International Journal of Algebra and Computation

In this paper, we give a lower bound of the number of maximal independent sets in a graph $G$ in terms of the Castelnuovo–Mumford regularity of its edge ideal. We also find two classes of graphs achieving this minimum value.

Let $G$ be a finite group. The set of all supercharacter theories of $G$ forms a lattice, where the join operation coincides with the join operation on the lattice of partitions of $G$, with partial order given by refinement. The meet operation is more complicated however, and seems difficult to describe. In this paper, we outline algorithms for determining the coarsest supercharacter theory whose associated partition is finer than a given partition. One of the primary applications is to compute the supercharacters and superclasses for the meet of two supercharacter theories.

Kudryavtseva and Mazorchuk exhibited Schur–Weyl duality between the rook monoid algebra $ℂR_n$ and the subalgebra $ℂI_k$ of the partition algebra $ℂA_k(n)$ acting on ${({ℂ}^n)}^{\otimes k}$. In this paper, we consider a subalgebra $ℂI_{k+\frac{1}{2}}$ of $ℂI_{k+1}$ such that there is Schur–Weyl duality between the actions of $ℂR_{n-1}$ and $ℂI_{k+\frac{1}{2}}$ on ${({ℂ}^n)}^{\otimes k}$. This paper studies the representation theory of partition algebras $ℂI_k$ and $ℂI_{k+\frac{1}{2}}$ for rook monoids inductively by considering the multiplicity free tower

We show that attaching a whisker (or a pendant) at the vertices of a cycle cover of a graph results in a new graph with the following property: all symbolic powers of its cover ideal are Koszul or, equivalently, componentwise linear. This extends previous work where the whiskers were added to all vertices or to the vertices of a vertex cover of the graph.

A numerical semigroup is a sub-semigroup of the natural numbers that has a finite complement. Some of the key properties of a numerical semigroup are its Frobenius number $F$, genus $g$ and type $t$. It is known that for any numerical semigroup $\frac{g}{F+1-g}\le t\le 2g-F$. Numerical semigroups with $t=2g-F$ are called almost symmetric, we introduce a new property that characterizes them. We give an explicit characterization of numerical semigroups with $t=\frac{g}{F+1-g}$. We show that for a fixed $\alpha$ the number of numerical semigroups with Frobenius number $F$ and type $F-\alpha$ is eventually constant for large $F$. The number of numerical semigroups with genus $g$ and type $g-\alpha$ is also eventually constant for large $g$.

Let $A$ be a reduced ring containing $\mathbb{Q}$ and let ${\xi }_1,{\xi }_2$ be commuting locally nilpotent derivations of $A$. In this paper, we give an algorithm to decide the local nilpotency of derivations of the form $\xi =a_1{\xi }_1+a_2{\xi }_2$, where $a_1,a_2$ are elements in $A$.

We build presentations for automata groups generated by Cayley machines of finite groups of nilpotency class two and prove that these automata groups are all cross-wired lamplighter groups.

Finite smooth digraphs, that is, finite directed graphs without sources and sinks, can be partially ordered via pp-constructability. We give a complete description of this poset and, in particular, we prove that it is a distributive lattice. Moreover, we show that in order to separate two smooth digraphs in our poset it suffices to show that the polymorphism clone of one of the digraphs satisfies a prime cyclic loop condition that is not satisfied by the polymorphism clone of the other. Furthermore, we prove that the poset of cyclic loop conditions ordered by their strength for clones is a distributive lattice, too.

By the density of a finite graph we mean its average vertex degree. For an $m$-generated group, the density of its Cayley graph in a given set of generators, is the supremum of densities taken over all its finite subgraphs. It is known that a group with $m$ generators is amenable if and only if the density of the corresponding Cayley graph equals $2m$.

A famous problem on the amenability of R. Thompson’s group $F$ is still open. Due to the result of Belk and Brown, it is known that the density of its Cayley graph in the standard set of group generators $\{x_0,x_1\}$, is at least $3.5$. This estimate has not been exceeded so far.

For the set of symmetric generators $S=\{x_1,{\bar{x}}_1\}$, where ${\bar{x}}_1=x_1{x}_0^{-1}$, the same example only gave an estimate of $3$. There was a conjecture that for this generating set equality holds. If so, $F$ would be non-amenable, and the symmetric generating set would have the doubling property. This would mean that for any finite set $X\subset F$, the inequality $\left|S^{\pm 1}X\right|\ge 2\left|X\right|$ holds.

In this paper, we disprove this conjecture showing that the density of the Cayley graph of $F$ in symmetric generators $S$ strictly exceeds $3$. Moreover, we show that even larger generating set $S_0=\{x_0,x_1,{\bar{x}}_1\}$ does not have doubling property.