arXiv subjects

Daniele Mundici

Publications and source records attributed to Daniele Mundici.

23 records · Page 2Linked to original sources

arXiv2011

Drawing Sound Conclusions from Unsound Premises

Given sets $Φ_1=\{ϕ_{11},...,ϕ_{1u(1)}\}, ...,Φ_{z}=\{ϕ_{z1},...,ϕ_{zu(z)}\}$ of boolean formulas, a formula $ω$ follows from the conjunction $\bigwedgeΦ_i= \bigwedge ϕ_{ij}$ iff $\neg ω\wedge \bigwedge_{i=1}^z Φ_i$ is unsatisfiable. Now assume that, given integers $0\leq e_i < u(i)$, we must check if $\neg ω\wedge \bigwedge_{i=1}^z Φ'_i$ remains unsatisfiable, where $Φ'_i\subseteq Φ_i$ is obtained by deleting $\,\,e_{i}$ arbitrarily chosen formulas of $Φ_i$, for each $i=1,...,z.$ Intuitively, does $ω$ {\it stably} follow, after removing $e_i$ random formulas from each $Φ_i$? We construct a quadratic reduction of this problem to the consequence problem in infinite-valued \luk\ logic Ł$_\infty$. In this way we obtain a self-contained proof that the Ł$_\infty$-consequence problem is coNP-complete.

cs.LO↗

arXiv2011

Measure theory in the geometry of $GL(n,\mathbb Z) \ltimes \mathbb Z^{n}$

The $n$-dimensional affine group over the integers is the group $\mathcal G_n$ of all affinities on $\mathbb R^{n}$ which leave the lattice $ \mathbb Z^{n}$ invariant. $\mathcal G_n$ yields a geometry in the classical sense of the Erlangen Program. In this paper we construct a $\mathcal G_n$-invariant measure on rational polyhedra in $\mathbb R^n$, i.e., finite unions of simplexes with rational vertices in $\mathbb R^n$, and prove its uniqueness. Our main tool is given by the Morelli-Włodarczyk factorization of birational toric maps in blow-ups and blow-downs (solution of the weak Oda conjecture).

math.GN↗

arXiv2010

Finitely presented lattice-ordered abelian groups with order-unit

Let $G$ be an $\ell$-group (which is short for ``lattice-ordered abelian group''). Baker and Beynon proved that $G$ is finitely presented iff it is finitely generated and projective. In the category $\mathcal U$ of {\it unital} $\ell$-groups---those $\ell$-groups having a distinguished order-unit $u$---only the $(\Leftarrow)$-direction holds in general. Morphisms in $\mathcal U$ are {\it unital $\ell$-homomorphisms,} i.e., hom\-o\-mor\-phisms that preserve the order-unit and the lattice structure. We show that a unital $\ell$-group $(G,u)$ is finitely presented iff it has a basis, i.e., $G$ is generated by an abstract Schauder basis over its maximal spectral space. Thus every finitely generated projective unital $\ell$-group has a basis $\mathcal B$. As a partial converse, a large class of projectives is constructed from bases satisfying $\bigwedge\mathcal B\not=0$. Without using the Effros-Handelman-Shen theorem, we finally show that the bases of any finitely presented unital $\ell$-group $(G,u)$ provide a direct system of simplicial groups with 1-1 positive unital homomorphisms, whose limit is $(G,u)$.

math.GR↗

arXiv2009

Classification of finitely generated lattice-ordered abelian groups with order-unit

A unital $\ell$-group $(G,u)$ is an abelian group $G$ equipped with a translation-invariant lattice-order and a distinguished element $u$, called order-unit, whose positive integer multiples eventually dominate each element of $G$. We classify finitely generated unital $\ell$-groups by sequences $\mathcal W = (W_{0},W_{1},...)$ of weighted abstract simplicial complexes, where $W_{t+1}$ is obtained from $W_{t}$ either by the classical Alexander binary stellar operation, or by deleting a maximal simplex of $W_{t}$. A simple criterion is given to recognize when two such sequences classify isomorphic unital $\ell$-groups. Many properties of the unital $\ell$-group $(G,u)$ can be directly read off from its associated sequence: for instance, the properties of being totally ordered, archimedean, finitely presented, simplicial, free.

math.GR↗

arXiv2009

Rational polyhedra and projective lattice-ordered abelian groups with order unit

An l-group G is an abelian group equipped with a translation invariant lattice order. Baker and Beynon proved that G is finitely generated projective iff it is finitely presented. A unital l-group is an l-group G with a distinguished order unit, i.e., an element u in G whose positive integer multiples eventually dominate every element of G. While every finitely generated projective unital l-group is finitely presented, the converse does not hold in general. Classical algebraic topology (a la Whitehead) will be combined in this paper with the Włodarczyk-Morelli solution of the weak Oda conjecture for toric varieties, to describe finitely generated projective unital l-groups.

math.AT↗