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 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.

**[1]**Pierre Antoine Grillet,*The free envelope of a finitely generated commutative semigroup*, Trans. Amer. Math. Soc.**149**(1970), 665–682. MR**0292975**, 10.1090/S0002-9947-1970-0292975-4**[2]**A. H. Clifford and G. B. Preston,*The algebraic theory of semigroups*. Vols. 1, 2, Math. Surveys, no. 7, Amer. Math. Soc., Providence, R. I., 1961, 1967. MR**24**#A2627;**36**#1558.**[3]**László Fuchs,*Infinite abelian groups. Vol. I*, Pure and Applied Mathematics, Vol. 36, Academic Press, New York-London, 1970. MR**0255673**

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DOI:
https://doi.org/10.1090/S0002-9947-1980-0549166-5

Article copyright:
© Copyright 1980
American Mathematical Society