We first show that the subgroup of the abelian real group $mathbb{R}$ generated by the coordinates of a point in $x = (x_1,dots,x_n)inmathbb{R}^n$ completely classifies the $mathsf{GL}(n,mathbb Z)$-orbit of $x$. This yields a short proof of J.S.Danis
theorem: the $mathsf{GL}(n,mathbb Z)$-orbit of $xinmathbb{R}^n$ is dense iff $x_i/x_jin mathbb{R} setminus mathbb Q$ for some $i,j=1,dots,n$. We then classify $mathsf{GL}(n,mathbb Z)$-orbits of rational affine subspaces $F$ of $mathbb{R}^n$. We prove that the dimension of $F$ together with the volume of a special parallelotope associated to $F$ yields a complete classifier of the $mathsf{GL}(n,mathbb Z)$-orbit of $F$.
Let $t_1,ldots,t_n$ be $ell$-group terms in the variables $X_1,ldots,X_m$. Let $hat t_1,ldots,hat t_n$ be their associated piecewise homogeneous linear functions. Let $G $ be the $ell$-group generated by $hat t_1, ldots,hat t_n$ in the free $m$-gener
ator $ell$-group $mathcal A_m.$ We prove: (i) the problem whether $G$ is $ell$-isomorphic to $mathcal A_n$ is decidable; (ii) the problem whether $G$ is $ell$-isomorphic to $mathcal A_l$ ($l$ arbitrary) is undecidable; (iii) for $m=n$, the problem whether ${hat t_1,ldots,hat t_n}$ is a {it free} generating set is decidable. In view of the Baker-Beynon duality, these theorems yield recognizability and unrecognizability results for the rational polyhedron associated to the $ell$-group $G$. We make pervasive use of fans and their stellar subdivisions.
In their recent seminal paper published in the Annals of Pure and Applied Logic, Dubuc and Poveda call an MV-algebra A strongly semisimple if all principal quotients of A are semisimple. All boolean algebras are strongly semisimple, and so are all fi
nitely presented MV-algebras. We show that for any 1-generator MV-algebra semisimplicity is equivalent to strong semisimplicity. Further, a semisimple 2-generator MV-algebra A is strongly semisimple if and only if its maximal spectral space m(A) does not have any rational Bouligand-Severi tangents at its rational points. In general, when A is finitely generated and m(A) has a Bouligand-Severi tangent then A is not strongly semisimple.
