No abstract

No abstract

We prove that a finitely presented Rees quotient of a free inverse semigroup has polynomial or exponential growth, and that the type of growth is algorithmically recognizable. We prove that such a semigroup has polynomial growth if and only if it satisfies a certain semigroup identity. However we give an example of such a semigroup which has exponential growth and satisfies some nontrivial identity in signature with involution.

Let ᑬ be a nonperiodic semigroup variety satisfying the nontrivial identity Z n = W , where Z n is the nth element of the so-called Zimin's sequence Z 1 = u 1 , Z n+1 = Z n u n+1 Z n (n = 1, 2, . . .) of unavoidable words. It is shown that every finitely generated (f.g.) semigroup S ∈ ᑬ has subexponential growth and every uniformly recurrent infinite word which is geodesic and lexicographically reduced relative to S is periodic. Furthermore, the length of the period is bounded above by a constant which depends only on the number n and on the number of generators of S. The first of these results gives a positive answer to the problem posed by M. Sapir. Some applications including examples of f.g. relatively free semigroups having intermediate growth and maximal (equals 1) superdimension are considered.