International Journal of Algebra and Computation

We classify generalized block graphs whose binomial edge ideals admit a unique extremal Betti number. We prove that the Castelnuovo–Mumford regularity of binomial edge ideals of generalized block graphs is bounded below by $m(G)+1$, where $m(G)$ is the number of minimal cut sets of the graph $G$ and obtain an improved upper bound for the regularity in terms of the number of maximal cliques and pendant vertices of $G$.

The solubility graph associated with a finite group $G$ is a simple graph whose vertices are the elements of $G$, and there is an edge between two distinct elements $x$ and $y$ if and only if $〈x,y〉$ is a soluble subgroup of $G$. We examine some properties of solubility graphs.

Using methods of computer algebra, especially, Gröbner bases for submodules of free modules over polynomial rings, we solve a classification problem in theory of algebraic operads: we show that the only nontrivial (possibly inhomogeneous) distributive law between the operad of Lie algebras and the operad of commutative associative algebras is given by the Livernet–Loday formula deforming the Poisson operad into the associative operad.

The paper investigates exterior and symmetric (co)homologies of groups. We introduce symmetric homology of groups and compute exterior and symmetric (co)homologies of some finite groups. We also compare the classical, exterior and symmetric (co)homologies. Finally, we derive restriction and corestriction homomorphisms for exterior cohomology.

We find sufficient conditions guaranteeing that for a quasivariety $M$ of structures of finite type containing a $\mathrm{\text{B}}$-class with respect to $M$, there exists a subquasivariety $K\subseteq M$ and a structure ${𝒜}\in K$ such that the problems whether a finite lattice embeds into the lattice $\mathrm{\text{Lv}}(K)$ of $K$-varieties and into the lattice ${Con}_K{𝒜}$ are undecidable.

In this paper we compute the center of many noncommutative algebras that can be interpreted as skew $PBW$ extensions. We show that, under some natural assumptions on the parameters that define the extension, either the center is trivial, or, it is of polynomial type. As an application, we provided new examples of noncommutative algebras that are cancellative.

We provide an explicit algorithm to compute a lifted Massey triple product relative to a defining system for a smooth projective plane curve $X$ defined by a homogeneous polynomial $G(\underline{x})$ over a field. The main idea is to use the description (due to Carlson and Griffiths) of the cup product for $H^1(X,ℂ)$ in terms of the multiplications inside the Jacobian ring of $G(\underline{x})$ and the Cech–deRham complex of $X$. Our algorithm gives a criterion whether a lifted Massey triple product vanishes or not in $H^2(X)$ under a particular nontrivial defining system of the Massey triple product and thus can be viewed as a generalization of the vanishing criterion of the cup product in $H^2(X)$ of Carlson and Griffiths. Based on our algorithm, we provide explicit numerical examples by running the computer program.

A good way of parameterizing zero-dimensional schemes in an affine space ${𝔸}_K^n$ has been developed in the last 20 years using border basis schemes. Given a multiplicity $\mu$, they provide an open covering of the Hilbert scheme ${\mathrm{\text{Hilb}}}^{\mu }({𝔸}_K^n)$ and can be described by easily computable quadratic equations. A natural question arises on how to determine loci which are contained in border basis schemes and whose rational points represent zero-dimensional $K$-algebras sharing a given property. The main focus of this paper is on giving effective answers to this general problem. The properties considered here are the locally Gorenstein, strict Gorenstein, strict complete intersection, Cayley–Bacharach, and strict Cayley–Bacharach properties. The key characteristic of our approach is that we describe these loci by exhibiting explicit algorithms to compute their defining ideals. All results are illustrated by nontrivial, concrete examples.

Let ${{𝒱}}_P$ be the variety generated by an order primal algebra of finite signature associated with a finite bounded poset $P$ that admits a near-unanimity operation. Let $\mathrm{\Lambda }$ be a finite set of linear identities that does not interpret in ${{𝒱}}_P$. Let ${{𝒱}}_{\mathrm{\Lambda }}$ be the variety defined by $\mathrm{\Lambda }$. We prove that ${{𝒱}}_P\vee {{𝒱}}_{\mathrm{\Lambda }}$ is $n$-permutable for some $n$. This implies that there is an $n$ such that $n$-permutability is not join-prime in the lattice of interpretability types of varieties. In fact, it follows that $n$-permutability where $n$ runs through the integers greater than 1 is not prime in the lattice of interpretability types of varieties.

We strengthen this result by making $P$ and $\mathrm{\Lambda }$ more special. We let $P$ be the 6-element bounded poset that is not a lattice and ${{𝒱}}_m$ the variety defined by the set of majority identities for a ternary operational symbol $m$. We prove in this case that ${{𝒱}}_P\vee {{𝒱}}_m$ is 5-permutable. This implies that $n$-permutability is not join-prime in the lattice of interpretability types of varieties whenever $n\ge 5$. We also provide an example demonstrating that ${{𝒱}}_P\vee {{𝒱}}_m$ is not 4-permutable.