Closure properties of knapsack semilinear groups

TL;DR

This paper investigates how certain group constructions preserve the semilinearity of solution sets for knapsack equations and analyzes how the complexity measure of these solutions varies with the length of the equations.

Contribution

It establishes that graph products, certain free products, HNN-extensions, and finite extensions preserve solution set semilinearity in groups, and studies the complexity dependence on equation length.

Findings

Group constructions preserve semilinearity of solution sets.

The magnitude of solution sets depends on the length of the knapsack equations.

Analysis of how group operations affect the complexity measure.

Abstract

We show that the following group constructions preserve the semilinearity of the solution sets for knapsack equations (equations of the form $g_1^{x_1} \cdots g_k^{x_k} = g$ in a group $G$, where the variables $x_1, \ldots, x_k$ take values in the natural numbers): graph products, amalgamated free products with finite amalgamated subgroups, HNN-extensions with finite associated subgroups, and finite extensions. Moreover, we study the dependence of the so-called magnitude for the solution set of a knapsack equation (the magnitude is a complexity measure for semi-linear sets) with respect to the length of the knapsack equation (measured in number of generators). We investigate, how this dependence changes under the above group operations.