International Journal of Algebra and Computation

Let $G$ be a connected graph on the vertex set $[n]$. Then $\mathrm{\text{depth}}(S/J_G)\le n+1$. In this paper, we prove that if $G$ is a unicyclic graph, then the depth of $S/J_G$ is bounded below by $n$. Also, we characterize $G$ with $\mathrm{\text{depth}}(S/J_G)=n$ and $\mathrm{\text{depth}}(S/J_G)=n+1$. We then compute one of the distinguished extremal Betti numbers of $S/J_G$. If $G$ is obtained by attaching whiskers at some vertices of the cycle of length $k$, then we show that $k-1\le \mathrm{\text{reg}}(S/J_G)\le k+1$. Furthermore, we characterize $G$ with $\mathrm{\text{reg}}(S/J_G)=k-1$, $\mathrm{\text{reg}}(S/J_G)=k$ and $\mathrm{\text{reg}}(S/J_G)=k+1$. In each of these cases, we classify the uniqueness of the extremal Betti number of these graphs.

We study the class of symmetric $n$-ary bands. These are $n$-ary semigroups $(X,F)$ such that $F$ is invariant under the action of permutations and idempotent, i.e., satisfies $F(x,\dots ,x)=x$ for all $x\in X$. We first provide a structure theorem for these symmetric $n$-ary bands that extends the classical (strong) semilattice decomposition of certain classes of bands. We introduce the concept of strong $n$-ary semilattice of $n$-ary semigroups and we show that the symmetric $n$-ary bands are exactly the strong $n$-ary semilattices of $n$-ary extensions of Abelian groups whose exponents divide $n-1$. Finally, we use the structure theorem to obtain necessary and sufficient conditions for a symmetric $n$-ary band to be reducible to a semigroup.

This paper concerns a number of diagram categories, namely the partition, planar partition, Brauer, partial Brauer, Motzkin and Temperley–Lieb categories. If ${𝒦}$ denotes any of these categories, and if $\sigma \in {{𝒦}}_{nm}$ is a fixed morphism, then an associative operation ${\star }_{\sigma }$ may be defined on ${{𝒦}}_{mn}$ by $\alpha {\star }_{\sigma }\beta =\alpha \sigma \beta$. The resulting semigroup ${{𝒦}}_{mn}^{\sigma }=({{𝒦}}_{mn},{\star }_{\sigma })$ is called a sandwich semigroup. We conduct a thorough investigation of these sandwich semigroups, with an emphasis on structural and combinatorial properties such as Green’s relations and preorders, regularity, stability, mid-identities, ideal structure, (products of) idempotents, and minimal generation. It turns out that the Brauer category has many remarkable properties not shared by any of the other diagram categories we study. Because of these unique properties, we may completely classify isomorphism classes of sandwich semigroups in the Brauer category, calculate the rank (smallest size of a generating set) of an arbitrary sandwich semigroup, enumerate Green’s classes and idempotents, and calculate ranks (and idempotent ranks, where appropriate) of the regular subsemigroup and its ideals, as well as the idempotent-generated subsemigroup. Several illustrative examples are considered throughout, partly to demonstrate the sometimes-subtle differences between the various diagram categories.

We show that the right ideal of a Novikov algebra generated by the square of a right nilpotent subalgebra is nilpotent. We also prove that a $G$-graded Novikov algebra $N$ over a field $K$ with solvable $0$-component $N_0$ is solvable, where $G$ is a finite additive abelean group and the characteristic of $K$ does not divide the order of the group $G$. We also show that any Novikov algebra $N$ with a finite solvable group of automorphisms $G$ is solvable if the algebra of invariants $N^G$ is solvable.

Let $G$ be a finite group. How minimal subgroups can be embedded in $G$ is a question of particular interest in studying the structure of $G$. A subgroup $H$ of $G$ is called $s$-permutable in $G$ if $HP=PH$ for all Sylow subgroups $P$ of $G$. A subgroup $H$ of $G$ is called $n$-embedded in $G$ if there exists a normal subgroup $T$ of $G$ such that $H^G=HT$ and $H\cap T\le H_{sG}$, where $H_{sG}$ is the subgroup of $H$ generated by all those subgroups of $H$ which are $s$-permutable in $G$. In this paper, we investigate the structure of the finite group $G$ with $n$-embedded subgroups.

We show that if a right-angled Artin group $A(\mathrm{\Gamma })$ has a non-trivial, minimal action on a tree $T$ with more than two ends, then $\mathrm{\Gamma }$ contains a separating subgraph $\mathrm{\Lambda }$ such that $A(\mathrm{\Lambda })$ stabilizes an edge in $T$.

Let $X$ and $Y$ be two generic traceless matrices of size $2\times 2$ with entries from a commutative associative polynomial algebra over a field $K$ of characteristic zero. Consider the associative unitary algebra $W$, and its Lie subalgebra $L$ generated by $X$ and $Y$ over the field $K$. It is well known that the center $C(W)=K[t,u,v]$ of $W$ is the polynomial algebra generated by the algebraically independent commuting elements $t=\mathrm{\text{tr}}(X^2)I_2$, $u=\mathrm{\text{tr}}(Y^2)I_2$, $v=\mathrm{\text{tr}}(XY)I_2$. We call a polynomial $p\in W$ symmetric, if $p(X,Y)=p(Y,X)$. The set of symmetric polynomials is equal to the algebra $W^{S_2}$ of invariants of symmetric group $S_2$. Similarly, we define the Lie algebra $L^{S_2}$ of symmetric polynomials in the Lie algebra $L$. We give the description of the algebras $W^{S_2}$ and $L^{S_2}$, and we provide finite sets of free generators for $W^{S_2}$, and ${[L,L]}^{S_2}$ as $K[t+u,tu,v]$-modules.

In a remarkable series of papers, Zlil Sela classified the first-order theories of free groups and torsion-free hyperbolic groups using geometric structures he called towers. It was later proved by Chloé Perin that if $H$ is an elementarily embedded subgroup (or elementary submodel) of a torsion-free hyperbolic group $G$, then $G$ is a tower over $H$. We prove a generalization of Perin’s result to toral relatively hyperbolic groups using JSJ and shortening techniques.

In this paper, we determine all partial actions and partial coactions of Taft and Nichols Hopf algebras on their base fields. Furthermore, we prove that all such partial (co)actions are symmetric.