[Paper Review] Rational Simplicial geometry and projective unital lattice-ordered abelian groups

Leonardo Manuel Cabrer|arXiv (Cornell University)|May 28, 2014

Advanced Algebra and Logic11 references3 citations

TL;DR

This paper establishes a geometric characterization of finitely generated projective unital lattice-ordered abelian groups by proving they are isomorphic to the group of real-valued piecewise linear functions with integer coefficients on a rational, contractible polyhedron $ P \subseteq [0,1]^n $ that contains an integer point and satisfies a local rational extension property. The key contribution is a converse to a known characterization, fully classifying such groups via topological and arithmetical conditions on $ P $.

ABSTRACT

A unital $\ell$-group is an abelian group equipped with a translation invariant lattice-order and with a distinguished strong unit, i.e. an element whose positive integer multiples eventually dominate every element of $G$.If $X$ is a compact subset of $R^n$, the set $M(X)$ of real-valued piecewise linear maps with integer coefficients, whose addition and lattice operations defined pointwise and whose distinguished element is the constant map $1$, is a unital $\ell$-group. In this paper we provide a geometric decription of finitely generated (regular) projective unital $\ell$-groups. We prove that a finitely unital $\ell$-group is projective if and only if it is isomorphic to $M(P)$ for some polyhedron $P$ which is rational, contractible, contains an integer point, and satisfies an elementary arithmetical-topological property.

Motivation & Objective

• To provide a complete geometric characterization of finitely generated projective unital lattice-ordered abelian groups.
• To resolve the open problem of identifying which rational polyhedra $ P \subseteq [0,1]^n $ yield projective unital $\ell$-groups $\mathscr{M}(P)$.
• To prove that projectivity of $\mathscr{M}(P)$ is equivalent to $ P $ being rational, contractible, containing an integer point, and satisfying a local rational extension condition.
• To extend previous results that only covered finite unions of simplexes to general rational polyhedra.
• To provide a solution to the sixth open problem in Mundici's list concerning the classification of projective MV-algebras.

Proposed method

• Use of simplicial geometry and triangulations to analyze the structure of rational polyhedra $ P \subseteq [0,1]^n $.
• Definition of a $\mathbb{Z}$-map as a piecewise linear map with integer coefficients that preserves the lattice and group structure.
• Construction of a continuous retraction $ \eta: [0,1]^n \to P $ using a regular triangulation $ \Delta_P $ of $ P $, ensuring compatibility with the lattice operations.
• Introduction of a weighted abstract simplicial complex $ \mathfrak{W} $ with weights derived from the denominator function $ \operatorname{den}(\eta(v)) $, ensuring strong regularity.
• Application of Theorem 4.6 to show that the geometric realization $ Q \subseteq [0,1]^k $ of $ \mathfrak{W} $ is a $\mathbb{Z}$-retract.
• Establishing a composition of $\mathbb{Z}$-maps $ \mu \circ \nu: [0,1]^k \to P $ and $ \xi: P \to Q $ such that $ (\mu \circ \nu) \circ \xi = \text{id}_P $, proving $ P $ is a $\mathbb{Z}$-retract.

Experimental results

Research questions

• RQ1 Which rational polyhedra $ P \subseteq [0,1]^n $ yield finitely generated projective unital $\ell$-groups $\mathscr{M}(P)$?
• RQ2 What topological and arithmetical conditions on $ P $ are necessary and sufficient for $ \mathscr{M}(P) $ to be projective?
• RQ3 Can the characterization of projective unital $\ell$-groups be extended beyond finite unions of simplexes to general rational polyhedra?
• RQ4 How does the local rational extension property—existence of rational segments extending from rational points—relate to projectivity?
• RQ5 Does the $\mathbb{Z}$-retract condition fully characterize projective unital $\ell$-groups in the finitely generated case?

Key findings

• A unital $\ell$-group $ (G,u) $ is finitely generated and projective if and only if it is isomorphic to $ \mathscr{M}(P) $ for some rational polyhedron $ P \subseteq [0,1]^n $ satisfying three conditions: (i) $ P $ is contractible, (ii) $ P \cap \{0,1\}^n \neq \emptyset $, and (iii) for every rational point $ v \in P $, there exists $ w \in \mathbb{Z}^n $ and $ \varepsilon > 0 $ such that the segment $ \operatorname{conv}(v, v + \varepsilon(w - v)) \subseteq P $.
• The condition (iii) is equivalent to strong regularity of the polyhedron $ P $, as established in Lemma 3.6.
• The result generalizes a previous characterization that only applied to finite unions of $ n $-simplexes, now covering all rational polyhedra.
• The proof establishes that such $ P $ is a $\mathbb{Z}$-retract of $[0,1]^n $, which is the key algebraic-geometric condition for projectivity.
• The characterization provides a complete solution to the sixth open problem in Mundici's list concerning the classification of projective MV-algebras.
• The result confirms that the category of unital $\ell$-groups is dually equivalent to the category of $\mathbb{Z}$-retracts in $[0,1]^n $, with $ \mathscr{M}(P) $ being projective precisely when $ P $ is a $\mathbb{Z}$-retract.