A theorem on free envelopes
Abstract  References  Similar Articles  Additional Information Abstract: The free envelope of a finite commutative semigroup was defined by Grillet [Trans. Amer. Math. Soc. 149 (1970), 665682] to be a finitely generated free commutative semigroup with identity and a homomorphism endowed with certain properties. Grillet raised the following question: does always generate a pure subgroup of the free Abelian group with the same basis as ? 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 such that f is positive on X and 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 where N is the nonnegative integers under addition. Then if and only if S is isomorphic to a unitary subsemigroup of a finitely generated free commutative semigroup with identity.
Retrieve articles in Transactions of the American Mathematical Society with MSC: 20M14 Retrieve articles in all journals with MSC: 20M14
Additional Information
