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Daniele Mundici

Publications and source records attributed to Daniele Mundici.

At least 19 recordsLinked to original sources
arXiv2026

Unital Specker $\ell$-groups and boolean multispaces

As a topological generalization of the notion of a multiset, a boolean multispace is a boolean space $X$ with a continuous function $u\colon X\to \mathbb Z_{>0}$, where $\mathbb Z_{>0}=\{1,2,\dots\}$ has the discrete topology. In this paper the category of boolean multispaces and continuous multiplicity-decreasing morphisms with respect to the divisibility order is shown to be dually equivalent to the category of unital Specker $\ell$-groups and unital $\ell$-homomorphisms. This result extends Stone duality, because unital Specker $\ell$-groups whose distinguished unit is singular are equivalent to boolean algebras. Boolean multispaces, in turn, are categorically equivalent to the Priestley duals of the MV-algebras corresponding to unital Specker $\ell$-groups via the $Γ$ functor. Via duality, we show that the category of unital Specker $\ell$-groups has finite colimits and finite products, but lacks some countable copowers and equalizers.

math.LO↗
arXiv2021

De Finetti for mathematics undergraduates

In 1931 de Finetti proved what is known as his Dutch Book Theorem. This result implies that the finite additivity {\it axiom} for the probability of the disjunction of two incompatible events becomes a {\it consequence} of de Finetti's logic-operational consistency notion. Working in the context of boolean algebras, we prove de Finetti's theorem. The mathematical background required is little more than that which is taught in high school. As a preliminary step we prove what de Finetti called ``the Fundamental Theorem of Probability'', his main contribution both to Boole's probabilistic inference problem on the object of probability theory, and to its modern reformulation known as the optimization version of the probabilistic satisfiability problem. In a final section, we give a self-contained combinatorial proof of de Finetti's exchangeability theorem.

math.PR↗
arXiv2021

Polytime reductions of AF-algebraic problems

We assess the computational complexity of several decision problems concerning (Murray-von Neumann) equivalence classes of projections of AF-algebras whose Elliott classifier is lattice-ordered. We construct polytime reductions among many of these problems.

cs.CC↗
arXiv2019

Basic geometry of the affine group over Z

The subject matter of this paper is the geometry of the affine group over the integers, $\mathsf{GL}(n,\mathbb{Z})\ltimes \mathbb{Z}^n$. Turing-computable complete $\mathsf{GL}(n,\mathbb{Z})\ltimes \mathbb{Z}^n$-orbit invariants are constructed for angles, segments, triangles and ellipses. In rational affine $\mathsf{GL}(n,\mathbb Q)\ltimes \mathbb Q^n$-geometry, ellipses are classified by the Clifford--Hasse--Witt invariant, via the Hasse-Minkowski theorem. We classify ellipses in $\mathsf{GL}(n,\mathbb{Z})\ltimes \mathbb{Z}^n$-geometry combining results by Apollonius of Perga and Pappus of Alexandria with the Hirzebruch-Jung continued fraction algorithm and the Morelli-Włodarczyk solution of the weak Oda conjecture on the factorization of toric varieties. We then consider {\it rational polyhedra}, i.e., finite unions of simplexes in $\mathbb R^n$ with rational vertices. Markov's unrecognizability theorem for combinatorial manifolds states the undecidability of the problem whether two rational polyhedra $P$ and $P'$ are continuously $\mathsf{GL}(n,\mathbb Q)\ltimes \mathbb Q^n$-equidissectable. The same problem for the continuous $\mathsf{GL}(n,\mathbb{Z})\ltimes \mathbb{Z}^n$-equi\-dis\-sect\-ability of $P$ and $P'$ is open. We prove the decidability of the problem whether two rational polyhedra $P,Q$ in $\mathbb R^n$ have the same $\mathsf{GL}(n,\mathbb{Z})\ltimes \mathbb{Z}^n$-orbit.

math.DS↗
arXiv2018

Approaching central projections in AF-algebras

Let $A$ be a unital AF-algebra whose Murray-von Neumann order of projections is a lattice. For any two equivalence classes $[p]$ and $[q]$ of projections we write $[p]\sqsubseteq [q]$ iff for every primitive ideal $\mathfrak p$ of $A$ either $p/\mathfrak p\preceq q/\mathfrak p\preceq (1- q)/\mathfrak p$ or $p/\mathfrak p\succeq q/\mathfrak p \succeq (1-q)/\mathfrak p.$ We prove that $p$ is central iff $[p]$ is $\sqsubseteq$-minimal iff $[p]$ is a characteristic element in $K_0(A)$. If, in addition, $A$ is liminary, then each extremal state of $K_0(A)$ is discrete, $K_0(A)$ has general comparability, and $A$ comes equipped with a centripetal transformation $[p]\mapsto [p]^\Game$ that moves $p$ towards the center. The number $n(p) $ of $\Game$-steps needed by $[p]$ to reach the center has the monotonicity property $[p]\sqsubseteq [q]\Rightarrow n(p)\leq n(q).$ Our proofs combine the $K_0$-theoretic version of Elliott's classification, the categorical equivalence $Γ$ between MV-algebras and unital $\ell$-groups, and Łoś ultraproduct theorem for first-order logic.

