Finite filtering semigroups

A semigroup is called filtering if each of its subsemigroups has the smallest (with respect to inclusion) generating set. It is proved in this article that every maximal chain of nonempty subsemigroups of a finite filtering semigroup has length equal to the order of the semigroup, and that filtering semigroups are characterized by this property in the class of finite semigroups. The main result is a characterization of the class of finite filtering semigroups by means of forbidden divisors, to which end the author finds all finite nonfiltering semigroups all of whose proper divisors are filtering semigroups.