Free monoids and forests of rational numbers

The Calkin-Wilf tree is an infinite binary tree whose vertices are the positive rational numbers. Each such number occurs in the tree exactly once and in the form $a/b$, where are $a$ and $b$ are relatively prime positive integers. This tree is associated with the matrices $L_1 = \left( \begin{matrix} 1 & 0 \\ 1 & 1 \end{matrix} \right)$ and $R_1 = \left( \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix} \right)$, which freely generate the monoid $SL_2(\mathbf{N}_0)$ of $2 \times 2$ matrices with determinant 1 and nonnegative integral coordinates. For other pairs of matrices $L_u$ and $R_v$ that freely generate submonoids of $GL_2(\mathbf{N}_0)$, there are forests of infinitely many rooted infinite binary trees that partition the set of positive rational numbers, and possess a remarkable symmetry property.Comment: 10 page

The submonoid and rational subset membership problems for graph groups

We show that the membership problem in a finitely generated submonoid of a graph group (also called a right-angled Artin group or a free partially commutative group) is decidable if and only if the independence graph (commutation graph) is a transitive forest. As a consequence we obtain the first example of a finitely presented group with a decidable generalized word problem that does not have a decidable membership problem for finitely generated submonoids. We also show that the rational subset membership problem is decidable for a graph group if and only if the independence graph is a transitive forest, answering a question of Kambites, Silva, and the second author. Finally we prove that for certain amalgamated free products and HNN-extensions the rational subset and submonoid membership problems are recursively equivalent. In particular, this applies to finitely generated groups with two or more ends that are either torsion-free or residually finite

A Characterization for Decidable Separability by Piecewise Testable Languages

The separability problem for word languages of a class $\mathcal{C}$ by languages of a class $\mathcal{S}$ asks, for two given languages $I$ and $E$ from $\mathcal{C}$, whether there exists a language $S$ from $\mathcal{S}$ that includes $I$ and excludes $E$, that is, $I \subseteq S$ and $S\cap E = \emptyset$. In this work, we assume some mild closure properties for $\mathcal{C}$ and study for which such classes separability by a piecewise testable language (PTL) is decidable. We characterize these classes in terms of decidability of (two variants of) an unboundedness problem. From this, we deduce that separability by PTL is decidable for a number of language classes, such as the context-free languages and languages of labeled vector addition systems. Furthermore, it follows that separability by PTL is decidable if and only if one can compute for any language of the class its downward closure wrt. the scattered substring ordering (i.e., if the set of scattered substrings of any language of the class is effectively regular). The obtained decidability results contrast some undecidability results. In fact, for all (non-regular) language classes that we present as examples with decidable separability, it is undecidable whether a given language is a PTL itself. Our characterization involves a result of independent interest, which states that for any kind of languages $I$ and $E$, non-separability by PTL is equivalent to the existence of common patterns in $I$ and $E$