International Journal of Algebra and Computation

The semi-direct product of arbitrary two monoids and a presentation for this product have received considerable attention, see for instance [12, 14, 15]. In [15], Wang defined a trivializer set of the Squier complex associated with this presentation. In this paper, as a main result, we discuss necessary and sufficient conditions for the standard presentation of the semi-direct product of any two monoids to be p-Cockcroft for any prime p or 0. Finally we present some applications of this main theorem.

Let F(x,y) be a relatively free algebra of rank 2 in some variety of algebras over a field K of characteristic 0. In this paper we consider the problem whether p(x,y) ∈ F(x,y) is a primitive element (i.e. an automorphic image of x): (i) If F(x,y)/(p(x,y)) ≅ F(z), the relatively free algebra of rank 1 (ii) If p(f,g) is primitive for some injective endomorphism (f,g) of F(x,y) (iii) If p(x,y) is primitive in a relatively free algebra of larger rank. These problems have positive solutions for polynomial algebras in two variables. We give the complete answer for the free metabelian associative and Lie algebras and some partial results for free associative algebras.

We develop a method of creating skew congruences on subpowers of finite algebras using groups of twin polynomials, and apply it to the investigation of residually small varieties generated by nilpotent algebras. We prove that a residually small variety generated by a finite nilpotent (in particular, a solvable E-minimal) algebra is weakly abelian. Conversely, we show in two special cases that a weakly abelian variety is residually bounded by a finite number: when it is generated by an E-minimal, or by a finite strongly nilpotent algebra. This establishes the RS-conjecture for E-minimal algebras.

Andrews and Curtis conjectured in 1965 that every balanced presentation of the trivial group can be transformed into a standard presentation by a finite sequence of elementary transformations. Recent computational work by Miasnikov and Myasnikov on this problem has been based on genetic algorithms. We show that a computational attack based on a breadth-first search of the tree of equivalent presentations is also viable, and seems to outperform that based on genetic algorithms. It allows us to extract shorter proofs (in some cases, provably shortest) and to consider the length thirteen case for two generators. We prove that, up to equivalence, there is a unique minimum potential counterexample.

In the 1960s, G. Grätzer introduced the notion of the minimal extension property (MEP) of a finite sequence in order to investigate p_n-sequences and free spectra of algebras. While there are many particular results on the MEP, stating that some sequences or families of sequences have the MEP, no general result has been obtained so far and the main general problems remain open.

In this paper we prove a fact about existence of finite complete systems of subalgebras, which generalizes a method used occasionally to prove the minimal extension property of a sequence. Using this we obtain some general results about minimal extensions and, along the way, a rather unexpected solution of one of the central open problems in the area.

In this note we give simple explicit examples of free subgroups of rank 2 in the group of infinite upper unitriangular matrices over integers. The proofs that the given subgroup is free are elementary. Using canonical projections mod p (p — any odd prime) we obtain also free subgroups in the group of finite state automata on alphabet of size p, extending results of Aleshin [1], Brunner–Sidki [2] and Olijnyk–Sushchansky [8, 9] in this direction. As application, we give the simple proofs of two classical results on free groups.

A Coxeter group is said to be rigid if any two systems for that group yield isomorphic Coxeter diagrams. A Coxeter group is said to be strongly rigid if moreover the generating sets of any two systems are conjugate. We determine a new class of Coxeter groups which are rigid, and introduce the notion of reflection independence, a generalization of strong rigidity.

We prove that any solvable subgroup K of automorphisms of the binary tree, which contains the binary adding machine is an extension of a torsion-free metabelian group by a finite 2-group. If the group K is moreover nilpotent then it is torsion-free abelian.