The free envelope of a finitely generated commutative semigroup

Abstract

Let S be a commutative semigroup. A quasi-universal free semigroup of S is a free commutative semigroup with identity F together with a homomorphism -q of S into F such that any homomorphism of S into a free commutative semigroup with identity factors through -q; if there is uniqueness in this factorization, we say that (F, -q) is a universal free semigroup of S. If S is finitely generated, there exists a smallest quasi-universal free semigroup of S; we call it the free envelope of S. Its construction and study is the first object of this paper, the second being the application of the free envelope to the study of cancellative and power-cancellative commutative semigroups. We construct the free envelope in the first section. Cancellative and powercancellative semigroups appear in ?2; we prove that a finitely generated commutative semigroup is embeddable into a free commutative semigroup with identity if and only if it has these properties and has either no identity or a trivial group of units; then it is embeddable into its free envelope. The study of this latter embedding gives, conversely, a number of interesting properties of the free envelope in the general situation. The dual of a finitely generated commutative semigroup S may be defined as the semigroup S* of all homomorphisms of S into the additive semigroup of all nonnegative integers, under pointwise addition. Using free envelopes, we prove in ?3 that S***-S* and investigate the relationship between S and S**. This yields in turn a number of results concerning universal free semigroups when they exist. A study of various dimensions completes the section. ?4 deals with embeddings Sc T such that every relation which holds in S can be deduced from the presentation of S in T without using any relation which may hold in T (in which case we say that T kills S). We show that, if S is finitely generated, the (inclusion) homomorphism of S into T can be extended to the free envelope of S; if furthermore T is cancellative, power-cancellative and without identity element, then T contains subsemigroups which kill S and are minimal with that property. This paper has benefited from numerous suggestions, by the members of the Tulane semigroup seminar, especially William R. Nico and A. H. Clifford, and by our referee, which we acknowledge gladly.