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.