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$\newcommand{\Z}{{\Bbb Z}} \newcommand{\Q}{{\Bbb Q}} \newcommand{\Hom}{{\rm Hom}} $ The following lemma is certainly known. Lemma (well-known). Let $B$ be a lattice (that is, a finitely generated ...

Suppose that $G$ is a group and $P:G\rightarrow G$ is a permutation of $G$. We would now like to apply a forgetful functor to the algebra $(G,\cdot,P,^{-1},e)$. Whenever $g\in G$, let $gP$ be the ...