International Journal of Algebra and Computation

For a given integer n, we define ωⁿ-semigroups as a generalization of ω-semigroups for languages of words of length less than ωⁿ⁺¹. When they are finite, those algebraic structures define the same sets as those recognized by Choueka automata. These sets are also equivalent to regular expressions in which an unary ω operator standing for the infinite repetition of a language is as free as the Kleene closure operator is. Naturally, the notion of syntactic congruence still works on ωⁿ-semigroups: among all ωⁿ-semigroups recognizing a regular language X, there exists an unique ωⁿ-semigroup of which all others are refinements.

Let G be a finitely generated group. The Bieri–Neumann–Strebel invariant Σ¹(G) of G determines, among other things, the distribution of finitely generated subgroups N◃G with G/N abelian. This invariant can be quite difficult to compute. Given a finite presentation 〈S:R〉 for G, there is an algorithm, introduced by Brown and extended by Bieri and Strebel, which determines a space Σ(R) that is always contained in, and is sometimes equal to, Σ¹(G). We refine this algorithm to one which involves the local structure of the universal cover of the standard 2-complex of a given presentation. Let Ψ(R) denote the space determined by this algorithm. We show that Σ(R) ⊆ Ψ ⊆ Σ¹(G) for any finitely presented group G, and if G admits a staggered presentation, then Ψ = Σ¹(G). By casting this algorithm in terms of connectivity properties of graphs, it is shown to be computationally feasible.

We depict the weight diagrams (alias, crystal graphs) of basic and adjoint representations of complex simple Lie algebras/algebraic groups and describe some of their uses.

Several authors have used the representation theory of symmetric groups and superalgebras to prove that certain classes of algebras are nilpotent. We show how to extend these techniques to facilitate the computation of the dimensions of relatively free algebras, and we prove two general theorems which formalize the techniques. The ideas described here have been used in the computation of the dimensions of the associated Lie rings of free Engel-4 groups of exponent 5.

The linear group GL(n, Z) is residually finite by virtue of its action on the (one-rooted) regular 2ⁿ-ary coset tree for Zⁿ. In this paper we construct finite state automata which effect this action. This shows that GL(n, Z) is embeddable in the group of finite state automorphisms of the one-rooted regular tree of valency 2ⁿ.