math.OA↗
arXiv2017

Word problems in Elliott monoids

Algorithmic issues concerning Elliott local semigroups are seldom considered in the literature, although these combinatorial structures completely classify AF algebras. In general, the addition operation of an Elliott local semigroup is {\it partial}, but for every AF algebra $\mathfrak B$ whose Murray-von Neumann order of projections is a lattice, this operation is uniquely extendible to the addition of an involutive monoid $E(\mathfrak B)$. Let $\mathfrak M_1$ be the Farey AF algebra introduced by the present author in 1988 and rediscovered by F. Boca in 2008. The freeness properties of the involutive monoid $E(\mathfrak M_1)$ yield a natural word problem for every AF algebra $\mathfrak B$ with singly generated $E(\mathfrak B)$, because $\mathfrak B$ is automatically a quotient of $\mathfrak M_1$. Given two formulas $ϕ$ and $ψ$ in the language of involutive monoids, the problem asks to decide whether $ϕ$ and $ψ$ code the same equivalence of projections of $\mathfrak B$. This mimics the classical definition of the word problem of a group presented by generators and relations. We show that the word problem of $\mathfrak M_1$ is solvable in polynomial time, and so is the word problem of the Behnke-Leptin algebras $\mathcal A_{n,k}$, and of the Effros-Shen algebras $\mathfrak F_θ$, for $θ\in [0,1]\setminus \mathbb Q$ a real algebraic number, or $θ= 1/e$. We construct a quotient of $\mathfrak M_1$ having a Gödel incomplete word problem, and show that no primitive quotient of $\mathfrak M_1$ is Gödel incomplete.

math.LO↗
arXiv2017

Diophantine approximation by almost equilateral triangles

A {\it two-dimensional continued fraction expansion} is a map $μ$ assigning to every $x \in\mathbb R^2\setminus\mathbb Q^2$ a sequence $μ(x)=T_0,T_1,\dots$ of triangles $T_n$ with vertices $x_{ni}=(p_{ni}/d_{ni},q_{ni}/d_{ni})\in\mathbb Q^2, d_{ni}>0, p_{ni}, q_{ni}, d_{ni}\in \mathbb Z,$ $i=1,2,3$, such that \begin{eqnarray*} \det \left(\begin{matrix} p_{n1}& q_{n1} &d_{n1}\\ p_{n2}& q_{n2} &d_{n2}\\ p_{n3}& q_{n3} &d_{n3} \end{matrix} \right) = \pm 1\,\,\, \,\,\,\mbox{and}\,\,\,\,\,\, \bigcap_n T_n = \{x\}. \end{eqnarray*} We construct a two-dimensional continued fraction expansion $μ^*$ such that for densely many (Turing computable) points $x$ the vertices of the triangles of $μ(x)$ strongly converge to $x$. Strong convergence depends on the value of $\lim_{n\to \infty}\frac{\sum_{i=1}^3\dist(x,x_{ni})}{(2d_{n1}d_{n2}d_{n3})^{-1/2}},$ ("dist" denoting euclidean distance) which in turn depends on the smallest angle of $T_n$. Our proofs combine a classical theorem of Davenport Mahler in diophantine approximation, with the algorithmic resolution of toric singularities in the equivalent framework of regular fans and their stellar operations.

math.NT↗
arXiv2015

Hopfian $\ell$-groups, MV-algebras and AF C$^*$-algebras

An algebra is said to be hopfian if it is not isomorphic to a proper quotient of itself. We describe several classes of hopfian and of non-hopfian unital lattice-ordered abelian groups and MV-algebras. Using Elliott classification and $K_0$-theory, we apply our results to other related structures, notably the Farey-Stern-Brocot AF C$^*$-algebra and all its primitive quotients, including the Behnke-Leptin C$^*$-algebras $\mathcal A_{k,q}$.

math.RA↗
arXiv2015

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 $x = (x_1,\dots,x_n)\in\mathbb{R}^n$ completely classifies the $\mathsf{GL}(n,\mathbb Z)$-orbit of $x$. This yields a short proof of J.S.Dani's theorem: the $\mathsf{GL}(n,\mathbb Z)$-orbit of $x\in\mathbb{R}^n$ is dense iff $x_i/x_j\in \mathbb{R} \setminus \mathbb Q$ 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 $F$.

math.DS↗
arXiv2015

Fans and generators of free abelian l-groups

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 $m$-generator $\ell$-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.

