of a subsemigroup of the full transformational semigroup over $n$ elements consisting only of nonpermutational transformations. As an application we gain the same upper bound for the syntactic complexity of (generalized) definite languages as well. 
Szabolcs Ivan, Judit Nagy-Gyorgy
cs.FL