Uniquely presented finitely generated commutative monoids.

*(English)*Zbl 1208.20052A well-known result of Rédei says that every finitely generated commutative monoid has a finite presentation. In this paper the authors give necessary and sufficient conditions for a finitely generated, combinatorially finite, cancellative, commutative monoid to have a unique minimal presentation. This class of monoids contains the affine semigroups, which are monoids isomorphic to a finitely generated submonoid of \(\mathbb{N}^r\) with \(r\) a positive integer.

The paper first recalls a general method of determining all minimal presentations of a finitely generated, commutative, cancellative, unit-free monoid. The authors then define the notion of a Betti element, which is an element satisfying a particular factorization condition making it important to the study of the monoid’s minimal presentations. There is a natural binary relation on elements of the monoid \(S\), defined by \(b\prec_Sa\) if \(a-b\in S\), which can be applied to Betti elements to establish an important criterion for \(S\) to have a unique minimal presentation.

Much of the paper focuses on the case of affine semigroups and the construction called ‘gluing’ which allows one to combine affine semigroups in a straightforward way. A main result is to relate the set of Betti elements of affine semigroups \(S_1\) and \(S_2\) to the affine semigroup \(S\) which is the gluing of the two, and to establish when \(S\) has a unique minimal presentation in terms of these Betti elements.

The last section of the paper examines when numerical semigroups, subsemigroups of \(\mathbb{N}\) with finite complement, have unique minimal presentations. Several important classes of semigroups are studied in this context, those generated by intervals, those with embedding dimension three, and semigroups of maximal embedding dimension.

The authors do a good job placing their work within the context of related literature on semigroups and give clear proofs and interesting examples throughout the paper.

The paper first recalls a general method of determining all minimal presentations of a finitely generated, commutative, cancellative, unit-free monoid. The authors then define the notion of a Betti element, which is an element satisfying a particular factorization condition making it important to the study of the monoid’s minimal presentations. There is a natural binary relation on elements of the monoid \(S\), defined by \(b\prec_Sa\) if \(a-b\in S\), which can be applied to Betti elements to establish an important criterion for \(S\) to have a unique minimal presentation.

Much of the paper focuses on the case of affine semigroups and the construction called ‘gluing’ which allows one to combine affine semigroups in a straightforward way. A main result is to relate the set of Betti elements of affine semigroups \(S_1\) and \(S_2\) to the affine semigroup \(S\) which is the gluing of the two, and to establish when \(S\) has a unique minimal presentation in terms of these Betti elements.

The last section of the paper examines when numerical semigroups, subsemigroups of \(\mathbb{N}\) with finite complement, have unique minimal presentations. Several important classes of semigroups are studied in this context, those generated by intervals, those with embedding dimension three, and semigroups of maximal embedding dimension.

The authors do a good job placing their work within the context of related literature on semigroups and give clear proofs and interesting examples throughout the paper.

Reviewer: Nathan Kaplan (Cambridge)

##### MSC:

20M14 | Commutative semigroups |

20M05 | Free semigroups, generators and relations, word problems |

20M13 | Arithmetic theory of semigroups |