Identities and a bounded height condition for semigroups

We consider two different types of bounded height condition for semigroups. The first one originates from the classical Shirshov's bounded height theorem for associative rings. The second which is weaker, in fact was introduced by Wolf and also used by Bass for calculating the growth of finitely generated (f.g.) nilpotent groups. Both conditions yield polynomial growth. We give the first two examples of f.g. semigroups which have bounded height and do not satisfy any nontrivial identity. One of these semigroups does not have bounded height in the sense of Shirshov and the other satisfies the classical bounded height condition. This develops further one of the main results of the author's paper (J. Algebra, 1993) where the first examples of f.g. semigroups of polynomial growth and without nontrivial identities were given.