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Auditory steady state responses (ASSRs) can be reliably recorded in sleeping neonates and can be used to estimate hearing thresholds in individuals with normal or impaired hearing. However, artifactual responses can contaminate recordings, particularly when recording bone conduction ASSRs. The aim of this investigation was to study the presence of electromagnetic artifact in bone conduction ASSRs and to investigate methods of reducing the amplitude of the artifact.

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)$ $ \alpha :\,S\, \to \,F(S)$ $ \alpha (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}}$f is positive on X and $ {f_{\left\vert H \right.}}$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)$N is the nonnegative integers under addition. Then $ S\, \cong \,{S^{{\ast}{\ast}}}$S is isomorphic to a unitary subsemigroup of a finitely generated free commutative semigroup with identity.