Diversity in inside factorial monoids (original) (raw)

Membership and elasticity in certain affine monoids

arXiv: Commutative Algebra, 2018

For affine monoids of dimension 2 with embedding dimension 2 and 3, we study the problem of determining when a vector is an element of the monoid, and the problem of determining the elasticity of a monoid element.

Factorization in monoids by stratification of atoms and the Elliott Problem

2020

In an additive factorial monoid each element can be represented as a linear combination of irreducible elements (atoms) with uniquely determined coefficients running over all natural numbers. In this paper we develop for a wide class of non-factorial monoids a concept of stratification for atoms which allows to represent each element as a linear combination of atoms where the coefficients are uniquely determined when restricted in a particular way. This wide class includes inside factorial monoids and in particular simplicial affine semigroups. In the latter case the question of uniqueness is related to a problem studied by E. B. Elliott in a paper from 1903. For the monoid of all nonnegative solutions of a certain linear Diophantine equation in three variables, Elliott considers"simple sets of solutions"(atoms of the monoid) and looks for a method that gives"every set once only". We show that for simplicial affine semigroups in two dimensions a stratification is...

Homogeneous finitely presented monoids of linear growth

2017

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 aba bab, where a,b,wa,b,wa,b,w are elements of MMM and $ $ denotes the monogenic semigroup generated by www. Moreover, the decomposition can be chosen in such a way that the sandwiches are either singletons or "free" ones (meaning that all elements awnba w^n bawnb in each sandwich are pairwise different). So, the minimal number of free sandwiches in such a decomposition is a numerical invariant of a homogeneous (and conjecturally, non-homogeneous) finitely presented monoid of linear growth.

A Theory of Transformation Monoids: Combinatorics and Representation Theory

The Electronic Journal of Combinatorics, 2010

The aim of this paper is to develop a theory of finite transformation monoids and in particular to study primitive transformation monoids. We introduce the notion of orbitals and orbital digraphs for transformation monoids and prove a monoid version of D. Higman's celebrated theorem characterizing primitivity in terms of connectedness of orbital digraphs. A thorough study of the module (or