Network: monoids and applications

Goals

The main goals of the network

Both numerical and affine semigroups were initially studied in association to the solutions of linear Diophantine equations. After monomial curves became a very productive source of examples for commutative algebra and algebraic geometry, their study experimented a revitalization in the last thirty years.
Factorization in terms of irreducible elements in integral domains has been carried out without using addition as a binary operation. This allows to endow domains with the monoid structure with respect to the product. As for monomial curves, the treatment of affine and numerical semigroups (as well as the tools which were developed for them) revealed to be very useful for general monoids.
The study of the properties of algebraic-geometry codes in terms of the Weierstrass semigroup associated with one point of a curve pushed forward even more the research on the properties of the Weierstrass semigroups.
Applications to combinatorial configurations of lines and points, as well as the connection between certain families of numerical semigroups and cyclotomic polynomials, provided even more motivation for the investigation of commutative monoids.
The main purpose of this network is to intensify the relationship between the different working groups whose current research is oriented to the theory of monoids in Spain. At the same time, we want our collaborators from abroad to take part into this experience, expecting our network to become an international reference in the field of commutative monoids.

Activities

Related activities

Since 2008 the members of the network have organized the following meetings on numerical semigroups and applications: Porto 2008, Granada 2010, Vila Real 2012, Cortona 2014, Levico Terme 2016, Cortona 2018 (see eventos). One of the main tasks of the network is to continue these activities.
We have also implemented software to make computations in this area. This software needs maintenance and continuous improvement.

Structure

Nodes of the network

The nodes of the network are: Barcelona, Cádiz, Castellón, Granada, La Laguna, and Valladolid (see nodes).
Every node gathers researchers both national and international sorted with respect to a set of particular specific subjects inside the network.

Origin

Birth of the network

In the last decade there has been an intense collaboration between several members of the network. Also several imns meetings took place since 2008 (see events) where the need of creating a network on numerical semigroups and applications was considered. It was in the International meeting on numerical semigroups with applications - Levico Terme 2016 where the formal creation of the network took place.