math.GT↗
arXiv2014

Interval MV-algebras and generalizations

For any MV-algebra $A$ we equip the set $I(A)$ of intervals in $A$ with pointwise Łukasiewicz negation $\neg x=\{\neg α\mid α\in x\}$, (truncated) Minkowski sum, $x\oplus y=\{α\oplus β\mid α\in x,\,\,β\in y\}$, pointwise Łukasiewicz conjunction $x\odot y=\neg(\neg x\oplus \neg y)$, the operators $Δx=[\min x,\min x]$, $\nabla x=[\max x,\max x]$, and distinguished constants $0=[0,0],\,\, 1=[1,1],\,\,\, \mathsf{i} = A$. We list a few equations satisfied by the algebra $\mathcal I(A)=(I(A),0,1,\mathsf{i},\neg,Δ,\nabla,\oplus,\odot)$, 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 Łukasiewicz 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.

math.LO↗
arXiv2014

The Euler valuation on MV-algebras

Every finitely presented MV-algebra A has a unique idempotent valuation E assigning value 1 to every basic element of A. For each element a of A, E(a) turns out to coincide with the Euler characteristic of the open set of maximal ideals m of A such that a/m is nonzero.

math.RA↗
arXiv2014

Severi-Bouligand tangents, Frenet frames and Riesz spaces

It was recently proved that a compact set $X\subseteq \mathbb R^2$ has an outgoing Severi-Bouligand tangent vector $u\not=0$ at $x\in X$ iff some principal ideal of the Riesz space $\mathcal R(X)$ of piecewise linear functions on $X$ is not an intersection of maximal ideals. "Outgoing" means $X\cap [x,x+u]=\{x\}$. Suppose now $X\subseteq \mathbb{R}^n$ and some principal ideal of $\mathcal R(X)$ is not an intersection of maximal ideals. We prove that this is equivalent to saying that $X$ contains a sequence $\{x_i\}$ whose Frenet $k$-frame $(u_1,\ldots,u_k)$ is an outgoing Severi-Bouligand tangent of $X$. When the $\{x_i\}$ are taken as sample points of a smooth curve $γ,$ the Frenet $k$-frames of $\{x_i\}$ and of $γ$ coincide. The computation of Frenet frames via sample sequences does not require the knowledge of any higher-order derivative of $γ$.

math.RA↗
arXiv2014

Classifying $GL(2,\mathbb Z) \ltimes \mathbb Z^{2}$-orbits by subgroups of $\mathbb R$

Let $\mathcal G_2$ denote the affine group $GL(2,\mathbb Z) \ltimes \mathbb Z^{2}$. For every point $x=(x_1,x_2) \in \R2$ let $\orb(x)=\{y\in\R2\mid y=γ(x)$ for some $γ\in \mathcal{G}_2 \}$. Let $G_{x}$ be the subgroup of the additive group $\mathbb R$ generated by $x_1,x_2, 1$. If $\rank(G_x)\in \{1,3\}$ then $\orb(x)=\{y\in\R2\mid G_y=G_x\}$. If $\rank(G_x)=2$, knowledge of $G_x$ is not sufficient in general to uniquely recover $\orb(x)$: rather, $G_x$ classifies precisely $\max(1,ϕ(d)/2)$ different orbits, where $d$ is the denominator of the smallest positive nonzero rational in $G_x$ and $ϕ$ is Euler function. To get a complete classification, polyhedral geometry provides an integer $c_x\geq 1$ such that $\orb(y)=\orb(x) $ iff $(G_{x},c_{x})=(G_{y},c_{y})$.

math.DS↗
arXiv2012

The differential semantics of Lukasiewicz syntactic consequence

The classical condition "$ϕ$ is a semantic consequence of $Θ$" in infinite-valued propositional Łukasiewicz logic Ł$_\infty$ is refined using enriched valuations that take into account the effect on $ϕ$ of the stability of the truth-value of all $θ\in Θ$ under small perturbations (or, measurement errors) of the models of $Θ$. The differential properties of the functions represented by $ϕ$ and by all $θ\in Θ$ naturally lead to a new notion of semantic consequence $\models_\partial$ that turns out to coincide with syntactic consequence $\vdash$.

math.LO↗
arXiv2012

Logic on the n-cube

We endow the partially ordered set of nonempty faces of the n-cube with a distinguished 0-dimensional face and three operations that naturally extend the Rota-Metropolis partial operations. While the structures thus obtained turn out to be term-equivalent to Post algebras of order 3, the inclusion order between faces coincides with the De Luca-Termini sharpening order, and yields a compact coNP-complete logic that tolerates a modicum of inconsistency and nonmonotonicity.

math.LO↗
arXiv2012

Bouligand-Severi Tangents in MV-Algebras

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 finitely 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.

math.LO↗