Related papers: The structure theorem for sets of length for numer…

A numerical semigroup $S$ is an additive subsemigroup of the non-negative integers, containing zero, with finite complement. Its multiplicity $m$ is its smallest nonzero element. The Apery set of $S$ is the set $\text{Ap}(S) = \{n \in S :…

We prove that every quasi-projective semi log canonical pair has a quasi-log structure with several good properties. It implies that various vanishing theorems, torsion-free theorem, and the cone and contraction theorem hold for semi log…

In an atomic, cancellative, commutative monoid $S$, the elasticity of an element provides a coarse measure of its non-unique factorizations by comparing the largest and smallest values in its set of factorization lengths (called its length…

An infinite structure has the finite length property (over a given field) if, for each of its finite powers, chains of equivariant subspaces in the corresponding free vector space are bounded in length. Prior work showed that the countable…

We prove that every length on a simple group over a locally compact field, is either bounded or proper.

The main motivation for this article is to explore the connections between the existence of certain combinatorial patterns (as in van der Corputs's theorem on arithmetic progressions of length $3$) with well-known tools and theorems for…

We use basic tools of descriptive set theory to prove that a closed set $\mathcal S$ of marked groups has $2^{\aleph_0}$ quasi-isometry classes provided every non-empty open subset of $\mathcal S$ contains at least two non-quasi-isometric…

Let H be a Krull monoid with finite class group G and suppose that every class contains a prime divisor. Then sets of lengths in H have a well-defined structure which just depends on the class group G. With methods from additive…

We call a finite, spanning set of a semi-simple real Lie algebra a distinguished set if it satisfies the following property: The Lie bracket of any two elements out of the set is, up to some constant, another element in the set; conversely,…

We study the structure of sets $S\subseteq\{0, 1\}^n$ with small sensitivity. The well-known Simon's lemma says that any $S\subseteq\{0, 1\}^n$ of sensitivity $s$ must be of size at least $2^{n-s}$. This result has been useful for proving…

We consider general structures where formulas have truth values in the real unit interval as in continuous model theory, but whose predicates and functions need not be uniformly continuous with respect to a distance predicate. Every general…

In this paper, we investigate zero-divisor, nilpotent, idempotent, unit, small, and irreducible elements in semiring extensions such as amount, content, and monoid semialgebras. We also introduce new concepts such as the prime avoidance…

H.Furstenberg and E.Glasner proved that for an arbitrary $k\in\mathbb{N}$, any piecewise syndetic set of integers contains a $k$-term arithmetic progression and the collection of such progressions is itself piecewise syndetic in…

A monoid $S$ is right coherent if every finitely generated subact of every finitely presented right $S$-act is finitely presented. This is a finiteness condition, and we investigate whether or not it is preserved under some standard…

We extend two well-known results in additive number theory, S\'ark\"ozy's theorem on square differences in dense sets and a theorem of Green on long arithmetic progressions in sumsets, to subsets of random sets of asymptotic density 0. Our…

Given two numerical semigroups $S$ and $T$ we say that $T$ is a multiple of $S$ if there exists an integer $d \in \mathbb{N} \setminus \{0\}$ such that $S = \{x \in \mathbb{N} \mid d x \in T\}$. In this paper we study the family of…

A group code structure of a linear code is a description of the code as one-sided or two-sided ideal of a group algebra of a finite group. In these realizations, the group algebra is identified with the ambient space, and the group elements…

The structure of transformation semigroups on a finite set is analyzed by introducing a hierarchy of functions mapping subsets to subsets. The resulting hierarchy of semigroups has a corresponding hierarchy of minimal ideals, or kernels.…

In this paper it is shown that for any network there is a uniquely determined network based on a structure tree that provides a convenient way of determining a minimal cut separating a pair $s, t$ where each of $s, t$ is either a vertex or…

This paper gives an explicit structure theorem for the symmetric group acting on the symmetric algebra of its natural module. Let $G$ be the symmetric group on $x_1,..., x_n$ and let $d_i$ be the $i^{\text{th}}$ elementary symmetric…