Related papers: A realization theorem for sets of lengths in numer…

Let $H$ be a Krull monoid with class group $G$ such that every class contains a prime divisor. Then every nonunit $a \in H$ can be written as a finite product of irreducible elements. If $a=u\_1 \cdot \ldots \cdot u\_k$, with irreducibles…

We establish a new sufficient condition under which a monoid is non-finitely based and apply this condition to Lee monoids $L_\ell^1$, obtained by adjoining an identity element to the semigroup generated by two idempotents $a$ and $b$…

Let $H$ be an atomic monoid. For $x \in H$, let $\mathsf{L}(x)$ denote the set of all possible lengths of factorizations of $x$ into irreducibles. The system of sets of lengths of $H$ is the set $\mathcal{L}(H) = \{\mathsf{L}(x) \mid x \in…

Let $\mathcal L^*$ be a family of finite subsets of $\mathbb N_0$ having the following properties. (a). $\{0\}, \{1\} \in \mathcal L^*$ and all other sets of $\mathcal L^*$ lie in $\mathbb N_{\ge 2}$. (b). If $L_1, L_2 \in \mathcal L^*$,…

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…

In this paper, an explicit construction of a countable parafree Lie algebra over $\mathbb Z/2$ with nonzero second homology is given. It is also shown that the cohomological dimension of the pronilpotent completion of a free noncyclic…

We show that the number of elements generating a squarefree monomial ideal up to radical can always be bounded above in terms of the number of its minimal monomial generators and the maximal height of its minimal primes.

Let L be the zero set of a nonconstant monic polynomial with complex coefficients. In the context of constructive mathematics without countable choice, it may not be possible to construct an element of L. In this paper we introduce a notion…

A Nikodym set $\mathcal{N}\subseteq(\mathbb{Z}/(N\mathbb{Z}))^n$ is a set containing $L\setminus\{x\}$ for every $x\in(\mathbb{Z}/(N\mathbb{Z}))^n$, where $L$ is a line passing through $x$. We prove that if $N$ is square-free, then the size…

We show that for any positive integer $N$, there are only finitely many holomorphic eta quotients of level $N$, none of which is a product of two holomorphic eta quotients other than 1 and itself. This result is an analog of Zagier's…

Let $R$ be a commutative Noetherian ring of dimension $d$, $M$ a commutative cancellative torsion-free monoid of rank $r$ and $P$ a finitely generated projective $R[M]$-module of rank $t$. $(1)$ Assume $M$ is $\Phi$-simplicial seminormal.…

We show that the set of realizations of a given dimension of a max-plus linear sequence is a finite union of polyhedral sets, which can be computed from any realization of the sequence. This yields an (expensive) algorithm to solve the…

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 study representation of square-free polynomials in the polynomial ring F[t] over a finite field F by polynomials in F[t][x]. This is a function field version of the well-studied problem of representing squarefree integers by integer…

The problem of determining the representation type of the full transformation monoid was resolved by Ponizovskii, Putcha, and Ringel. In this paper, we present a similar result for the monoid of binary relations, a partial result for the…

If a finitely generated monoid M is defined by a finite number of degree-preserving relations, then it has linear growth if and only if it can be decomposed into a finite disjoint union of subsets (which we call "sandwiches") of the form…

Up to isomorphism, there exist two non-isomorphic two-element monoids. We show that the identities of the free product of every pair of such monoids admit no finite basis.

For every infinite cardinal number $\kappa$, $\kappa$-monoids and their realization have recently been introduced and studied by Nazemian and Smertnig. A $\kappa$-monoid $H$ has a realization to a ring $R$ if there exists an element $x \in…

Let $\mathcal{A}$ be a finite set of integers, and let $h\mathcal{A}$ denote the $h$-fold sumset of $\mathcal{A}$. Let $(h\mathcal{A})^{(t)}$ be subset of $h\mathcal{A}$ consisting of all integers that have at least $t$ representations as a…

For a commutative cancellative monoid $M$, we introduce the notion of the length density of both a nonunit $x\in M$, denoted $\mathrm{LD}(x)$, and the entire monoid $M$, denoted $\mathrm{LD}(M)$. This invariant is related to three widely…