A theorem on free envelopes

Abstract:

The free envelope of a finite commutative semigroup was defined by Grillet [Trans. Amer. Math. Soc. 149 (1970), 665-682] to be a finitely generated free commutative semigroup $F(S)$ with identity and a homomorphism $\alpha : S \to F(S)$ endowed with certain properties. Grillet raised the following question: does $\alpha (S)$ always generate a pure subgroup of the free Abelian group with the same basis as $F(S)$? We prove this is indeed the case. It follows as a result of two lemmas. Lemma 1: Given a full rank proper subgroup H of a finitely generated free Abelian group F and a basis X of F there exists a surjective homomorphism $f: F \to {\textbf {Z}}$ such that f is positive on X and ${f_{\left | H \right .}}$ is not surjective. Lemma 2: A finitely generated totally cancellative reduced subsemigroup of a finitely generated free Abelian group F is contained in the positive cone of some basis of F. The following duality theorem is also proved. Let ${S^{\ast }} \cong \operatorname {Hom} (S, N)$ where N is the nonnegative integers under addition. Then $S \cong {S^{{\ast }{\ast }}}$ if and only if S is isomorphic to a unitary subsemigroup of a finitely generated free commutative semigroup with identity.