Classifying $mathsf{GL}(n,mathbb Z)$-orbits of points and rational subspaces

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 $F$.

Interval MV-algebras and generalizations

For any MV-algebra $A$ we equip the set $I(A)$ of intervals in $A$ with pointwise L ukasiewicz negation $ eg x={ eg alphamid alphain x}$, (truncated) Minkowski sum, $xoplus y={alphaoplus betamid alpha in x,,,betain y}$, pointwise L ukasiewicz conjunction $xodot y= eg( eg xoplus eg y)$, the operators $Delta x=[min x,min x]$, $ abla x=[max x,max x]$, and distinguished constants $0=[0,0],,, 1=[1,1],,,, mathsf{i} = A$. We list a few equations satisfied by the algebra $mathcal I(A)=(I(A),0,1,mathsf{i}, eg,Delta, abla,oplus,odot)$, call IMV-algebra every model ofthese equations, and show that, conversely, every IMV-algebra is isomorphic to the IMV-algebra $mathcal I(B)$ of all intervals in some MV-algebra $B$. We show that IMV-algebras are categorically equivalent to MV-algebras, and give a representation of free IMV-algebras. We construct L ukasiewicz interval logic, with its coNP-complete consequence relation, which we prove to be complete for $mathcal I([0,1])$-valuations. For any class $mathsf{Q}$ of partially ordered algebras with operations that are monotone or antimonotone in each variable, we consider the generalization $mathcal I_{mathsf{Q}}$ of the MV-algebraic functor $mathcal I$, and give necessary and sufficient conditions for $mathcal I_{mathsf{Q}}$ to be a categorical equivalence. These conditions are satisfied, e.g., by all subquasivarieties of residuated lattices.