Introduction to semigroup theory

We present a survey of results related to the Milnor's problem on group growth We discuss the cases of polynomial growth, exponential but not uniformly exponential growth, but the main part of the article is devoted to the intermediate (between polynomial and exponential) growth case A number of related topics (growth of manifolds, amenability, asymptotic behavior of random walks) is considered, and a number of open problems is suggested

Milnor's Problem on the Growth of Groups and its Consequences

Identities in upper triangular tropical matrix semigroups and the bicyclic monoid

In this paper, we study the so-called diagram groups. Our main result is that finitely generated diagram groups are free if and only if they do not contain any subgroup isomorphic to $$\mathbb {Z}^2$$
 . As an immediate corollary, we get that hyperbolic diagram groups are necessarily free, answering a question of Guba and Sapir.

Hyperbolic diagram groups are free

A general approach to designing multiple classifiers represents them as a combination of several binary classifiers in order to enable correction of classification errors and increase reliability. This method is explained, for example, in Witten and Frank (Data Mining: Practical Machine Learning Tools and Techniques, 2005, Sect. 7.5). The aim of this paper is to investigate representations of this sort based on Brandt semigroups. We give a formula for the maximum number of errors of binary classifiers, which can be corrected by a multiple classifier of this type. Examples show that our formula does not carry over to larger classes of semigroups.

/pdf/a-formula-for-multiple-classifiers-in-data-mining-based-on-1vavir1yb0.pdf

A formula for multiple classifiers in data mining based on Brandt semigroups

We prove that up to isomorphism $\\langle a,b\\,\\vert\\, ab=0\\rangle$ is the unique principal Rees quotient of a free inverse semigroup that is not trivial or monogenic with zero, satisfying a nontrivial identity in signature with involution.

/pdf/principal-rees-quotients-of-free-inverse-semigroups-1jqdyirguj.pdf

Principal rees quotients of free inverse semigroups

In this paper we discuss the problem: how large can the intermediate growth of nilpotent and relatively free semigroups be. We construct a sequence {Sn} of finitely-generated semigroups such that the growth of the semigroup Sn+1 (n = 1,2,…) is intermediate and larger than or equal to the growth of exp(m/φn(m)), where φn(m) is the nth iteration of ln m. All semigroups Sn are nilpotent in the sense of Malcev. We also find the sequence of relatively free semigroups with the same types of growth.

Types of growth and identities of semigroups

For any positive integer n > 1 we construct an example of inverse semigroup with n generators and n - 1 defining relations which has cubic growth and at least n generators in any presentation. This semigroup has the same set of identities as the free monogenic inverse semigroup. In particular, we give the first example of a one relation nonmonogenic inverse semigroup having polynomial growth. We also prove that for any positive integer n there exists an inverse semigroup ϒn of deficiency 1 and rank n + 1 such that ϒn has exponential growth and it does not contain nonmonogenic free inverse subsemigroups. Furthermore, ϒn satisfies the identity [[x, y], [z, t]]2 = [[x, y], [z, t]] of quasi-solvability and it contains a free subsemigroup of rank 2.

On growth, identities and free subsemigroups for inverse semigroups of deficiency one

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.

Identities and a bounded height condition for semigroups

We prove that a finitely presented Rees quotient of a free inverse semigroup has polynomial growth if and only if it has bounded height. This occurs if and only if the set of nonzero reduced words has bounded Shirshov height and all nonzero reduced but not cyclically reduced words are nilpotent. This occurs also if and only if the set of nonzero geodesic words have bounded Shirshov height. We also give a simple sufficient graphical condition for polynomial growth, which is necessary when all zero relators are reduced. As a final application of our results, we give an inverse semigroup analogue of a classical result that characterizes polynomial growth of finitely presented Rees quotients of free semigroups in terms of connections between non-nilpotent elements and primitive words that label loops of the Ufnarovsky graph of the presentation.

/pdf/growth-of-finitely-presented-rees-quotients-of-free-inverse-2mo3bvhhwf.pdf

Lev M. Shneerson

Papers

Principal rees quotients of free inverse semigroups

Types of growth and identities of semigroups

On growth, identities and free subsemigroups for inverse semigroups of deficiency one

Identities and a bounded height condition for semigroups

Growth of finitely presented rees quotients of free inverse semigroups