International Journal of Algebra and Computation

We show that, given an equation over a finitely generated free group, the set of all solutions in reduced words forms an effectively constructible EDT0L language. In particular, the set of all solutions in reduced words is an indexed language in the sense of Aho. The language characterization we give, as well as further questions about the existence or finiteness of solutions, follow from our explicit construction of a finite directed graph which encodes all the solutions. Our result incorporates the recently invented recompression technique of Jeż, and a new way to integrate solutions of linear Diophantine equations into the process. As a byproduct of our techniques, we improve the complexity from quadratic nondeterministic space in previous works to $NSPACE(n\mathrm{\text{log}}n)$ here.

Leech’s (co)homology groups of finite cyclic monoids are computed.

This paper concerns several ring theoretic properties related to matrices and polynomials. The basic properties of $\pi$-reversible and power-Armendariz are studied. We provide a method by which one can always construct a power-Armendariz ring but neither symmetric nor Armendariz from given any symmetric ring. We investigate next various interesting relations among ring theoretic properties containing $\pi$-reversibility and power-Armendariz condition.

In the general context of presentations of monoids, we study normalization process that are determined by their restriction to length-two words. Garside’s greedy normal forms and quadratic convergent rewriting systems, in particular those associated with the plactic monoids, are typical examples. Having introduced a parameter, called the class and measuring the complexity of the normalization of length-three words, we analyze the normalization of longer words and describe a number of possible behaviors. We fully axiomatize normalizations of class $(4,3)$, show the convergence of the associated rewriting systems, and characterize those deriving from a Garside family.

Let $g$ be an element of a group $G$. For a positive integer $n$, let $E_n(g)$ be the subgroup generated by all commutators $[\dots [[x,g],g],\dots ,g]$ over $x\in G$, where $g$ is repeated $n$ times. We prove that if $G$ is a profinite group such that for every $g\in G$ there is $n=n(g)$ such that $E_n(g)$ is finite, then $G$ has a finite normal subgroup $N$ such that $G/N$ is locally nilpotent. The proof uses the Wilson–Zelmanov theorem saying that Engel profinite groups are locally nilpotent. In the case of a finite group $G$, we prove that if, for some $n$, $|E_n(g)|\le m$ for all $g\in G$, then the order of the nilpotent residual ${\gamma }_{\infty }(G)$ is bounded in terms of $m$.

We study varieties of associative algebras over a finite field and varieties of associative rings satisfying semigroup or adjoint semigroup identities. We characterize these varieties in terms of “forbidden algebras” and discuss some corollaries of the characterizations.

An $n$-ary operation $f$ is called symmetric if, for all permutations $\pi$ of $\{1,\dots ,n\}$, it satisfies the identity $f(x_1,x_2,\dots ,x_n)=f(x_{\pi (1)},x_{\pi (2)},\dots ,x_{\pi (n)})$. We show that, for each finite algebra ${𝒜}$, either it has symmetric term operations of all arities or else some finite algebra in the variety generated by ${𝒜}$ has two automorphisms without a common fixed point. We also show this two-automorphism condition cannot be replaced by a single fixed-point-free automorphism.

We characterize absorption in finite idempotent algebras by means of Jónsson absorption and cube term blockers. As an application we show that it is decidable whether a given subset is an absorbing subuniverse of an algebra given by the tables of its basic operations.

An affine algebraic variety $X$ is rigid if the algebra of regular functions $𝕂[X]$ admits no nonzero locally nilpotent derivation. We prove that a factorial trinomial hypersurface is rigid if and only if every exponent in the trinomial is at least $2$.

In this paper and its sequel we continue our study of nilpotent symplectic alternating algebras. In particular we give a full classification of such algebras of dimension 10 over any field. It is known that symplectic alternating algebras over GF(3) correspond to a special rich class ${𝒞}$ of 2-Engel 3-groups of exponent 27 and under this correspondence we will see that the nilpotent algebras correspond to a subclass of ${𝒞}$ that are those groups in ${𝒞}$ that have an extra group theoretical property that we refer to as being powerfully nilpotent and can be described also in the context of $p$-groups where $p$ is an arbitrary prime.

In this paper, we finish our classification of nilpotent symplectic alternating algebras of dimension 10 over any field $𝔽$.