A theorem on free envelopes
Chester C. John · Transactions of the American Mathematical Society · 1980
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 ) F(S) with identity and a homomorphism α : S → F ( S ) \alpha :\,S\, \to \,F(S) endowed with certain properties. Grillet raised the following question: does α ( S ) \alpha (S) always generate a pure subgroup of the free Abelian group with the same basis as F ( S ) 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 → Z f:\,F\, \to \,{\textbf {Z}} such that f is positive on X and f | H {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 ∗ ≅ Hom ( S , N ) {S^{\ast }}\, \cong \,\operatorname {Hom} (S,\,N) where N is the nonnegative integers under addition. Then