Uniquely presented finitely generated commutative monoids

A finitely generated commutative monoid is uniquely presented if it has only a minimal presentation. We give necessary and sufficient conditions for finitely generated, combinatorially finite, cancellative, commutative monoids to be uniquely presented. We use the concept of gluing to construct commutative monoids with this property. Finally for some relevant families of numerical semigroups we describe the elements that are uniquely presented.Comment: 13 pages, typos corrected, references update

On the rational subset problem for groups

We use language theory to study the rational subset problem for groups and monoids. We show that the decidability of this problem is preserved under graph of groups constructions with finite edge groups. In particular, it passes through free products amalgamated over finite subgroups and HNN extensions with finite associated subgroups. We provide a simple proof of a result of Grunschlag showing that the decidability of this problem is a virtual property. We prove further that the problem is decidable for a direct product of a group G with a monoid M if and only if membership is uniformly decidable for G-automata subsets of M. It follows that a direct product of a free group with any abelian group or commutative monoid has decidable rational subset membership.Comment: 19 page

Factorization invariants in numerical monoids

Nonunique factorization in commutative monoids is often studied using factorization invariants, which assign to each monoid element a quantity determined by the factorization structure. For numerical monoids (co-finite, additive submonoids of the natural numbers), several factorization invariants have received much attention in the recent literature. In this survey article, we give an overview of the length set, elasticity, delta set, $\omega$-primality, and catenary degree invariants in the setting of numerical monoids. For each invariant, we present current major results in the literature and identify the primary open questions that remain

Excision for deformation K-theory of free products

Associated to a discrete group $G$, one has the topological category of finite dimensional (unitary) $G$-representations and (unitary) isomorphisms. Block sums provide this category with a permutative structure, and the associated $K$-theory spectrum is Carlsson's deformation $K$-theory of G. The goal of this paper is to examine the behavior of this functor on free products. Our main theorem shows the square of spectra associated to $G*H$ (considered as an amalgamated product over the trivial group) is homotopy cartesian. The proof uses a general result regarding group completions of homotopy commutative topological monoids, which may be of some independent interest.Comment: 32 pages, 1 figure. Final version: The title has changed, and the paper has been substantially revised to improve clarit