Arithmetic in free lattice-ordered groups

Let \(F_\lambda\) be the free lattice-ordered group of rank \(\lambda\) in the variety of all lattice-ordered groups. We prove that each positive element has only finitely many components, where a component of \(f\ge1\) is a factor \(c\) satisfying \(1\le c\le f\) and \(c\wedge fc^{-1}=1\). Components introduce no new free generators: if an element belongs to the free lattice-ordered subgroup on a subset of the basis, all its components belong to that subgroup. For every rank \(\lambda\ge2\), we then define standard arithmetic in \(F_\lambda\) with one free-generator parameter \(y\). The numerical domain is \(\{(y\vee1)^m:m\in\N\}\); finite chains of components under conjugation provide codes for arbitrary finite subsets of this domain. It follows that the elementary theory of \(F_\lambda\), in the original lattice-group language, is hereditarily undecidable. Undecidability already occurs among \(\Sigma_3\) sentences when \(y\) is named and among \(\Pi_5\) sentences in the original language. For each \(k\ge1\), we also construct a formula \(\chi_k(z,t)\), independent of the rank, which is not equivalent modulo \(\Th(F_\lambda)\) to any Boolean combination of \(\Sigma_k\) formulas with free variables among \(z,t\). After naming \(y\), the formula \(\chi_k(z,y)\) has no such equivalent whose only possible free variable is \(z\). Paper source and revision history: GitHub repository.