International Journal of Algebra and Computation

We determine exactly which graph products, also known as Right Angled Artin Groups, embed into Richard Thompson’s group $V$. It was shown by Bleak and Salazar-Díaz that ${\mathbb{Z}}^2\ast \mathbb{Z}$ was an obstruction. We show that this is the only obstruction. This is shown by proving a graph theory result giving an alternate description of simple graphs without an appropriate induced subgraph.

For a given linear system on a curve, the number of divisors of a certain type contained in this system is known as the formula of de Jonquières. In this paper, we give an algorithm for getting a general de Jonquières formula for a family of nodal curves. Using this algorithm we obtain one of such formulas for a particular case; on a single partition $(m)$ of $m$ for a one parameter family of nodal curves. Moreover we retrieve the classical de Jonquières’ formula for a single curve using the method developed in this paper.

We describe an algorithm for deciding whether or not a given finitely generated torsion-free nilpotent group is decomposable as the direct product of nontrivial subgroups.

We establish a new hereditary characterization of finite idempotent Taylor algebras. This generalizes the algebraic constructions which figured in the recent successful characterization of the correctness of the bounded width algorithm for constraint satisfaction problems by finite idempotent algebras generating congruence meet-semidistributive varieties. We introduce the cyclic reduct of a finite Taylor algebra and prove a collapse of Mal’cev conditions.

The consistency problem for a class of algebraic structures asks for an algorithm to decide, for any given conjunction of equations, whether it admits a non-trivial satisfying assignment within some member of the class. For the variety of all groups, this is the complement of the triviality problem, shown undecidable by by Adyan [Algorithmic unsolvability of problems of recognition of certain properties of groups. (Russian) Dokl. Akad. Nauk SSSR (N.S.) 103 (1955) 533–535] and Rabin [Recursive unsolvability of group theoretic problems, Ann. of Math. (2) 67 (1958) 172–194]. For the class of finite groups, it amounts to the triviality problem for profinite completions, shown undecidable by Bridson and Wilton [The triviality problem for profinite completions, Invent. Math. 202 (2015) 839–874]. We derive unsolvability of the consistency problem for the class of (finite) modular lattices and various subclasses; in particular, the class of all subspace lattices of finite-dimensional vector spaces over a fixed or arbitrary field of characteristic $0$ and expansions thereof, e.g. the class of subspace ortholattices of finite-dimensional Hilbert spaces. The lattice results are used to prove unsolvability of the consistency problem for (finite) rings with unit and (finite) representable relation algebras. These results in turn apply to equations between simple expressions in Grassmann–Cayley algebra and to functional and embedded multivalued dependencies in databases.

We study the relative Gorenstein projective global dimension of a ring with respect to a weakly Wakamatsu tilting module $C$. We prove that this relative global dimension is finite if and only if the injective dimension of every module in Add$(C)$ and the ${\mathcal{𝒫}}_C$-projective dimension of every injective module are both finite (indeed these three dimensions have a common upper bound). When _RC satisfies some extra conditions we prove that the relative Gorenstein projective global dimension of $R$ is always bounded above by the ${\mathcal{𝒫}}_C$-projective global dimension of $R$, these two dimensions being equal when the class of all $C$-Gorenstein projective $R$-modules is contained in the Bass class of $R$ relative to $C$. Of course we also give the dual results concerning the relative Gorenstein injective global dimension.

Let $F$ be a field of characteristic 0 and let $R=M_2(F)$. The algebra $R$ admits a natural grading $R=R_0\oplus R_1$ by the cyclic group ${\mathbb{Z}}_2$ of order 2. In this paper, we describe the ${\mathbb{Z}}_2$-graded A-identities for $R$. Recall that an A-identity for an algebra is a multilinear polynomial identity for that algebra which is a linear combination of the monomials $x_{\sigma (1)}\cdots x_{\sigma (n)}$ where $\sigma$ runs over all even permutations of $\{1,\dots ,n\}$ that is $\sigma \in A_n$, the $n$th alternating group. We first introduce the notion of an A-identity in the case of graded polynomials, then we describe the graded A-identities for $R$, and finally we compute the corresponding graded A-codimensions.