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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

A theorem on free envelopes


Author: Chester C. John
Journal: Trans. Amer. Math. Soc. 257 (1980), 255-259
MSC: Primary 20M14
MathSciNet review: 549166
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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\vert 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.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1980-0549166-5
PII: S 0002-9947(1980)0549166-5
Article copyright: © Copyright 1980 American Mathematical Society