Narrow Results

We show that analogues of results from [4, 5] can be obtained if the relation is replaced by . We also give an example of a semigroup S in which does not follow from S being finitely generated, periodic, inverse, E-unitary and not even from the finiteness of its maximal subgroups.

We use Schlage-Puchta's concept of p-deficiency and Lackenby's property of p-largeness to show that a group having a finite presentation with p-deficiency greater than 1 is large, which implies that Schlage-Puchta's infinite finitely generated p-groups are not finitely presented. We also show that for all primes p at least 7, any group having a presentation of p-deficiency greater than 1 is Golod–Shafarevich, and has a finite index subgroup which is Golod–Shafarevich for the remaining primes. We also generalize a result of Grigorchuk on Coxeter groups to odd primes.

We continue our program of extending key techniques from geometric group theory to semigroup theory, by studying monoids acting by isometric embeddings on spaces equipped with asymmetric, partially defined distance functions. The canonical example of such an action is a cancellative monoid acting by translation on its Cayley graph. Our main result is an extension of the Švarc–Milnor lemma to this setting.

For $n$ a positive integer and $K$ a finite set of finite algebras, let $L(n,K)$ denote the largest $n$-generated subdirect product whose subdirect factors are algebras in $K$. When $K$ is the set of all $n$-generated subdirectly irreducible algebras in a locally finite variety ${𝒱}$, then $L(n,K)$ is the free algebra $F_{{𝒱}}(n)$ on $n$ free generators for ${𝒱}$. For a finite algebra $A$ the algebra $L(n,\{A\})$ is the largest $n$-generated subdirect power of $A$. For every $n$ and finite $A$ we provide an upper bound on the cardinality of $L(n,\{A\})$. This upper bound depends only on $n$ and these basic parameters: the cardinality of the automorphism group of $A$, the cardinalities of the subalgebras of $A$, and the cardinalities of the equivalence classes of certain equivalence relations arising from congruence relations of $A$. Using this upper bound on $n$-generated subdirect powers of $A$, as $A$ ranges over the $n$-generated subdirectly irreducible algebras in ${𝒱}$, we obtain an upper bound on $|F_{{𝒱}}(n)|$. And if all the $n$-generated subdirectly irreducible algebras in ${𝒱}$ have congruence lattices that are chains, then we characterize in several ways those ${𝒱}$ for which this upper bound is obtained.

A semigroup $S$ is right noetherian if every right congruence on $S$ is finitely generated. In this paper, we present some fundamental properties of right noetherian semigroups, discuss how semigroups relate to their substructures with regard to the property of being right noetherian, and investigate whether this property is preserved under various semigroup constructions.

A Lie ideal of a division ring $A$ is an additive subgroup $L$ of $A$ such that the Lie product $[l,a]=la-al$ of any two elements $l\in L,a\in A$ is in $L$ or $[l,a]\in L$. The main concern of this paper is to present some properties of Lie ideals of $A$ which may be interpreted as being dual to known properties of normal subgroups of $A^{\ast }$. In particular, we prove that if $A$ is a finite-dimensional division algebra with center $F$ and $\mathrm{\text{char}}F\ne 2$, then any finitely generated $\mathbb{Z}$-module Lie ideal of $A$ is central. We also show that the additive commutator subgroup $[A,A]$ of $A$ is not a finitely generated $\mathbb{Z}$-module. Some other results about maximal additive subgroups of $A$ and $M_n(A)$ are also presented.

Let $A$ be an associative algebra over a field of characteristic $\ne 2$ that is generated by a finite collection of nilpotent elements. We prove that all Lie derived powers of $A$ are finitely generated Lie algebras.

Let $A$ be a finitely generated associative algebra over a field $F$ of characteristic $\ne 2$ and let $A^{(-)}=(A,[a,b]=ab-ba)$ be its associated Lie algebra. In this paper, we investigate relations between the growth functions of $A$ and the Lie algebra $[A,A]$. We prove that if A is generated by a finite collection of nilpotent elements, then the growth functions are asymptotically equivalent.

Let $A$ be a finitely generated associative algebra over a field of characteristic different from 2. Herstein asked when the Lie algebra $[A,A]$ is finitely generated. Recently, it was shown that for a finitely generated nil algebra $A$ all derived powers of $A$ are finitely generated Lie algebras. Let $K$ be the Lie algebra of skew-symmetric elements of an associative algebra with involution. We consider all derived powers of the Lie algebra $K$ and prove that for any finitely generated associative nil algebra with an involution, all derived powers of $K$ are finitely generated Lie algebras.