Atlas: On the type of growth of semigroups by Lev Shneerson

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International Conference on Modern Algebra in conjunction with the 17th annual Shanks Lectures
May 21-24, 2002
Vanderbilt University
Nashville, TN, USA

Organizers
Jonathan Farley, Ralph Freese, Matthew Gould, Peter Jipsen, George McNulty, Miklos Maroti, Alexander Ol'shanskii, Steven Tschantz, Constantine Tsinakis, Matthew Valeriote

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On the type of growth of semigroups
by
Lev Shneerson
Hunter College, The City University of New York

In the paper [1] we gave a method for constructing different types of examples of finitely generated (f.g.) semigroups having intermediate growth. In particular, we found examples of nil-semigroups with the identity x² = 0 having arbitrarily small intermediate growth and relatively free nilpotent semigroup whose growth is intermediate, but smaller than the growth of Hardy-Ramanujan function. Here we discuss the following questions:

QUESTION 1. How large can the intermediate growth of semigroups be?

QUESTION 2. How large can the intermediate growth of relatively free semigroups be?

THEOREM 1. Let f(m) be a monotone non-decreasing mapping from N into R⁺ such that f(m)=o(c^m) for any c>1. Then there exists a 2- generated semigroup whose growth is intermediate, but larger than the growth of the function f.

Define the sequence { q_n(m) } of the increasing mappings from the tails of N into R⁺ by the rule:

q₁(m) = ln m, q_k+1(m) = ln q_k(m).

THEOREM 2. For any natural number k there exists a relatively free semigroup whose growth is intermediate, larger than the growth of the function exp(m/q_k(m)).

REFERENCES

1. L.M. Shneerson, Relatively free semigroups of intermediate growth, J. Algebra, 235 (2001) 484-546.

Date received: April 27, 2002