Groups which satisfy a Thue–Morse identity

In this paper, we study 3-Thue–Morse groups, but these are the groups satisfying the semigroup identity [Formula: see text]. We prove that if [Formula: see text] is a 3-Thue–Morse group then [Formula: see text] is soluble for every [Formula: see text] and [Formula: see text] in [Formula: see text]. Furthermore, if [Formula: see text] is an Engel group without involution then we show that [Formula: see text] is locally nilpotent.

On non-Hopfian groups of fractions

The group of fractions of a semigroup S, if exists, can be written as G = SS−1. If S is abelian, then G must be abelian. We say that a semigroup identity is transferable if being satisfied in S it must be satisfied in G = SS−1. One of problems posed by G.Bergman in 1981 asks whether the group G must satisfy every semigroup identity which is satisfied in S, that is whether every semigroup identity is transferable. The first non-transferable identities were constructed in 2005 by S.V.Ivanov and A.M. Storozhev. A group G is called Hopfian if each epimorphizm G → G is the automorphism. The residually finite groups are Hopfian, however there are many problems concerning the Hopfian property e.g. of infinite Burnside groups, of finitely generated relatively free groups [11, Problem 15]. We prove here that if G = SS−1 is an n-generator group of fractions of a relatively free semigroup S, satisfying m-variable (m < n) non-transferable identity, then G is the non-Hopfian group.

Polyhedral convex cones and the equational theory of the bicyclic semigroup

To any given balanced semigroup identity U ≈ W a number of polyhedral convex cones are associated. In this setting an algorithm is proposed which determines whether the given identity is satisfied in the bicylic semigroup or in the semigroup . The semigroups BC and E deserve our attention because a semigroup variety contains a simple semigroup which is not completely simple (respectively, which is idempotent free) if and only if this variety contains BC (respectively, E). Therefore, for a given identity U ≈ W it is decidable whether or not the variety determined by U ≈ W contains a simple semigroup which is not completely simple (respectively, which is idempotent free).

COMPLEXITY OF SEMIGROUP IDENTITY CHECKING

We consider the [Formula: see text] problem, whose instance is a finite semigroup S and an identity I, and the question is whether I is satisfied in S. We show that the question concerning computational complexity of this problem is much harder, when restricted to commutative semigroups. We provide a relatively simple proof that in general the problem is co-NP-complete, and demonstrate, using some structure theory, that for a fixed commutative semigroup the problem can be solved in polynomial time. The complexity status of the general [Formula: see text] problem remains open.

Semigroup identities on units of integral group rings

Let U(RG) be the group of units of a group ring RG over a commutative ring R with 1. We say that a group is an SIT-group if it is an extension of a group which satisfies a semigroup identity by a torsion group. It is a consequence of the main result that if G is torsion and R = Z, then U(RG) is an SIT-group if and only if G is either abelian or a Hamiltonian 2-group. If R is a local ring of characteristic 0 only the first alternative can occur.

Nilpotent semigroups of matrices

The class of nilpotent semigroups was introduced, via a semigroup identity, independently in [8] and [9], cf. [11], Every nilpotent cancellative semigroup S was shown to have a group of classical fractions G which is nilpotent and of the same nilpotency class as S. Groups, and linear groups in particular, satisfying certain related semigroup identities, introduced in [19], have been recently studied in [1], [15] and [18]. In particular, finitely generated residually finite groups satisfying a semigroup identity must be almost nilpotent, [18]. On the other hand, it was recently shown in [14] that a finitely generated linear semigroup S ⊆ Mn(K), over a field K, with no free non-commutative subsemigroups satisfies an identity and for every maximal subgroup H of Mn(K) the subgroup gp(S ∩ H) of H generated by S ∩ H is almost nilpotent. A natural question that arises here is to decide which of these semigroups are nilpotent. Because of the powerful classical theory of nilpotent linear groups, cf. [21], one can also ask whether such semigroups can be approached via group theoretical methods. We note that the very special case of nilpotent connected algebraic monoids has been recently considered in [4]. In [5] the structure of semigroup algebras of nilpotent semigroups was studied, in particular via prime Goldie homomorphic images, leading naturally to nilpotent subsemigroups of the matrix monoids Mn(D) over division rings D.

GROWTH AND EXISTENCE OF IDENTITIES IN A CLASS OF FINITELY PRESENTED INVERSE SEMIGROUPS WITH ZERO

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.